[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/mutation-jordan-algebra-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/mutation-jordan-algebra-wikipedia\/","headline":"Mutation (Jordan algebra) – Wikipedia","name":"Mutation (Jordan algebra) – Wikipedia","description":"In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by","datePublished":"2019-08-08","dateModified":"2019-08-08","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/57a56367aee395af8fe3f138bae1e9d0d61dc8aa","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/57a56367aee395af8fe3f138bae1e9d0d61dc8aa","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/mutation-jordan-algebra-wikipedia\/","wordCount":65645,"articleBody":"In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first introduced by Max Koecher in his Jordan algebraic approach to Hermitian symmetric spaces and bounded symmetric domains of tube type. Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compact subgroup. Both groups act transitively on the compactification. The theory has been extended to cover all Hermitian symmetric spaces using the theory of Jordan pairs or Jordan triple systems. Koecher obtained the results in the more general case directly from the Jordan algebra case using the fact that only Jordan pairs associated with period two automorphisms of Jordan algebras are required.Table of ContentsDefinitions[edit]Quadratic Jordan algebras[edit]Inverses[edit]Elementary properties of proper mutations[edit]Nonunital mutations[edit]Hua’s identity[edit]Bergman operator[edit]Quasi-invertibility[edit]Equivalence relation[edit]Structure groups[edit]Geometric properties of quotient space[edit]M\u00f6bius transformations[edit]Holomorphic symmetry group[edit]Exchange relations[edit]Lie algebra of holomorphic vector fields[edit]Compact real form[edit]Noncompact real form[edit]Jordan algebras with involution[edit]Examples[edit]See also[edit]References[edit]Definitions[edit]Let A be a unital Jordan algebra over a field k of characteristic \u2260 2.[1] For a in A define the Jordan multiplication operator on A byL(a)b=ab{displaystyle displaystyle {L(a)b=ab}}and the quadratic representation Q(a) byQ(a)=2L(a)2\u2212L(a2).{displaystyle Q(a)=2L(a)^{2}-L(a^{2}).,}It satisfiesQ(1)=I.{displaystyle Q(1)=I.,}the fundamental identityQ(Q(a)b)=Q(a)Q(b)Q(a){displaystyle displaystyle {Q(Q(a)b)=Q(a)Q(b)Q(a)}}the commutation or homotopy identityQ(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a),{displaystyle displaystyle {Q(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a),}}whereR(a,b)c=2Q(a,c)b,Q(x,y)=12(Q(x+y)\u2212Q(x)\u2212Q(y)).{displaystyle R(a,b)c=2Q(a,c)b,,,,Q(x,y)={frac {1}{2}}(Q(x+y)-Q(x)-Q(y)).}In particular if a or b is invertible thenR(a,b)=2Q(a)Q(a\u22121,b)=2Q(a,b\u22121)Q(b).{displaystyle displaystyle {R(a,b)=2Q(a)Q(a^{-1},b)=2Q(a,b^{-1})Q(b).}}It follows that A with the operations Q and R and the identity element defines a quadratic Jordan algebra, where a quadratic Jordan algebra consists of a vector space A with a distinguished element 1 and a quadratic map of A into endomorphisms of A, a \u21a6 Q(a), satisfying the conditions:Q(1) = idQ(Q(a)b) = Q(a)Q(b)Q(a) (“fundamental identity”)Q(a)R(b,a) = R(a,b)Q(a) (“commutation or homotopy identity”), where R(a,b)c = (Q(a + c) \u2212 Q(a) \u2212 Q(c))bThe Jordan triple product is defined by{a,b,c}=(ab)c+(cb)a\u2212(ac)b,{displaystyle {a,b,c}=(ab)c+(cb)a-(ac)b,,}so thatQ(a)b={a,b,a},Q(a,c)b={a,b,c},R(a,b)c={a,b,c}.{displaystyle Q(a)b={a,b,a},,,,Q(a,c)b={a,b,c},,,,R(a,b)c={a,b,c}.,}There are also the formulasQ(a,b)=L(a)L(b)+L(b)L(a)\u2212L(ab),R(a,b)=[L(a),L(b)]+L(ab).{displaystyle Q(a,b)=L(a)L(b)+L(b)L(a)-L(ab),,,,R(a,b)=[L(a),L(b)]+L(ab).,}For y in A the mutation Ay is defined to the vector space A with multiplicationa\u2218b={a,y,b}.{displaystyle acirc b={a,y,b}.,}If Q(y) is invertible, the mutual is called a proper mutation or isotope.Quadratic Jordan algebras[edit]Let A be a quadratic Jordan algebra over a field k of characteristic \u2260 2. Following Jacobson (1969), a linear Jordan algebra structure can be associated with A such that, if L(a) is Jordan multiplication, then the quadratic structure is given by Q(a) = 2L(a)2 \u2212 L(a2).Firstly the axiom Q(a)R(b,a) = R(a,b)Q(a) can be strengthened toQ(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a).{displaystyle displaystyle {Q(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a).}}Indeed, applied to c, the first two terms give2Q(a)Q(b,c)a=2Q(Q(a)c,a)b.{displaystyle displaystyle {2Q(a)Q(b,c)a=2Q(Q(a)c,a)b.}}Switching b and c then givesQ(a)R(b,a)c=2Q(Q(a)b,a)c.{displaystyle displaystyle {Q(a)R(b,a)c=2Q(Q(a)b,a)c.}}Now letL(a)=12R(a,1).{displaystyle displaystyle {L(a)={frac {1}{2}}R(a,1).}}Replacing b by a and a by 1 in the identity above givesR(a,1)=R(1,a)=2Q(a,1).{displaystyle displaystyle {R(a,1)=R(1,a)=2Q(a,1).}}In particularL(a)=Q(a,1),L(1)=Q(1,1)=I.{displaystyle displaystyle {L(a)=Q(a,1),,,,L(1)=Q(1,1)=I.}}The Jordan product is given bya\u2218b=L(a)b=12R(a,1)b=Q(a,b)1,{displaystyle displaystyle {acirc b=L(a)b={frac {1}{2}}R(a,1)b=Q(a,b)1,}}so thata\u2218b=b\u2218a.{displaystyle displaystyle {acirc b=bcirc a.}}The formula above shows that 1 is an identity. Defining a2 by a\u2218a = Q(a)1, the only remaining condition to be verified is the Jordan identity[L(a),L(a2)]=0.{displaystyle displaystyle {[L(a),L(a^{2})]=0.}}In the fundamental identityQ(Q(a)b)=Q(a)Q(b)Q(a),{displaystyle displaystyle {Q(Q(a)b)=Q(a)Q(b)Q(a),}}Replace a by a + t1, set b = 1 and compare the coefficients of t2 on both sides:Q(a)=2Q(a,1)2\u2212Q(a2,1)=2L(a)2\u2212L(a2).{displaystyle displaystyle {Q(a)=2Q(a,1)^{2}-Q(a^{2},1)=2L(a)^{2}-L(a^{2}).}}Setting b = 1 in the second axiom givesQ(a)L(a)=L(a)Q(a),{displaystyle displaystyle {Q(a)L(a)=L(a)Q(a),}}and therefore L(a) must commute with L(a2).Inverses[edit]Let A be a unital Jordan algebra over a field k of characteristic \u2260 2. An element a in a unital Jordan algebra A is said to be invertible if there is an element b such that ab = 1 and a2b = a.[2]Properties.[3]a is invertible if and only if there is an element b such that Q(a)b = a and Q(a)b2 =1. In this case ab = 1 and a2b = a.If ab = 1 and a2b = a, then Q(a)b = 2a(ab) \u2212 (a2)b = a. The Jordan identity [L(x),L(x2)] = 0 can be polarized by replacing x by x + ty and taking the coefficient of t. This gives[L(x2),L(y)]+2[L(xy),L(x)]=0.{displaystyle displaystyle {[L(x^{2}),L(y)]+2[L(xy),L(x)]=0.}}Taking x = a or b and y = b or a shows that L(a2) commutes with L(b) and L(b2) commutes with L(a). Hence (b2)(a2) = 1. Applying L(b) gives b2a = b. Hence Q(a)b2 = 1. Conversely if Q(a)b = a and Q(a)b2 = 1, then the second relation gives Q(a)Q(b)2Q(a) = I. So both Q(a) and Q(b) are invertible. The first gives Q(a)Q(b)Q(a) = Q(a) so that Q(a) and Q(b) are each other’s inverses. Since L(b) commutes with Q(b) it commutes with its inverse Q(a). Similarly L(a) commutes with Q(b). So (a2)b = L(b)a2 = Q(a)b = a and ab = L(b)Q(a)b= Q(a)Q(b)1= 1.a is invertible if and if only Q(a) defines a bijection on A. In that case a\u22121 = Q(a)\u22121a. In this case Q(a)\u22121 = Q(a\u22121).Indeed, if a is invertible then the above implies Q(a) is invertible with inverse Q(b). Any inverse b satisfies Q(a)b = a, so b = Q(a)\u22121a. Conversely if Q(a) is invertible let b = Q(a)\u22121a. ThenQ(a)b = a. The fundamental identity then implies that Q(b) and Q(a) are each other’s inverses so that Q(a)b2 = Q(a)Q(b)1=1.If an inverse exists it is unique. If a is invertible, its inverse is denoted by a\u22121.This follows from the formula a\u22121 = Q(a)\u22121a.a is invertible if and only if 1 lies in the image of Q(a).Suppose that Q(a)c = 1. Then by the fundamental identity Q(a) is invertible, so a is invertible.Q(a)b is invertible if and only if a and b are invertible, in which case (Q(a)b)\u22121 = Q(a\u22121)b\u22121.This is an immediate consequence of the fundamental identity and the fact that STS is invertible if and only S and T are invertible.If a is invertible, then Q(a)L(a\u22121) = L(a).In the commutation identity Q(a)R(b,a) = Q(Q(a)b,a), set b = c2 with c = a\u22121. Then Q(a)b = 1 and Q(1,a) = L(a). Since L(a) commutes with L(c2), R(b,a) = L(c) = L(a\u22121).a is invertible if and only if there is an element b such that ab = 1 and [L(a),L(b)] = 0 (a and b “commute”). In this case b = a\u22121.If L(a) and L(b) commute, then ba = 1 implies b(a2) = a. Conversely suppose that a is invertible with inverse b. Then ab = 1. Morevoer L(b) commutes with Q(b) and hence its inverse Q(a). So it commutes withL(a) = Q(a)L(b).When A is finite-dimensional over k, an element a is invertible if and only if it is invertible in k[a], in which case a\u22121 lies in k[a].The algebra k[a] is commutative and associative, so if b is an inverse there ab =1 and a2b = a. Conversely Q(a) leaves k[a] invariant. So if it is bijective on A it is bijective there. Thus a\u22121 = Q(a)\u22121a lies in k[a].Elementary properties of proper mutations[edit]The mutation Ay is unital if and only if y is invertible in which case the unit is given by y\u22121.The mutation Ay is a unital Jordan algebra if y is invertibleThe quadratic representation of Ay is given by Qy(x) = Q(x)Q(y).In fact [4]multiplication in the algebra Ay is given bya\u2218b={a,y,b},{displaystyle displaystyle {acirc b={a,y,b},}}so by definition is commutative. It follows thata\u2218b=Ly(a)b,{displaystyle displaystyle {acirc b=L_{y}(a)b,}}withLy(a)=[L(a),L(y)]+L(ay).{displaystyle displaystyle {L_{y}(a)=[L(a),L(y)]+L(ay).}}If e satisfies a \u2218 e = a, then taking a = 1 givesye=1.{displaystyle displaystyle {ye=1.}}Taking a = e givese(ya)=y(ea){displaystyle displaystyle {e(ya)=y(ea)}}so that L(y) and L(e) commute. Hence y is invertible and e = y\u22121.Now for y invertible setQy(a)=Q(a)Q(y),Ry(a,b)=R(a,Q(y)b).{displaystyle displaystyle {Q_{y}(a)=Q(a)Q(y),,,,R_{y}(a,b)=R(a,Q(y)b).}}ThenQy(e)=Qy(y\u22121)=Q(y\u22121)Q(y)=I.{displaystyle displaystyle {Q_{y}(e)=Q_{y}(y^{-1})=Q(y^{-1})Q(y)=I.}}Moreover,Qy(a)Qy(b)Qy(a)=Q(a)Q(y)Q(b)Q(y)Q(a)Q(y)=Q(a)Q(Q(y)b)Q(a)Q(y)=Q(Q(a)Q(y)b)Q(y)=Qy(Qy(a)b).{displaystyle displaystyle {Q_{y}(a)Q_{y}(b)Q_{y}(a)=Q(a)Q(y)Q(b)Q(y)Q(a)Q(y)=Q(a)Q(Q(y)b)Q(a)Q(y)=Q(Q(a)Q(y)b)Q(y)=Q_{y}(Q_{y}(a)b).}}FinallyQ(y)R(c,Q(y)d)Q(y)\u22121=R(Q(y)c,d),{displaystyle displaystyle {Q(y)R(c,Q(y)d)Q(y)^{-1}=R(Q(y)c,d),}}sinceQ(y)R(c,Q(y)d)x=2Q(y)Q(c,x)Q(y)d=2Q(Q(y)c,Q(y)x)d=R(Q(y)c,d)Q(y)x.{displaystyle displaystyle {Q(y)R(c,Q(y)d)x=2Q(y)Q(c,x)Q(y)d=2Q(Q(y)c,Q(y)x)d=R(Q(y)c,d)Q(y)x.}}HenceQy(a)Ry(b,a)=Q(a)Q(y)R(b,Q(y)a)=Q(y\u22121)Q(Q(y)a)R(b,Q(y)a)=Q(y)\u22121R(Q(y)a,b)Q(Q(y)a)=Ry(a,b)Qy(a).{displaystyle displaystyle {Q_{y}(a)R_{y}(b,a)=Q(a)Q(y)R(b,Q(y)a)=Q(y^{-1})Q(Q(y)a)R(b,Q(y)a)=Q(y)^{-1}R(Q(y)a,b)Q(Q(y)a)=R_{y}(a,b)Q_{y}(a).}}Thus (A,Qy,y\u22121) is a unital quadratic Jordan algebra. It therefore corresponds to a linear Jordan algebra with the associated Jordan multiplication operator M(a) given byM(a)b=12Ry(a,e)b=12R(a,Q(y)e)b=12R(a,y)b={a,y,b}=Ly(a)b.{displaystyle displaystyle {M(a)b={frac {1}{2}}R_{y}(a,e)b={frac {1}{2}}R(a,Q(y)e)b={frac {1}{2}}R(a,y)b={a,y,b}=L_{y}(a)b.}}This shows that the operators Ly(a) satisfy the Jordan identity so that the proper mutation or isotope Ay is a unital Jordan algebra. The correspondence with quadratic Jordan algebras shows that its quadratic representation is given by Qy.Nonunital mutations[edit]The definition of mutation also applies to non-invertible elements y. If A is finite-dimensional over R or C, invertible elements a in A are dense, since invertibility is equivalent to the condition that det Q(a) \u2260 0. So by continuity the Jordan identity for proper mutations implies the Jordan identity for arbitrary mutations. In general the Jordan identity can be deduced from Macdonald’s theorem for Jordan algebras because it involves only two elements of the Jordan algebra. Alternatively, the Jordan identity can be deduced by realizing the mutation inside a unital quadratic algebra.[5]For a in A define a quadratic structure on A1 = A \u2295 k byQ1(a\u2295\u03b11)(b\u2295\u03b21)=\u03b12\u03b21\u2295[\u03b12a+\u03b12b+2\u03b1\u03b2a+\u03b1{a,y,b}+\u03b2Q(a)y+Q(a)Q(y)b].{displaystyle displaystyle {Q_{1}(aoplus alpha 1)(boplus beta 1)=alpha ^{2}beta 1oplus [alpha ^{2}a+alpha ^{2}b+2alpha beta a+alpha {a,y,b}+beta Q(a)y+Q(a)Q(y)b].}}It can then be verified that (A1, Q1, 1) is a unital quadratic Jordan algebra. The unital Jordan algebra to which it corresponds has Ay as an ideal, so that in particular Ay satisfies the Jordan identity. The identities for a unital quadratic Jordan algebra follow from the following compatibility properties of the quadratic map Qy(a) = Q(a)Q(y) and the squaring mapSy(a) = Q(a)y:Ry(a,a) = Ly(Sy(a)).[Qy(a),Ly(a)] = 0.Qy(a)Sy(a) = Sy(Sy(a)).Qy\u2218 Sy = Sy \u2218 Qy.Qy(a) Qy(b) Sy(a) = Sy(Qy(a)b).Qy(Qy(a)b) = Qy(a) Qy(b) Qy(a).Hua’s identity[edit]Let A be a unital Jordan algebra. If a, b and a \u2013 b are invertible, then Hua’s identity holds:[6]a\u22121=(a\u2212b)\u22121+(a\u2212Q(a)b\u22121)\u22121=(a\u2212b)\u22121+Q(a)\u22121(a\u22121\u2212b\u22121)\u22121.{displaystyle displaystyle {a^{-1}=(a-b)^{-1}+(a-Q(a)b^{-1})^{-1}=(a-b)^{-1}+Q(a)^{-1}(a^{-1}-b^{-1})^{-1}.}}In particular if x and 1 \u2013 x are invertible, then so too is 1 \u2013 x\u22121 with(1\u2212x)\u22121+(1\u2212x\u22121)\u22121=1.{displaystyle displaystyle {(1-x)^{-1}+(1-x^{-1})^{-1}=1.}}To prove the identity for x, set y = (1 \u2013 x)\u22121. Then L(y) = Q(1 \u2013 x)\u22121L(1 \u2013 x). Thus L(y) commutes with L(x) and Q(x). Since Q(y) = Q(1 \u2013 x)\u22121, it also commutes with L(x) and Q(x). Since L(x\u22121) = Q(x)\u22121L(x), L(y) also commutes with L(x\u22121) and Q(x\u22121).It follows that (x\u22121 \u2013 1)xy =(1 \u2013 x) y = 1. Moreover, y \u2013 1 = xy since (1 \u2013 x)y = 1. So L(xy) commutes with L(x) and hence L(x\u22121 \u2013 1). Thus 1 \u2013 x\u22121 has inverse 1 \u2013 y.Now let Aa be the mutation of A defined by a. The identity element of Aa is a\u22121. Moreover, an invertible element c in A is also invertible in Aa with inverse Q(a)\u22121c\u22121.Let x = Q(a)\u22121b in Aa. It is invertible in A, as is a\u22121 \u2013 Q(a)\u22121b = Q(a)\u22121(a \u2013 b). So by the special case of Hua’s identity for x in Aaa\u22121=Q(a)\u22121(a\u22121\u2212Q(a)\u22121b)\u22121+Q(a)\u22121(a\u22121\u2212b\u22121)\u22121=(a\u2212b)\u22121+(a\u2212Q(a)b\u22121)\u22121.{displaystyle displaystyle {a^{-1}=Q(a)^{-1}(a^{-1}-Q(a)^{-1}b)^{-1}+Q(a)^{-1}(a^{-1}-b^{-1})^{-1}=(a-b)^{-1}+(a-Q(a)b^{-1})^{-1}.}}Bergman operator[edit]If A is a unital Jordan algebra, the Bergman operator is defined for a, b in A by[7]B(a,b)=I\u2212R(a,b)+Q(a)Q(b).{displaystyle displaystyle {B(a,b)=I-R(a,b)+Q(a)Q(b).}}If a is invertible thenB(a,b)=Q(a)Q(a\u22121\u2212b);{displaystyle displaystyle {B(a,b)=Q(a)Q(a^{-1}-b);}}while if b is invertible thenB(a,b)=Q(a\u2212b\u22121)Q(b).{displaystyle displaystyle {B(a,b)=Q(a-b^{-1})Q(b).}}In fact if a is invertibleQ(a)Q(a\u22121 \u2212 b) = Q(a)[Q(a\u22121 \u2212 2Q(a\u22121,b) + Q(b)]=I \u2212 2Q(a)Q(a\u22121,b) + Q(a)Q(b)=I \u2212 R(a,b) + Q(a)Q(b)and similarly if b is invertible.More generally the Bergman operator satisfies a version of the commutation or homotopy identity:B(a,b)Q(a)=Q(a)B(b,a)=Q(a\u2212Q(a)b){displaystyle displaystyle {B(a,b)Q(a)=Q(a)B(b,a)=Q(a-Q(a)b)}}and a version of the fundamental identity:Q(B(a,b)c)=B(a,b)Q(c)B(b,a).{displaystyle displaystyle {Q(B(a,b)c)=B(a,b)Q(c)B(b,a).}}There is also a third more technical identity:2Q(B(a,b)c,a\u2212Q(a)b)=B(a,b)(2Q(a,c)\u2212R(c,b)Q(a))=(2Q(a,c)\u2212Q(a)R(b,c))B(b,a).{displaystyle displaystyle {2Q(B(a,b)c,a-Q(a)b)=B(a,b)(2Q(a,c)-R(c,b)Q(a))=(2Q(a,c)-Q(a)R(b,c))B(b,a).}}Quasi-invertibility[edit]Let A be a finite-dimensional unital Jordan algebra over a field k of characteristic \u2260 2.[8] For a pair (a,b) with a and a\u22121 \u2212 b invertible defineab=(a\u22121\u2212b)\u22121.{displaystyle displaystyle {a^{b}=(a^{-1}-b)^{-1}.}}In this case the Bergman operator B(a,b) = Q(a)Q(a\u22121 \u2212 b) defines an invertible operator on A andab=B(a,b)\u22121(a\u2212Q(a)b).{displaystyle displaystyle {a^{b}=B(a,b)^{-1}(a-Q(a)b).}}In factB(a,b)\u22121(a\u2212Q(a)b)=Q(ab)Q(a\u22121)(a\u2212Q(a)b)=Q(ab)(ab)\u22121=ab.{displaystyle displaystyle {B(a,b)^{-1}(a-Q(a)b)=Q(a^{b})Q(a^{-1})(a-Q(a)b)=Q(a^{b})(a^{b})^{-1}=a^{b}.}}Moreover, by definition a\u22121 \u2212 b \u2212 c is invertible if and only if (ab)\u22121 \u2212 c is invertible. In that caseab+c=(ab)c.{displaystyle displaystyle {a^{b+c}=(a^{b})^{c}.}}Indeed,ab+c=((a\u22121\u2212b)\u2212c)\u22121=((ab)\u22121\u2212c)\u22121=(ab)c.{displaystyle displaystyle {a^{b+c}=((a^{-1}-b)-c)^{-1}=((a^{b})^{-1}-c)^{-1}=(a^{b})^{c}.}}The assumption that a be invertible can be dropped since ab can be defined only supposing that the Bergman operator B(a,b) is invertible. The pair (a,b) is then said to be quasi-invertible. In that case ab is defined by the formulaab=B(a,b)\u22121(a\u2212Q(a)b).{displaystyle displaystyle {a^{b}=B(a,b)^{-1}(a-Q(a)b).}}If B(a,b) is invertible, then B(a,b)c = 1 for some c. The fundamental identity implies that B(a,b)Q(c)B(b,a) = I. So by finite-dimensionality B(b,a) is invertible. Thus (a,b) is invertible if and only if(b,a) is invertible and in this caseab=a+Q(a)ba.{displaystyle displaystyle {a^{b}=a+Q(a)b^{a}.}}In factB(a,b)(a + Q(a)ba) = a \u2212 2R(a,b)a + Q(a)Q(b)a + Q(a)(b \u2212 Q(b)a) = a \u2212 Q(a)b,so the formula follows by applying B(a,b)\u22121 to both sides.As before (a,b+c) is quasi-invertible if and only if (ab,c) is quasi-invertible; and in that caseab+c=(ab)c.{displaystyle displaystyle {a^{b+c}=(a^{b})^{c}.}}If k = R or C, this would follow by continuity from the special case where a and a\u22121 \u2212 b were invertible. In general the proof requires four identities for the Bergman operator:B(a,b)Q(ab)=Q(ab)B(b,a)=Q(a){displaystyle displaystyle {B(a,b)Q(a^{b})=Q(a^{b})B(b,a)=Q(a)}}B(a,b)Q(ab,c)+Q(a)R(b,c)=Q(ab,c)B(b,a)+R(c,b)Q(a)=Q(a,c){displaystyle displaystyle {B(a,b)Q(a^{b},c)+Q(a)R(b,c)=Q(a^{b},c)B(b,a)+R(c,b)Q(a)=Q(a,c)}}B(a,b)R(ab,c)=R(a,c)\u22122Q(a)Q(b,c){displaystyle displaystyle {B(a,b)R(a^{b},c)=R(a,c)-2Q(a)Q(b,c)}}B(a,b)B(ab,c)=B(a,b+c){displaystyle displaystyle {B(a,b)B(a^{b},c)=B(a,b+c)}}In fact applying Q to the identity B(a,b)ab = a \u2212 Q(a)b yieldsB(a,b)Q(ab)B(b,a)=B(a,b)Q(a)=Q(a)B(b,a).{displaystyle displaystyle {B(a,b)Q(a^{b})B(b,a)=B(a,b)Q(a)=Q(a)B(b,a).}}The first identity follows by cancelling B(a,b) and B(b,a). The second identity follows by similar cancellation inB(a,b)Q(ab,c)B(b,a) = Q(B(a,b)ab,B(a,b)c) = Q(a \u2212 Q(a)b,B(a,b)c) = B(a,b)(Q(a,c) \u2212 R(c,b)Q(a)) = (Q(a,c) \u2212 Q(a)R(b,c))B(b,a).The third identity follows by applying the second identity to an element d and then switching the roles of c and d. The fourth follows becauseB(a,b)B(ab,c) = B(a,b)(I \u2212 R(ab,c) + Q(ab)Q(c)) = I \u2212 R(a,b + c) + Q(a) Q(b + c) = B(a,b+c).In fact (a,b) is quasi-invertible if and only if a is quasi-invertible in the mutation Ab. Since this mutation might not necessarily unital this means that when an identity is adjoint 1 \u2212 a becomes invertible in Ab \u2295 k1. This condition can be expressed as follows without mentioning the mutation or homotope:(a,b) is quasi-invertible if and only if there is an element c such that B(a,b)c = a \u2212 Q(a)b and B(a,b)Q(c)b = Q(a)b. In this case c = ab.In fact if (a,b) is quasi-invertible, then c = ab satisfies the first identity by definition. The second follows because B(a,b)Q(ab) = Q(a). Conversely the conditions state that in Ab \u2295 k1 the conditions imply that 1 + c is the inverse of 1 \u2212 a. On the other hand,( 1 \u2212 a) \u2218 x = B(a,b)x for x in Ab. Hence B(a,b) is invertible.Equivalence relation[edit]Let A be a finite-dimensional unital Jordan algebra over a field k of characteristic \u2260 2.[9]Two pairs (ai,bi) with ai invertible are said to be equivalent if (a1)\u22121 \u2212 b1 + b2 is invertible and a2 = (a1)b1 \u2212 b2.This is an equivalence relation, since if a is invertible a0 = a so that a pair (a,b) is equivalent to itself. It is symmetric since from the definition a1 = (a2)b2 \u2212 b1. It is transitive. For suppose that (a3,b3) is a third pair with (a2)\u22121 \u2212 b2 + b3 invertible and a3 = (a2)b2 \u2212 b3. From the abovea1\u22121\u2212b1+b3=(a1\u22121\u2212b1+b2)\u2212b2+b3=a2\u22121\u2212b2+b3{displaystyle displaystyle {a_{1}^{-1}-b_{1}+b_{3}=(a_{1}^{-1}-b_{1}+b_{2})-b_{2}+b_{3}=a_{2}^{-1}-b_{2}+b_{3}}}is invertible anda3=a2b2\u2212b3=(a1b1\u2212b2)b2\u2212b3=a1b1\u2212b3.{displaystyle displaystyle {a_{3}=a_{2}^{b_{2}-b_{3}}=(a_{1}^{b_{1}-b_{2}})^{b_{2}-b_{3}}=a_{1}^{b_{1}-b_{3}}.}}As for quasi-invertibility, this definition can be extended to the case where a and a\u22121 \u2212 b are not assumed to be invertible.Two pairs (ai,bi) are said to be equivalent if (a1, b1 \u2212 b2) is quasi-invertible and a2 = (a1)b1 \u2212 b2. When k = R or C, the fact that this more general definition also gives an equivalence relation can deduced from the invertible case by continuity. For general k, it can also be verified directly:The relation is reflexive since (a,0) is quasi-invertible and a0 = a.The relation is symmetric, since a1 = (a2)b2 \u2212 b1.The relation is transitive. For suppose that (a3,b3) is a third pair with (a2, b2 \u2212 b3) quasi-invertible and a3 = (a2)b2 \u2212 b3. In this caseB(a1,b1\u2212b3)=B(a1,b1\u2212b2)B(a2,b2\u2212b3),{displaystyle displaystyle {B(a_{1},b_{1}-b_{3})=B(a_{1},b_{1}-b_{2})B(a_{2},b_{2}-b_{3}),}}so that (a1,b1 \u2212 b3) is quasi-invertible witha3=a2b2\u2212b3=(a1b1\u2212b2)b2\u2212b3=a1b1\u2212b3.{displaystyle displaystyle {a_{3}=a_{2}^{b_{2}-b_{3}}=(a_{1}^{b_{1}-b_{2}})^{b_{2}-b_{3}}=a_{1}^{b_{1}-b_{3}}.}}The equivalence class of (a,b) is denoted by (a:b).Structure groups[edit]Let A be a finite-dimensional complex semisimple unital Jordan algebra. If T is an operator on A, let Tt be its transpose with respect to the trace form. ThusL(a)t = L(a), Q(a)t = Q(a), R(a,b)t = R(b,a) and B(a,b)t = B(b,a). The structure group of A consists of g in GL(A) such thatQ(ga)=gQ(a)gt.{displaystyle displaystyle {Q(ga)=gQ(a)g^{t}.}}They form a group \u0393(A). The automorphism group Aut A of A consists of invertible complex linear operators g such that L(ga) = gL(a)g\u22121 and g1 = 1. Since an automorphism g preserves the trace form, g\u22121 = gt.The structure group is closed under taking transposes g \u21a6 gt and adjoints g \u21a6 g*.The structure group contains the automorphism group. The automorphism group can be identified with the stabilizer of 1 in the structure group.If a is invertible, Q(a) lies in the structure group.If g is in the structure group and a is invertible, ga is also invertible with (ga)\u22121 = (gt)\u22121a\u22121.The structure group \u0393(A) acts transitively on the set of invertible elements in A.Every g in \u0393(A) has the form g = h Q(a) with h an automorphism and a invertible.The complex Jordan algebra A is the complexification of a real Euclidean Jordan algebra E, for which the trace form defines an inner product. There is an associated involution a \u21a6 a* on A which gives rise to a complex inner product on A. The unitary structure group \u0393u(A) is the subgroup of \u0393(A) consisting of unitary operators, so that \u0393u(A) = \u0393(A) \u2229 U(A). The identity component of \u0393u(A) is denoted by K. It is a connected closed subgroup of U(A).The stabilizer of 1 in \u0393u(A) is Aut E.Every g in \u0393u(A) has the form g = h Q(u) with h in Aut E and u invertible in A with u* = u\u22121.\u0393(A) is the complexification of \u0393u(A).The set S of invertible elements u in A such that u* = u\u22121 can be characterized equivalently either as those u for which L(u) is a normal operator with uu* = 1 or as those u of the form exp ia for some a in E. In particular S is connected.The identity component of \u0393u(A) acts transitively on SGiven a Jordan frame (ei) and v in A, there is an operator u in the identity component of \u0393u(A) such that uv = \u03a3 \u03b1iei with \u03b1i \u2265 0. If v is invertible, then \u03b1i > 0.The structure group \u0393(A) acts naturally on X.[10] For g in \u0393(A), setg(a,b)=(ga,(gt)\u22121b).{displaystyle displaystyle {g(a,b)=(ga,(g^{t})^{-1}b).}}Then (x,y) is quasi-invertible if and only if (gx,(gt)\u22121y) is quasi-invertible andg(xy)=(gx)(gt)\u22121y.{displaystyle displaystyle {g(x^{y})=(gx)^{(g^{t})^{-1}y}.}}In fact the covariance relations for g with Q and the inverse imply thatgB(x,y)g\u22121=B(gx,(gt)\u22121y){displaystyle displaystyle {gB(x,y)g^{-1}=B(gx,(g^{t})^{-1}y)}}if x is invertible and so everywhere by density. In turn this implies the relation for the quasi-inverse. If a is invertible then Q(a) lies in \u0393(A) and if (a,b) is quasi-invertible B(a,b) lies in \u0393(A). So both types of operators act on X.The defining relations for the structure group show that it is a closed subgroup of g0{displaystyle {mathfrak {g}}_{0}} of GL(A). Since Q(ea) = e2L(a), the corresponding complex Lie algebra contains the operators L(a). The commutators [L(a),L(b)] span the complex Lie algebra of derivations of A. The operators R(a,b) = [L(a),L(b)] + L(ab) span g0{displaystyle {mathfrak {g}}_{0}} andsatisfy R(a,b)t = R(b,a) and[R(a,b),R(c,d)]=R(R(a,b)c,d) \u2212 R(c,R(b,a)d).Geometric properties of quotient space[edit]Let A be a finite-dimensional complex unital Jordan algebra which is semisimple, i.e. the trace form Tr L(ab) is non-degenerate. Let X be the quotient of A\u00d7A by the equivalence relation. Let Xb be the subset of X of classes (a:b). The map \u03c6b:Xb \u2192 A, (a:b) \u21a6 a is injective. A subset U of X is defined to be open if and only if U \u2229 Xb is open for all b.The transition maps of the atlas with charts \u03c6b are given by\u03c6cb=\u03c6c\u2218\u03c6b\u22121:\u03c6b(Xb\u2229Xc)\u2192\u03c6c(Xb\u2229Xc).{displaystyle displaystyle {varphi _{cb}=varphi _{c}circ varphi _{b}^{-1}:varphi _{b}(X_{b}cap X_{c})rightarrow varphi _{c}(X_{b}cap X_{c}).}}and are injective and holomorphic since\u03c6cb(a)=ab\u2212c{displaystyle displaystyle {varphi _{cb}(a)=a^{b-c}}}with derivative\u03c6cb\u2032(a)=B(a,b\u2212c)\u22121.{displaystyle displaystyle {varphi _{cb}^{prime }(a)=B(a,b-c)^{-1}.}}This defines the structure of a complex manifold on X because \u03c6dc \u2218 \u03c6cb = \u03c6db on \u03c6b(Xb \u2229 Xc \u2229 Xd).Given a finite set of points (ai:bi) in X, they are contained in a common Xb.Indeed, all the polynomial functions pi(b) = det B(ai,bi \u2212 b) are non-trivial since pi(bi) = 1. Therefore, there is a b such that pi(b) \u2260 0 for all i, which is precisely the criterion for (ai:bi) to lie in Xb.Loos (1977) uses the Bergman operators to construct an explicit biholomorphism between X and a closed smooth algebraic subvariety of complex projective space.[11] This implies in particular that X is compact. There is a more direct proof of compactness using symmetry groups.Given a Jordan frame (ei) in E, for every a in A there is a k in U = \u0393u(A) such that a=k(\u03a3 \u03b1iei)with \u03b1i \u2265 0 (and \u03b1i > 0 if a is invertible).In fact, if (a,b) is in X then it is equivalent to k(c,d) with c and d in the unital Jordan subalgebra Ae = \u2295 Cei, which is the complexification of Ee = \u2295 Rei.Let Z be the complex manifold constructed for Ae. Because Ae is a direct sum of copies of C, Z is just a product of Riemann spheres, one for each ei. In particular it is compact. There is a natural map of Z into X which is continuous. Let Y be the image of Z. It is compact and therefore coincides with the closure of Y0 = Ae \u2282 A = X0. The set U\u22c5Y is the continuous image of the compact set U \u00d7 Y. It is therefore compact. On the other hand, U\u22c5Y0 = X0, so it contains a dense subset of X and must therefore coincide with X. So X is compact.The above argument shows that every (a,b) in X is equivalent to k(c,d) with c and d in Ae and k in\u0393u(A). The mapping of Z into X is in fact an embedding. This is a consequence of (x,y) being quasi-invertible in Ae if and only if it is quasi-invertible in A. Indeed, if B(x,y) is injective on A, its restriction to Ae is also injective. Conversely, the two equations for the quasi-inverse in Ae imply that it is also a quasi-inverse in A.M\u00f6bius transformations[edit]Let A be a finite-dimensional complex semisimple unital Jordan algebra. The group SL(2,C) acts by M\u00f6bius transformation on the Riemann sphere C \u222a {\u221e}, the one-point compactification of C. If g in SL(2,C) is given by the matrixg=(\u03b1\u03b2\u03b3\u03b4),{displaystyle displaystyle {g={begin{pmatrix}alpha &beta \\gamma &delta end{pmatrix}},}}theng(z)=(\u03b1z+\u03b2)(\u03b3z+\u03b4)\u22121.{displaystyle displaystyle {g(z)=(alpha z+beta )(gamma z+delta )^{-1}.}}There is a generalization of this action of SL(2,C) to A and its compactification X. In order to define this action, note that SL(2,C) is generated by the three subgroups of lower and upper unitriangular matrices and the diagonal matrices. It is also generated by the lower (or upper) unitriangular matrices, the diagonal matrices and the matrixJ=(01\u221210).{displaystyle displaystyle {J={begin{pmatrix}0&1\\-1&0end{pmatrix}}.}}The matrix J corresponds to the M\u00f6bius transformation j(z) = \u2212z\u22121 and can be writtenJ=(10\u221211)(1101)(10\u221211).{displaystyle displaystyle {J={begin{pmatrix}1&0\\-1&1end{pmatrix}}{begin{pmatrix}1&1\\0&1end{pmatrix}}{begin{pmatrix}1&0\\-1&1end{pmatrix}}.}}The M\u00f6bius transformations fixing \u221e are just the upper triangular matrices. If g does not fix \u221e, it sends \u221e to a finite point a. But then g can be composed with an upper unitriangular to send a to 0 and then with J to send 0 to infinity.For an element a of A, the action of g in SL(2,C) is defined by the same formulag(a)=(\u03b1a+\u03b21)(\u03b3a+\u03b41)\u22121.{displaystyle displaystyle {g(a)=(alpha a+beta 1)(gamma a+delta 1)^{-1}.}}This defines an element of C[a] provided that \u03b3a + \u03b41 is invertible in A. The action is thus defined everywhere on A if g is upper triangular. On the other hand, the action on X is simple to define for lower triangular matrices.[12]For diagonal matrices g with diagonal entries \u03b1 and \u03b1\u22121, g(a,b) = (\u03b12a, \u03b1\u22122b) is a well-defined holomorphic action on A2 which passes to the quotient X. On X0 = A it agrees with the M\u00f6bius action.For lower unitriangular matrices, with off-diagonal parameter \u03b3, define g(a,b) = (a,b \u2212 \u03b31). Again this is holomorphic on A2 and passes to the quotient X. When b = 0 and \u03b3 \u2260 0,g(a:0)=(a:\u2212\u03b3)=(a\u2212\u03b3:0)=(a(\u03b3a+1)\u22121:0){displaystyle displaystyle {g(a:0)=(a:-gamma )=(a^{-gamma }:0)=(a(gamma a+1)^{-1}:0)}}if \u03b3a + 1 is invertible, so this is an extension of the M\u00f6bius action.For upper unitriangular matrices, with off-diagonal parameter \u03b2, the action on X0 = (A:0) is defined by g(a,0) = (a + \u03b21). Loos (1977) showed that this defined a complex one-parameter flow on A. The corresponding holomorphic complex vector field extended to X, so that the action on the compact complex manifold X could be defined by the associated complex flow. A simpler method is to note that the operator J can be implemented directly using its intertwining relations with the unitary structure group.In fact on the invertible elements in A, the operator j(a) = \u2212a\u22121 satisfies j(ga) = (gt)\u22121j(a). To define a biholomorphism j on X such that j \u2218 g = (gt)\u22121 \u2218 j, it is enough to define these for (a:b) in some suitable orbit of \u0393(A) or \u0393u(A). On the other hand, as indicated above, given a Jordan frame (ei) in E, for every a in A there is a k in U = \u0393u(A) such that a=k(\u03a3 \u03b1iei) with \u03b1i \u2265 0.The computation of j in the associative commutative algebra Ae is straightforward since it is a direct product. For c = \u03a3 \u03b1iei and d = \u03a3 \u03b2iei, the Bergman operator on Ae has determinant det B(c,d) = \u03a0(1 \u2212 \u03b1i\u03b2i)2. In particular det B(c,d \u2212 \u03bb) \u2260 0 for some \u03bb \u2260 0. So that (c,d) is equivalent to (x,\u03bb). Let \u03bc = \u2212\u03bb\u22121. On A, for a dense set of a, the pair (a,\u03bb) is equivalent to (b,0) with b invertible. Then (\u2212b\u22121,0) is equivalent to (\u03bc \u2212 \u03bc2a,\u03bc). Since a \u21a6 \u03bc \u2212 \u03bc2a is holomorphic it follows that j has a unique continuous extension to X such that j \u2218 g = (gt)\u22121 \u2218 j for g in \u0393(A), the extension is holomorphic and for \u03bb \u2260 0, \u03bc = \u2212\u03bb\u22121j(a,\u03bb)=(\u03bc\u2212\u03bc2a,\u03bc).{displaystyle displaystyle {j(a,lambda )=(mu -mu ^{2}a,mu ).}}The holomorphic transformations corresponding to upper unitriangular matrices can be defined using the fact that they are the conjugates by J of lower unitriangular matrices, for which the action is already known. A direct algebraic construction is given in Dineen, Mackey & Mellon (1999).This action of SL(2,C) is compatible with inclusions. More generally if e1, …, em is a Jordan frame, there is an action of SL(2,C)m on Ae which extends to A. If c = \u03a3 \u03b3iei and b = \u03a3 \u03b2iei, then S(c) and T(b) give the action of the product of the lower and upper unitriangular matrices. If a = \u03a3 \u03b1iei is invertible, the corresponding product of diagonal matrices act as W = Q(a).[13] In particular the diagonal matrices give an action of (C*)m and Tm.Holomorphic symmetry group[edit]Let A be a finite-dimensional complex semisimple unital Jordan algebra. There is a transitive holomorphic action of a complex matrix group G on the compact complex manifold X. Koecher (1967) described G analogously to SL(2,C) in terms of generators and relations. G acts on the corresponding finite-dimensional Lie algebra of holomorphic vector fields restricted to X0 = A, so that G is realized as a closed matrix group. It is the complexification of a compact Lie group without center, so a semisimple algebraic group. The identity component H of the compact group acts transitively on X, so that X can be identified as a Hermitian symmetric space of compact type.[14]The group G is generated by three types of holomorphic transformation on X:Operators W corresponding to elements W in \u0393(A) given by W(a,b) = (Wa, (Wt)\u22121b). These were already described above. On X0 = A, they are given by a \u21a6 Wa.Operators Sc defined by Sc(a,b) = (a,b + c). These are the analogue of lower unitriangular matrices and form a subgroup isomorphic to the additive group of A, with the given parametrization. Again these act holomorphically on A2 and the action passes to the quotient X. On A the action is given by a \u21a6 ac if (a,c) is quasi-invertible.The transformation j corresponding to J in SL(2,C). It was constructed above as part of the action of PSL(2,C) = SL(2,C)\/{\u00b1I} on X. On invertible elements in A it is given by a \u21a6 \u2212a\u22121.The operators W normalize the group of operators Sc. Similarly the operator j normalizes the structure group, j \u2218 W = (Wt)\u22121 \u2218 j. The operators Tc = j \u2218 S\u2212c \u2218 j also form a group of holomorphic transformations isomorphic to the additive group of A. They generalize the upper unitriangular subgroup of SL(2,C). This group is normalized by the operators W of the structure group. The operator Tc acts on A as a \u21a6 a + c.If c is a scalar the operators Sc and Tc coincide with the operators corresponding to lower and upper unitriangular matrices in SL(2,C). Accordingly, there is a relation j = S1 \u2218 T1 \u2218 S1 and PSL(2,C) is a subgroup of G. Loos (1977) defines the operators Tc in terms of the flow associated to a holomorphic vector field on X, while Dineen, Mackey & Mellon (1999) give a direct algebraic description.G acts transitively on X.Indeed, SbTa(0:0) = (a:b).Let G\u22121 and G+1 be the complex Abelian groups formed by the symmetries Tc and Sc respectively. Let G0 = \u0393(A).G=G0G+1G\u22121G+1=G0G\u22121G+1G\u22121.{displaystyle displaystyle {G=G_{0}G_{+1}G_{-1}G_{+1}=G_{0}G_{-1}G_{+1}G_{-1}.}}The two expressions for G are equivalent as follows by conjugating by j.For a invertible, Hua’s identity can be rewrittenQ(a)=Ta\u2218j\u2218Ta\u22121\u2218j\u2218Ta\u2218j.{displaystyle displaystyle {Q(a)=T_{a}circ jcirc T_{a^{-1}}circ jcirc T_{a}circ j.}}Moreover, j = S1 \u2218 T1 \u2218 S1 andSc = j \u2218 T\u2212c \u2218 j.[15]The convariance relations show that the elements of G fall into setsG0G1, G0G1jG1, G0G1jG1jG1, G0G1jG1jG1jG1. …The first expression for G follows once it is established that no new elements appear in the fourth or subsequent sets. For this it suffices to show that[16]j \u2218 G1 \u2218j \u2218 G1 \u2218j \u2286 G0G1 \u2218j \u2218 G1 \u2218 j \u2218 G1.For then if there are three or more occurrences of j, the number can be recursively reduced to two. Given a, b in A, pick \u03bb \u2260 0 so that c = a \u2212 \u03bb and d = b \u2212 \u03bb\u22121 are invertible. ThenjTajTbj=jTcT\u03bbjT\u03bb\u22121Td\u2218j=\u03bb2jTcjT\u2212\u03bbjTd\u00a0j=\u03bb2T\u2212c\u22121jQ(c\u22121)T\u2212c\u22121\u2212\u03bb\u2212d\u22121jQ(d\u22121)jT\u2212d\u22121,{displaystyle displaystyle {jT_{a}jT_{b}j=jT_{c}T_{lambda }jT_{lambda ^{-1}}T_{d}circ j=lambda ^{2}jT_{c}jT_{-lambda }jT_{d} j=lambda ^{2}T_{-c^{-1}}jQ(c^{-1})T_{-c^{-1}-lambda -d^{-1}}jQ(d^{-1})jT_{-d^{-1}},}}which lies in G0G1 \u2218 j \u2218 G1 \u2218 j \u2218 G1.The stabilizer of (0:0) in G is G0G\u22121.It suffices to check that if SaTb(0:0) = (0:0), then b = 0. If so (b:0) = (0: \u2212a) = (0:0), so b = 0.Exchange relations[edit]For a invertible, Hua’s identity can be rewrittenQ(a)=Ta\u2218j\u2218Ta\u22121\u2218j\u2218Ta\u2218j.{displaystyle displaystyle {Q(a)=T_{a}circ jcirc T_{a^{-1}}circ jcirc T_{a}circ j.}}Since j = S1 \u2218 T1 \u2218 S1, the operators Q(a) belong to the group generated by G\u00b11.[17]For quasi-invertible pairs (a,b), there are the “exchange relations”[18]This identity shows that B(a,b) is in the group generated by G\u00b11. Taking inverses, it is equivalent to the identity TaSb = SbaB(a,b)Tab.To prove the exchange relations, it suffices to check that it valid when applied to points the dense set of points (c:0) in X for which (a+c,b) is quasi-invertible. It then reduces to the identity:(a+c)b=ab+B(a,b)\u22121c(ba).{displaystyle displaystyle {(a+c)^{b}=a^{b}+B(a,b)^{-1}c^{(b^{a})}.}}In fact, if (a,b) is quasi-invertible, then (a + c,b) is quasi-invertible if and only if (c,ba) is quasi-invertible. This follows because (x,y) is quasi-invertible if and only if (y,x) is. Moreover, the above formula holds in this case.For the proof, two more identities are required:B(c+b,a)=B(c,ab)B(b,a){displaystyle displaystyle {B(c+b,a)=B(c,a^{b})B(b,a)}}R(a,b)=R(ab,b\u2212Q(b)a)=R(a\u2212Q(a)b,ba){displaystyle displaystyle {R(a,b)=R(a^{b},b-Q(b)a)=R(a-Q(a)b,b^{a})}}The first follows from a previous identity by applying the transpose. For the second, because of the transpose, it suffices to prove the first equality. Setting c = b \u2212 Q(b)a in the identity B(a,b)R(ab,c) =R(a,c) \u2212 Q(a)Q(b,c) yieldsB(a,b)R(ab,b \u2212 Q(b)c) = B(a,b)R(a,b),so the identity follows by cancelling B(a,b).To prove the formula, the relations (a + c)b = B(a,c)\u22121(a + c \u2212 Q(a + c)b)and ab + B(a,b)\u22121c(ba) = B(a + c,b)\u22121(B(c,ba) (a \u2212 Q(a)b) + c \u2212 Q(c)ba) show that it is enough to prove thata + c \u2212 Q(a + c)b = B(c,ba) (a \u2212 Q(a)b) + c \u2212 Q(c)ba.Indeed, B(c,ba) (a \u2212 Q(a)b) + c \u2212 Q(c)ba = a + c \u2212 Q(a)b + 2R(c,ba)(a \u2212 Q(a)b) \u2212 Q(c)[ ba \u2212 Q(ba)(a \u2212 Q(a)b)]. On the other hand,2R(c,ba)(a \u2212 Q(a)b) = 2R(c,a \u2212 Q(a)b)ba = R(a,b)c = 2Q(a,c)b and ba \u2212 Q(ba)(a \u2212 Q(a)b) = ba \u2212 Q(b)B(a,b)\u22121(a \u2212 Q(a)b) = ba \u2212 Q(b)ab = b. So B(c,ba) (a \u2212 Q(a)b) + c \u2212 Q(c)ba = a + c \u2212 Q(a)b \u2212 2Q(a,c)b \u2212 Q(c)b = a + c \u2212 Q(a + c)b.Now set \u03a9 = G+1G0G\u22121. Then the exchange relations imply that SbTa lies in \u03a9 if and only if (a,b) is quasi-invertible; and that g lies in \u03a9 if and only if g(0:0) is in X0.[19]In fact if SbTa lies in \u03a9, then (a,b) is equivalent to (x,0), so it a quasi-invertible pair; the converse follows from the exchange relations. Clearly \u03a9(0:0) = G1(0:0) = X0. The converse follows from G = G\u22121G1G0G\u22121 and the criterion for SbTa to lie in \u03a9.Lie algebra of holomorphic vector fields[edit]The compact complex manifold X is modelled on the space A. The derivatives of the transition maps describe the tangent bundle through holomorphic transition functions Fbc:Xb \u2229 Xc \u2192 GL(A). These are given by Fbc(a,b) = B(a,b \u2212 c), so the structure group of the corresponding principal fiber bundle reduces to \u0393(A), the structure group of A.[20] The corresponding holomorphic vector bundle with fibre A is the tangent bundle of the complex manifold X. Its holomorphic sections are just holomorphic vector fields on X. They can be determined directly using the fact that they must be invariant under the natural adjoint action of the known holomorphic symmetries of X. They form a finite-dimensional complex semisimple Lie algebra. The restriction of these vector fields to X0 can be described explicitly. A direct consequence of this description is that the Lie algebra is three-graded and that the group of holomorphic symmetries of X, described by generators and relations in Koecher (1967) and Loos (1979), is a complex linear semisimple algebraic group that coincides with the group of biholomorphisms of X.The Lie algebras of the three subgroups of holomorphic automorphisms of X give rise to linear spaces of holomorphic vector fields on X and hence X0 = A.The structure group \u0393(A) has Lie algebra g0{displaystyle {mathfrak {g}}_{0}} spanned by the operators R(x,y). These define a complex Lie algebra of linear vector fields a \u21a6 R(x,y)a on A.The translation operators act on A as Tc(a) = a + c. The corresponding one-parameter subgroups are given by Ttc and correspond to the constant vector fields a \u21a6 c. These give an Abelian Lie algebra g\u22121{displaystyle {mathfrak {g}}_{-1}} of vector fields on A.The operators Sc defined on X by Sc(a,b) = (a,b \u2212 c). The corresponding one-parameter groups Stc define quadratic vector fields a \u21a6 Q(a)c on A. These give an Abelian Lie algebra g1{displaystyle {mathfrak {g}}_{1}} of vector fields on A.Letg=g\u22121\u2295g0\u2295g1.{displaystyle displaystyle {{mathfrak {g}}={mathfrak {g}}_{-1}oplus {mathfrak {g}}_{0}oplus {mathfrak {g}}_{1}.}}Then, defining gi=(0){displaystyle {mathfrak {g}}_{i}=(0)} for i \u2260 \u22121, 0, 1, g{displaystyle {mathfrak {g}}} forms a complex Lie algebra with[gp,gq]\u2286gp+q.{displaystyle displaystyle {[{mathfrak {g}}_{p},{mathfrak {g}}_{q}]subseteq {mathfrak {g}}_{p+q}}.}This gives the structure of a 3-graded Lie algebra. For elements (a,T,b) in g{displaystyle {mathfrak {g}}}, the Lie bracket is given by[(a1,T1,b1),(a2,T2,b2)]=(T1a2\u2212T2a1,[T1,T2]+R(a1,b2)\u2212R(a2,b1),T2tb1\u2212T1tb2){displaystyle displaystyle {[(a_{1},T_{1},b_{1}),(a_{2},T_{2},b_{2})]=(T_{1}a_{2}-T_{2}a_{1},[T_{1},T_{2}]+R(a_{1},b_{2})-R(a_{2},b_{1}),T_{2}^{t}b_{1}-T_{1}^{t}b_{2})}}The group PSL(2,C) of M\u00f6bius transformations of X normalizes the Lie algebra g{displaystyle {mathfrak {g}}}. The transformation j(z) = \u2212z\u22121 corresponding to the Weyl group element J induces the involutive automorphism \u03c3 given by\u03c3(a,T,b)=(b,\u2212Tt,a).{displaystyle displaystyle {sigma (a,T,b)=(b,-T^{t},a).}}More generally the action of a M\u00f6bius transformationg=(\u03b1\u03b2\u03b3\u03b4){displaystyle displaystyle {g={begin{pmatrix}alpha &beta \\gamma &delta end{pmatrix}}}}can be described explicitly. In terms of generators diagonal matrices act as(\u03b100\u03b1\u22121)(a,T,b)=(\u03b12a,T,\u03b1\u22122b),{displaystyle displaystyle {{begin{pmatrix}alpha &0\\0&alpha ^{-1}end{pmatrix}}(a,T,b)=(alpha ^{2}a,T,alpha ^{-2}b),}}upper unitriangular matrices act as(1\u03b201)(a,T,b)=(a+\u03b2T(1)\u2212\u03b22b,T\u2212\u03b2L(a),b),{displaystyle displaystyle {{begin{pmatrix}1&beta \\0&1end{pmatrix}}(a,T,b)=(a+beta T(1)-beta ^{2}b,T-beta L(a),b),}}and lower unitriangular matrices act as(10\u03b31)(a,T,b)=(a,T\u2212\u03b3L(b),b\u2212\u03b3Tt(1)\u2212\u03b32a).{displaystyle displaystyle {{begin{pmatrix}1&0\\gamma &1end{pmatrix}}(a,T,b)=(a,T-gamma L(b),b-gamma T^{t}(1)-gamma ^{2}a).}}This can be written uniformly in matrix notation as(g(T)g(a)g(b)g(T)t)=g(TabTt)g\u22121.{displaystyle displaystyle {{begin{pmatrix}g(T)&g(a)\\g(b)&g(T)^{t}end{pmatrix}}=g{begin{pmatrix}T&a\\b&T^{t}end{pmatrix}}g^{-1}.}}In particular the grading corresponds to the action of the diagonal subgroup of SL(2,C), even with |\u03b1| = 1, so a copy of T.The Killing form is given byB((a1,T1,b1),(a2,T2,b2))=(a1,b2)+(b1,a2)+\u03b2(T1,T2),{displaystyle displaystyle {mathbf {B} ((a_{1},T_{1},b_{1}),(a_{2},T_{2},b_{2}))=(a_{1},b_{2})+(b_{1},a_{2})+beta (T_{1},T_{2}),}}where \u03b2(T1,T2) is the symmetric bilinear form defined by\u03b2(R(a,b),R(c,d))=(R(a,b)c,d)=(R(c,d)a,b),{displaystyle displaystyle {beta (R(a,b),R(c,d))=(R(a,b)c,d)=(R(c,d)a,b),}}with the bilinear form (a,b) corresponding to the trace form: (a,b) = Tr L(ab).More generally the generators of the group G act by automorphisms on g{displaystyle {mathfrak {g}}} asW(a,T,b)=(Wa,WTW\u22121,(Wt)\u22121b),{displaystyle displaystyle {W(a,T,b)=(Wa,WTW^{-1},(W^{t})^{-1}b),}}J(a,T,b)=(\u2212b,\u2212Tt,\u2212a),{displaystyle displaystyle {J(a,T,b)=(-b,-T^{t},-a),}}Tx(a,T,b)=(a+Tx\u2212Q(x)b,T\u2212R(x,b),b),{displaystyle displaystyle {T_{x}(a,T,b)=(a+Tx-Q(x)b,T-R(x,b),b),}}Sy(a,T,b)=(a,T\u2212R(a,y),b\u2212Tty\u2212Q(y)a).{displaystyle displaystyle {S_{y}(a,T,b)=(a,T-R(a,y),b-T^{t}y-Q(y)a).}}The Killing form is nondegenerate on g{displaystyle {mathfrak {g}}}.The nondegeneracy of the Killing form is immediate from the explicit formula. By Cartan’s criterion, g{displaystyle {mathfrak {g}}} is semisimple. In the next section the group G is realized as the complexification of a connected compact Lie group H with trivial center, so semisimple. This gives a direct means to verify semisimplicity. The group H also acts transitively on X.g{displaystyle {mathfrak {g}}} is the Lie algebra of all holomorphic vector fields on X.To prove that g{displaystyle {mathfrak {g}}} exhausts the holomorphic vector fields on X, note the group T acts on holomorphic vector fields. The restriction of such a vector field to X0 = A gives a holomorphic map of A into A. The power series expansion around 0 is a convergent sum of homogeneous parts of degree m \u2265 0. The action of T scales the part of degree m by \u03b12m \u2212 2. By taking Fourier coefficients with respect to T, the part of degree m is also a holomorphic vector field. Since conjugation by J gives the inverse on T, it follows that the only possible degrees are 0, 1 and 2. Degree 0 is accounted for by the constant fields. Since conjugation by J interchanges degree 0 and degree 2, it follows that g\u00b11{displaystyle {mathfrak {g}}_{pm 1}} account for all these holomorphic vector fields. Any further holomorphic vector field would have to appear in degree 1 and so would have the form a \u21a6 Ma for some M in End A. Conjugation by J would give another such map N. Moreover, etM(a,0,0)= (etMa,0,0). But thenetM(0,0,b)=JetNJ(0,0,b)=JetN(b,0,0)=(0,0,etNb).{displaystyle displaystyle {e^{tM}(0,0,b)=Je^{tN}J(0,0,b)=Je^{tN}(b,0,0)=(0,0,e^{tN}b).}}Set Ut = etM and Vt = etB. ThenQ(Uta)b=UtQ(a)V\u2212tb.{displaystyle displaystyle {Q(U_{t}a)b=U_{t}Q(a)V_{-t}b.}}It follows that Ut lies in \u0393(A) for all t and hence that M lies in g0{displaystyle {mathfrak {g}}_{0}}. So g{displaystyle {mathfrak {g}}} is exactly the space of holomorphic vector fields on X.Compact real form[edit]The action of G on g{displaystyle {mathfrak {g}}} is faithful.Suppose g = WTxSyTz acts trivially on g{displaystyle {mathfrak {g}}}. Then SyTz must leave the subalgebra (0,0,A) invariant. Hence so must Sy. This forces y = 0, so that g = WTx + z. But then Tx+z must leave the subalgebra (A,0,0) invariant, so that x + z = 0 and g = W. If W acts trivially, W = I.[21]The group G can thus be identified with its image in GL g{displaystyle {mathfrak {g}}}.Let A = E + iE be the complexification of a Euclidean Jordan algebra E. For a = x + iy, set a* = x \u2212 iy. The trace form on E defines a complex inner product on A and hence an adjoint operation. The unitary structure group \u0393u(A) consists of those g in \u0393(A) that are in U(A), i.e. satisfy gg*=g*g = I. It is a closed subgroup of U(A). Its Lie algebra consists of the skew-adjoint elements in g0{displaystyle {mathfrak {g}}_{0}}. Define a conjugate linear involution \u03b8 on g{displaystyle {mathfrak {g}}} by\u03b8(a,T,b)=(b\u2217,\u2212T\u2217,a\u2217).{displaystyle displaystyle {theta (a,T,b)=(b^{*},-T^{*},a^{*}).}}This is a period 2 conjugate-linear automorphism of the Lie algebra. It induces an automorphism of G, which on the generators is given by\u03b8(Sa)=Ta\u2217,\u03b8(j)=j,\u03b8(Tb)=Sb\u2217,\u03b8(W)=(W\u2217)\u22121.{displaystyle displaystyle {theta (S_{a})=T_{a^{*}},,,,theta (j)=j,,,,theta (T_{b})=S_{b^{*}},,,,theta (W)=(W^{*})^{-1}.}}Let H be the fixed point subgroup of \u03b8 in G. Let h{displaystyle {mathfrak {h}}} be the fixed point subalgebra of \u03b8 in g{displaystyle {mathfrak {g}}}. Define a sesquilinear form on g{displaystyle {mathfrak {g}}} by (a,b) = \u2212B(a,\u03b8(b)). This defines a complex inner product on g{displaystyle {mathfrak {g}}} which restricts to a real inner product on h{displaystyle {mathfrak {h}}}. Both are preserved by H. Let K be the identity component of \u0393u(A). It lies in H. Let Ke = Tm be the diagonal torus associated with a Jordan frame in E. The action of SL(2,C)m is compatible with \u03b8 which sends a unimodular matrix (\u03b1\u03b2\u03b3\u03b4){displaystyle {begin{pmatrix}alpha &beta \\gamma &delta end{pmatrix}}} to (\u03b4\u00af\u2212\u03b3\u00af\u2212\u03b2\u00af\u03b1\u00af){displaystyle {begin{pmatrix}{overline {delta }}&-{overline {gamma }}\\-{overline {beta }}&{overline {alpha }}end{pmatrix}}}. In particular this gives a homomorphism of SU(2)m into H.Now every matrix M in SU(2) can be written as a productM=(\u03b6100\u03b61\u22121)(cos\u2061\u03c6sin\u2061\u03c6\u2212sin\u2061\u03c6cos\u2061\u03c6)(\u03b6200\u03b62\u22121).{displaystyle displaystyle {M={begin{pmatrix}zeta _{1}&0\\0&zeta _{1}^{-1}end{pmatrix}}{begin{pmatrix}cos varphi &sin varphi \\-sin varphi &cos varphi end{pmatrix}}{begin{pmatrix}zeta _{2}&0\\0&zeta _{2}^{-1}end{pmatrix}}.}}The factor in the middle gives another maximal torus in SU(2) obtained by conjugating by J. If a = \u03a3 \u03b1iei with |\u03b1i| = 1, then Q(a) gives the action of the diagonal torus T = Tm and corresponds to an element of K \u2286 H. The element J lies in SU(2)m and its image is a M\u00f6bius transformation j lying in H. Thus S = j \u2218 T \u2218 j is another torus in H and T \u2218 S \u2218 T coincides with the image of SU(2)m.H acts transitively on X. The stabilizer of (0:0) is K. Furthermore H = KSK, so that H is a connected closed subgroup of the unitary group on g{displaystyle {mathfrak {g}}}. Its Lie algebra is h{displaystyle {mathfrak {h}}}.Since Z = SU(2)m(0:0) for the compact complex manifold corresponding to Ae, if follows that Y = T S (0:0), where Y is the image of Z. On the other hand, X = KY, so thatX = KTS(0:0) = KS(0:0). On the other hand, the stabilizer of (0:0) in H is K, since the fixed point subgroup of G0G\u22121 under \u03b8 is K. Hence H = KSK. In particular H is compact and connected since both K and S are. Because it is a closed subgroup of U g{displaystyle {mathfrak {g}}}, it is a Lie group. It contains K and hence its Lie algebra contains the operators (0,T,0) with T* = \u2212T. It contains the image of SU(2)m and hence the elements (a,0,a*) with a in Ae. Since A = KAe and (kt)\u22121(a*) = (ka)*, it follows that the Lie algebra h1{displaystyle {mathfrak {h}}_{1}} of H contains (a,0,a*) for all a in A. Thus it contains h{displaystyle {mathfrak {h}}}.They are equal because all skew-adjoint derivations of h{displaystyle {mathfrak {h}}} are inner. In fact, since H normalizes h{displaystyle {mathfrak {h}}} and the action by conjugation is faithful, the map of h1{displaystyle {mathfrak {h}}_{1}} into the Lie algebra d{displaystyle {mathfrak {d}}} of derivations of h{displaystyle {mathfrak {h}}} is faithful. In particular h{displaystyle {mathfrak {h}}} has trivial center. To show that h{displaystyle {mathfrak {h}}} equals h1{displaystyle {mathfrak {h}}_{1}}, it suffices to show that d{displaystyle {mathfrak {d}}} coincides with h{displaystyle {mathfrak {h}}}. Derivations on h{displaystyle {mathfrak {h}}} are skew-adjoint for the inner product given by minus the Killing form. Take the invariant inner product on d{displaystyle {mathfrak {d}}} given by \u2212Tr D1D2. Since h{displaystyle {mathfrak {h}}} is invariant under d{displaystyle {mathfrak {d}}} so is its orthogonal complement. They are both ideals in d{displaystyle {mathfrak {d}}}, so the Lie bracket between them must vanish. But then any derivation in the orthogonal complement would have 0 Lie bracket with h{displaystyle {mathfrak {h}}}, so must be zero. Hence h{displaystyle {mathfrak {h}}} is the Lie algebra of H. (This also follows from a dimension count since dim X = dim H \u2212 dim K.)G is isomorphic to a closed subgroup of the general linear group on g{displaystyle {mathfrak {g}}}.The formulas above for the action of W and Sy show that the image of G0G\u22121 is closed in GL g{displaystyle {mathfrak {g}}}. Since H acts transitively on X and the stabilizer of (0:0) in G is G0G\u22121, it follows that G =HG0G\u22121. The compactness of H and closedness of G0G\u22121 implies that G is closed in GL g{displaystyle {mathfrak {g}}}.G is a connected complex Lie group with Lie algebra g{displaystyle {mathfrak {g}}}. It is the complexification of H.G is a closed subgroup of GL g{displaystyle {mathfrak {g}}} so a real Lie group. Since it contains Gi with i = 0 or \u00b11, its Lie algebra contains g{displaystyle {mathfrak {g}}}. Since g{displaystyle {mathfrak {g}}} is the complexification of h{displaystyle {mathfrak {h}}}, like h{displaystyle {mathfrak {h}}} all its derivations are inner and it has trivial center. Since the Lie algebra of G normalizes g{displaystyle {mathfrak {g}}} and o is the only element centralizing g{displaystyle {mathfrak {g}}}, as in the compact case the Lie algebra of G must be g{displaystyle {mathfrak {g}}}. (This can also be seen by a dimension count since dim X = dim G \u2212 dim G0G\u22121.) Since it is a complex subspace, G is a complex Lie group. It is connected because it is the continuous image of the connected set H \u00d7 G0G\u22121.Since g{displaystyle {mathfrak {g}}} is the complexification of h{displaystyle {mathfrak {h}}}, G is the complexification of H.Noncompact real form[edit]For a in A the spectral norm ||a|| is defined to be max \u03b1i if a = u \u03a3 \u03b1iei with \u03b1i \u2265 0 and u in K. It is independent of choices and defines a norm on A. Let D be the set of a with ||a|| < 1 and let H* be the identity component of the closed subgroup of G carrying D onto itself. It is generated by K, the M\u00f6bius transformations in PSU(1,1) and the image of SU(1,1)m corresponding to a Jordan frame. Let \u03c4 be the conjugate-linear period 2 automorphism of g{displaystyle {mathfrak {g}}} defined by\u03c4(a,T,b)=(\u2212a\u2217,\u2212T\u2217,\u2212b\u2217).{displaystyle displaystyle {tau (a,T,b)=(-a^{*},-T^{*},-b^{*}).}}Let h\u2217{displaystyle {mathfrak {h}}^{*}} be the fixed point algebra of \u03c4. It is the Lie algebra of H*. It induces a period 2 automorphism of G with fixed point subgroup H*. The group H* acts transitively on D. The stabilizer of0 is K.[22]The noncompact real semisimple Lie group H* acts on X with an open orbit D. As with the action of SU(1,1) on the Riemann sphere, it has only finitely many orbits. This orbit structure can be explicitly described when the Jordan algebra A is simple. Let X0(r,s) be the subset of A consisting of elements a = u \u03a3 \u03b1iai with exactly r of the \u03b1i less than one and exactly s of them greater than one. Thus 0 \u2264 r + s \u2264 m. These sets are the intersections of the orbits X(r,s) of H* with X0. The orbits with r + s = m are open. There is a unique compact orbit X(0,0). It is the Shilov boundary S of D consisting of elements eix with x in E, the underlying Euclidean Jordan algebra. X(p,q) is in the closure of X(r,s) if and only if p \u2264 r and q \u2264 s.In particular S is in the closure of every orbit.[23]Jordan algebras with involution[edit]The preceding theory describes irreducible Hermitian symmetric spaces of tube type in terms of unital Jordan algebras. In Loos (1977) general Hermitian symmetric spaces are described by a systematic extension of the above theory to Jordan pairs. In the development of Koecher (1969) harvtxt error: no target: CITEREFKoecher1969 (help), however, irreducible Hermitian symmetric spaces not of tube type are described in terms of period two automorphisms of simple Euclidean Jordan algebras. In fact any period 2 automorphism defines a Jordan pair: the general results of Loos (1977) on Jordan pairs can be specialized to that setting.Let \u03c4 be a period two automorphism of a simple Euclidean Jordan algebra E with complexification A. There are corresponding decompositions E = E+ \u2295 E\u2212 and A = A+ \u2295 A\u2212 into \u00b11 eigenspaces of \u03c4. Let V \u2261 A\u03c4 = A\u2212. \u03c4 is assumed to satisfy the additional condition that the trace form on V defines an inner product. For a in V, define Q\u03c4(a) to be the restriction of Q(a) to V. For a pair (a,b) in V2, define B\u03c4(a,b) and R\u03c4(a,b) to be the restriction of B(a,b) and R(a,b) to V. Then V is simple if and only if the only subspaces invariant under all the operators Q\u03c4(a) and R\u03c4(a,b) are (0) and V.The conditions for quasi-invertibility in A show that B\u03c4(a,b) is invertible if and only if B(a,b) is invertible. The quasi-inverse ab is the same whether computed in A or V. A space of equivalence classes X\u03c4 can be defined on pairs V2. It is a closed subspace of X, so compact. It also has the structure of a complex manifold, modelled on V. The structure group \u0393(V) can be defined in terms of Q\u03c4 and it has as a subgroup the unitary structure group \u0393u(V) = \u0393(V) \u2229 U(V) with identity component K\u03c4. The group K\u03c4 is the identity component of the fixed point subgroup of \u03c4 in K. Let G\u03c4 be the group of biholomorphisms of X\u03c4 generated by W in G\u03c4,0, the identity component of \u0393(V), and the Abelian groupsG\u03c4,\u22121 consisting of the Sa and G\u03c4,+1 consisting of the Tb witha and b in V. It acts transitively on X\u03c4 with stabilizer G\u03c4,0G\u03c4,\u22121 andG\u03c4 = G\u03c4,0G\u03c4,\u22121G\u03c4,+1G\u03c4,\u22121. The Lie algebra g\u03c4{displaystyle {mathfrak {g}}_{tau }} of holomorphic vector fields on X\u03c4 is a 3-graded Lie algebra,g\u03c4=g\u03c4,+1\u2295g\u03c4,0\u2295g\u03c4,\u22121.{displaystyle displaystyle {{mathfrak {g}}_{tau }={mathfrak {g}}_{tau ,+1}oplus {mathfrak {g}}_{tau ,0}oplus {mathfrak {g}}_{tau ,-1}.}}Restricted to V the components are generated as before by the constant functions into V, by the operators R\u03c4(a,b) and by the operators Q\u03c4(a). The Lie brackets are given by exactly the same formula as before.The spectral decomposition in E\u03c4 and V is accomplished using tripotents, i.e. elements e such that e3 = e. In this case f = e2 is an idempotent in E+. There is a Pierce decomposition E = E0(f) \u2295 E1\/2(f) \u2295 E1(f) into eigenspaces of L(f). The operators L(e) andL(f) commute, so L(e) leaves the eigenspaces above invariant.In fact L(e)2 acts as 0 on E0(f), as 1\/4 on E1\/2(f) and 1 on E1(f). This induces a Pierce decomposition E\u03c4 = E\u03c4,0(f) \u2295 E\u03c4,1\/2(f) \u2295 E\u03c4,1(f). The subspace E\u03c4,1(f) becomes a Euclidean Jordan algebra with unit f under the mutation Jordan product x \u2218 y = {x,e,y}.Two tripotents e1 and e2 are said to be orthogonal if all the operators [L(a),L(b)] = 0 when a and b are powers of e1 and e2 and if the corresponding idempotents f1 and f2 are orthogonal. In this case e1 and e2 generate a commutative associative algebra and e1e2 = 0, since (e1e2,e1e2) =(f1,f2) =0. Let a be in E\u03c4. Let F be the finite-dimensional real subspace spanned by odd powers of a. The commuting self-adjoint operators L(x)L(y) with x, y odd powers of a act on F, so can be simultaneously diagonalized by an orthonormal basis ei. Since (ei)3 is a positive multiple of ei, rescaling if necessary, ei can be chosen to be a tripotent. They form an orthogonal family by construction. Since a is in F, it can be written a = \u03a3 \u03b1iei with \u03b1i real. These are called the eigenvalues of a (with respect to \u03c4). Any other tripotent e in F has the form a = \u03a3 \u03b5iei with \u03b5i = 0, \u00b11, so the ei are up to sign the minimal tripotents in F.A maximal family of orthogonal tripotents in E\u03c4 is called a Jordan frame. The tripotents are necessarily minimal. All Jordan frames have the same number of elements, called the rank of E\u03c4. Any two frames are related by an element in the subgroup of the structure group of E\u03c4 preserving the trace form. For a given Jordan frame (ei), any element a in V can be written in the form a = u \u03a3 \u03b1iei with \u03b1i \u2265 0 and u an operator in K\u03c4. The spectral norm of a is defined by ||a|| = sup \u03b1i and is independent of choices. Its square equals the operator norm of Q\u03c4(a). Thus V becomes a complex normed space with open unit ball D\u03c4.Note that for x in E, the operator Q(x) is self-adjoint so that the norm ||Q(x)n|| = ||Q(x)||n. Since Q(x)n = Q(xn), it follows that ||xn|| = ||x||n. In particular the spectral norm of x = \u03a3 \u03b1iei in A is the square root of the spectral norm of x2 = \u03a3 (\u03b1i)2fi. It follows that the spectral norm of x is the same whether calculated in A or A\u03c4. Since K\u03c4 preserves both norms, the spectral norm on A\u03c4 is obtained by restricting the spectral norm on A.For a Jordan frame e1, …, em, let Ve = \u2295 C ei. There is an action of SL(2,C)m on Ve which extends to V. If c = \u03a3 \u03b3iei and b = \u03a3 \u03b2iei, then S(c) and T(b) give the action of the product of the lower and upper unitriangular matrices. If a = \u03a3 \u03b1iei with \u03b1i \u2260 0, then the corresponding product of diagonal matrices act as W = B\u03c4(a, e \u2212 a), where e = \u03a3 ei.[24] In particular the diagonal matrices give an action of (C*)m and Tm.As in the case without an automorphism \u03c4, there is an automorphism \u03b8 of G\u03c4. The same arguments show that the fixed point subgroup H\u03c4 is generated by K\u03c4 and the image of SU(2)m. It is a compact connected Lie group. It acts transitively on X\u03c4; the stabilizer of (0:0) is K\u03c4. Thus X\u03c4 = H\u03c4\/K\u03c4, a Hermitian symmetric space of compact type.Let H\u03c4* be the identity component of the closed subgroup of G\u03c4 carrying D\u03c4 onto itself. It is generated by K\u03c4 and the image of SU(1,1)m corresponding to a Jordan frame. Let \u03c1 be the conjugate-linear period 2 automorphism of g\u03c4{displaystyle {mathfrak {g}}_{tau }} defined by\u03c1(a,T,b)=(\u2212a\u2217,\u2212T\u2217,\u2212b\u2217).{displaystyle displaystyle {rho (a,T,b)=(-a^{*},-T^{*},-b^{*}).}}Let h\u03c4\u2217{displaystyle {mathfrak {h}}_{tau }^{*}} be the fixed point algebra of \u03c1. It is the Lie algebra of H\u03c4*. It induces a period 2 automorphism of G with fixed point subgroup H\u03c4*. The group H\u03c4* acts transitively on D\u03c4. The stabilizer of0 is K\u03c4*.[25]H\u03c4*\/K\u03c4 is the Hermitian symmetric space of noncompact type dual to H\u03c4\/K\u03c4.The Hermitian symmetric space of non-compact type have an unbounded realization, analogous the upper half-plane in C. M\u00f6bius transformations in PSL(2,C) corresponding to the Cayley transform and its inverse give biholomorphisms of the Riemann sphere exchanging the unit disk and the upper halfplane. When the Hermitian symmetric space is of tube type the same M\u00f6bius transformations map the disk D in A onto the tube domain T = E + iC were C is the open self-dual convex cone of squares in the Euclidean Jordan algebra E.For Hermitian symmetric space not of tube type there is no action of PSL(2,C) on X, so no analogous Cayley transform. A partial Cayley transform can be defined in that case for any given maximal tripotent e = \u03a3 \u03b5iei in E\u03c4. It takes the disk D\u03c4 in A\u03c4 = A\u03c4,1(f) \u2295 A\u03c4,1\/2(f) onto a Siegel domain of the second kind.In this case E\u03c4,1(f) is a Euclidean Jordan algebra and there is symmetric E\u03c4,1(f)-valued bilinear form on E\u03c4,1\/2(f) such that the corresponding quadratic form q takes values in its positive cone C\u03c4. The Siegel domain consists of pairs (x + iy,u + iv) such that y \u2212 q(u) \u2212 q(v) lies in C\u03c4.The quadratic form q on E\u03c4,1\/2(f) and the squaring operation on E\u03c4,1(f) are given byx \u21a6 Q\u03c4(x)e. The positive cone C\u03c4 corresponds to x with Q\u03c4(x) invertible.[26]Examples[edit]For simple Euclidean Jordan algebras E with complexication A, the Hermitian symmetric spaces of compact type X can be described explicitly as follows, using Cartan’s classification.[27]Type In. A is the Jordan algebra of n \u00d7 n complex matrices Mn(C) with the operator Jordan product x \u2218 y = 1\/2(xy + yx). It is the complexification of E = Hn(C), the Euclidean Jordan algebra of self-adjoint n \u00d7 n complex matrices. In this case G = PSL(2n,C) acting on A with g=(abcd){displaystyle g={begin{pmatrix}a&b\\c&dend{pmatrix}}} acting as g(z) = (az + b)(cz + d)\u22121. Indeed, this can be verified directly for diagonal, upper and lower unitriangular matrices which correspond to the operators W, Sc and Tb. The subset \u03a9 corresponds to the matrices g with d invertible. In fact consider the space of linear maps from Cn to C2n = Cn \u2295 Cn. It is described by a pair (T1|T2) with Ti in Mn(C). This is a module for GL(2n,C) acting on the target space. There is also an action of GL(n,C) induced by the action on the source space. The space of injective maps U is invariant and GL(n,C) acts freely on it. The quotient is the Grassmannian M consisting of n-dimensional subspaces of C2n. Define a map of A2 into M by sending (a,b) to the injective map (a|I \u2212 bta). This map induces an isomorphism of X onto M.In fact let V be an n-dimensional subspace of Cn \u2295 Cn. If it is in general position, i.e. it and its orthogonal complement have trivial intersection with Cn \u2295 (0) and(0) \u2295 Cn, it is the graph of an invertible operator T.So the image corresponds to (a|I \u2212 bta) with a = I and bt = I \u2212 T.At the other extreme,V and its orthogonal complement U can be written as orthogonal sums V = V1 \u2295 V2, U = U1 \u2295 U2, where V1 and U1 are the intersections with Cn \u2295 (0) and V2 and U2 with (0) \u2295 Cn. Then dim V1 = dim U2 and dim V2 = dim U1. Moreover, Cn \u2295 (0) = V1 \u2295 U1 and (0) \u2295 Cn = V2 \u2295 U2. The subspace V corresponds to the pair (e|I \u2212 e), where e is the orthogonal projection of Cn \u2295 (0) onto V1. So a = e and b = I.The general case is a direct sum of these two cases. V can be written as an orthogonal sum V = V0 \u2295 V1 \u2295 V2 where V1 and V2 are the intersections with Cn \u2295 (0) and(0) \u2295 Cn and V0 is their orthogonal complement in V. Similarly the orthogonal complement U of V can be written U = U0 \u2295 U1 \u2295 U2.Thus Cn \u2295 (0) = V1 \u2295 U1 \u2295 W1 and (0) \u2295 Cn = V2 \u2295 U2 \u2295 W2, where Wi are orthogonal complements. The direct sum (V1 \u2295 U1) \u2295 (V2 \u2295 U2) \u2286 Cn \u2295 Cn is of the second kind and its orthogonal complement of the first.Maps W in the structure group correspond to h in GL(n,C), with W(a) = haht. The corresponding map on M sends (x|y) to(hx|(ht)\u22121y). Similarly the map corresponding to Sc sends (x|y) to(x|y + c), the map corresponding to Tb sends (x|y) to(x + b|y) and the map corresponding to J sends(x|y) to (y|\u2212x). It follows that the map corresponding to g sends(x|y) to(ax + by|cx + dy).On the other hand, if y is invertible,(x|y) is equivalent to(xy\u22121|I), whence the formula for the fractional linear transformation.Type IIIn. A is the Jordan algebra of n \u00d7 n symmetric complex matrices Sn(C) with the operator Jordan product x \u2218 y = 1\/2(xy + yx). It is the complexification of E = Hn(R), the Euclidean Jordan algebra of n \u00d7 n symmetric real matrices. On C2n = Cn \u2295 Cn, define a nondegenerate alternating bilinear form by \u03c9(x1 \u2295 y1, x2 \u2295 y2) = x1 \u2022 y2 \u2212 y1 \u2022 x2. In matrix notation if J=(0I\u2212I0){displaystyle J={begin{pmatrix}0&I\\-I&0end{pmatrix}}},\u03c9(z1,z2)=zJzt.{displaystyle displaystyle {omega (z_{1},z_{2})=zJz^{t}.}}Let Sp(2n,C) denote the complex symplectic group, the subgroup of GL(2n,C) preserving \u03c9. It consists of g such that gJgt = J and is closed under g \u21a6 gt. If g=(abcd){displaystyle g={begin{pmatrix}a&b\\c&dend{pmatrix}}} belongs to Sp(2n,C) theng\u22121=(dt\u2212ct\u2212btat).{displaystyle displaystyle {g^{-1}={begin{pmatrix}d^{t}&-c^{t}\\-b^{t}&a^{t}end{pmatrix}}.}}It has center {\u00b1I}. In this case G = Sp(2n,C)\/{\u00b1I} acting on A as g(z) = (az + b)(cz + d)\u22121. Indeed, this can be verified directly for diagonal, upper and lower unitriangular matrices which correspond to the operators W, Sc and Tb. The subset \u03a9 corresponds to the matrices g with d invertible. In fact consider the space of linear maps from Cn to C2n = Cn \u2295 Cn. It is described by a pair (T1|T2) with Ti in Mn(C). This is a module for Sp(2n,C) acting on the target space. There is also an action of GL(n,C) induced by the action on the source space. The space of injective maps U with isotropic image, i.e. \u03c9 vanishes on the image, is invariant. Moreover, GL(n,C) acts freely on it. The quotient is the symplectic Grassmannian M consisting of n-dimensional Lagrangian subspaces of C2n. Define a map of A2 into M by sending (a,b) to the injective map (a|I \u2212 ba). This map induces an isomorphism of X onto M.In fact let V be an n-dimensional Lagrangian subspace of Cn \u2295 Cn. Let U be a Lagrangian subspace complementing V. If they are in general position, i.e. they have trivial intersection with Cn \u2295 (0) and(0) \u2295 Cn, than V is the graph of an invertible operator T with Tt = T. So the image corresponds to (a|I \u2212 ba) with a = I and b = I \u2212 T.At the other extreme,V and U can be written as direct sums V = V1 \u2295 V2, U = U1 \u2295 U2, where V1 and U1 are the intersections with Cn \u2295 (0) and V2 and U2 with (0) \u2295 Cn. Then dim V1 = dim U2 and dim V2 = dim U1. Moreover, Cn \u2295 (0) = V1 \u2295 U1 and (0) \u2295 Cn = V2 \u2295 U2. The subspace V corresponds to the pair (e|I \u2212 e), where e is the projection of Cn \u2295 (0) onto V1. Note that the pair (Cn \u2295 (0), (0) \u2295 Cn) is in duality with respect to \u03c9 and the identification between them induces the canonical symmetric bilinear form on Cn. In particular V1 is identified with U2 and V2 with U1. Moreover, they are V1 and U1 are orthogonal with respect to the symmetric bilinear form on (Cn \u2295 (0). Hence the idempotent e satisfies et = e. So a = e and b = I lie in A and V is the image of (a|I \u2212 ba).The general case is a direct sum of these two cases. V can be written as a direct sum V = V0 \u2295 V1 \u2295 V2 where V1 and V2 are the intersections with Cn \u2295 (0) and(0) \u2295 Cn and V0 is a complement in V. Similarly U can be written U = U0 \u2295 U1 \u2295 U2.Thus Cn \u2295 (0) = V1 \u2295 U1 \u2295 W1 and (0) \u2295 Cn = V2 \u2295 U2 \u2295 W2, where Wi are complements. The direct sum (V1 \u2295 U1) \u2295 (V2 \u2295 U2) \u2286 Cn \u2295 Cn is of the second kind. It has a complement of the first kind.Maps W in the structure group correspond to h in GL(n,C), with W(a) = haht. The corresponding map on M sends (x|y) to(hx|(ht)\u22121y). Similarly the map corresponding to Sc sends (x|y) to(x|y + c), the map corresponding to Tb sends (x|y) to(x + b|y) and the map corresponding to J sends(x|y) to (y|\u2212x). It follows that the map corresponding to g sends(x|y) to(ax + by|cx + dy).On the other hand, if y is invertible,(x|y) is equivalent to(xy\u22121|I), whence the formula for the fractional linear transformation.Type II2n. A is the Jordan algebra of 2n \u00d7 2n skew-symmetric complex matrices An(C) and Jordan product x \u2218 y = \u22121\/2(x J y + y J x) where the unit is given by J=(0I\u2212I0){displaystyle J={begin{pmatrix}0&I\\-I&0end{pmatrix}}}. It is the complexification of E = Hn(H), the Euclidean Jordan algebra of self-adjoint n \u00d7 n matrices with entries in the quaternions. This is discussed in Loos (1977) and Koecher (1969) harvtxt error: no target: CITEREFKoecher1969 (help).Type IVn. A is the Jordan algebra Cn \u2295 C with Jordan product (x,\u03b1) \u2218 (y,\u03b2) = (\u03b2x + \u03b1y,\u03b1\u03b2 + x\u2022y). It is the complexication of the rank 2 Euclidean Jordan algebra defined by the same formulas but with real coefficients. This is discussed in Loos (1977).Type VI. The complexified Albert algebra. This is discussed in Faulkner (1972), Loos (1978) and Drucker (1981).The Hermitian symmetric spaces of compact type X for simple Euclidean Jordan algebras E with period two automorphism can be described explicitly as follows, using Cartan’s classification.[28]Type Ip,q. Let F be the space of q by p matrices over R with p \u2260 q. This corresponds to the automorphism of E = Hp + q(R) given by conjugating by the diagonal matrix with p diagonal entries equal to 1 and q to \u22121. Without loss of generality p can be taken greater than q. The structure is given bythe triple product xytz. The space X can be identified with the Grassmannian of p-dimensional subspace of Cp + q = Cp \u2295 Cq. This has a natural embedding in C2p = Cp \u2295 Cp by adding 0’s in the last p \u2212 q coordinates. Since any p-dimensional subspace of C2p can be represented in the form [I \u2212 ytx|x], the same is true for subspaces lying in Cp + q. The last p \u2212 q rows of x must vanish and the mapping does not change if the last p \u2212 q rows of y are set equal to zero. So a similar representation holds for mappings, but now with q by p matrices. Exactly as when p = q, it follows that there is an action of GL(p + q, C) by fractional linear transformations.[29]Type IIn F is the space of real skew-symmetric m by m matrices. After removing a factor of \u221a-1, this corresponds to the period 2 automorphism given by complex conjugation on E = Hn(C).Type V. F is the direct sum of two copies of the Cayley numbers, regarded as 1 by 2 matrices. This corresponds to the canonical period 2 automorphism defined by any minimal idempotent in E = H3(O).See also[edit]^ See:^ See:^ See:^ See:^ See:^ McCrimmon 1978, pp.\u00a0616\u2013617^ Loos 1975, pp.\u00a020\u201322^ In the main application in Loos (1977), A is finite dimensional. In that case invertibility of operators on A is equivalent to injectivity or surjectivity. The general case is treated in Loos (1975) and McCrimmond (2004) harvtxt error: no target: CITEREFMcCrimmond2004 (help).^ Loos 1977^ Loos & 77, pp.\u00a08.3\u20138.4 harvnb error: no target: CITEREFLoos77 (help)^ Loos 1977, p.\u00a07.1\u22127.15^ See:^ Loos 1977, pp.\u00a09.4\u20139.5^ See:^ Koecher 1967, p.\u00a0144^ Koecher 1967, p.\u00a0145^ Koecher 1967, p.\u00a0144^ Loos 1977, p.\u00a08.9-8.10^ Loos 1977^ See:^ Koecher 1967, p.\u00a0164^ See:^ See:^ Loos 1977, pp.\u00a09.4\u20139.5^ See:^ Loos 1977, pp.\u00a010.1\u201310.13^ Loos 1978, pp.\u00a0125\u2013128^ Koecher 1969 harvnb error: no target: CITEREFKoecher1969 (help)^ See:References[edit]Dineen, S.; Mackey, M.; Mellon, P. (1999), “The density property for JB\u2217-triples”, Studia Math., 137: 143\u2013160, hdl:10197\/7056Drucker, D. (1978), “Exceptional Lie algebras and the structure of Hermitian symmetric spaces”, Mem. Amer. Math. Soc., 16 (208)Drucker, D. (1981), “Simplified descriptions of the exceptional bounded symmetric domains”, Geom. Dedicata, 10 (1\u20134): 1\u201329, doi:10.1007\/bf01447407, S2CID\u00a0120210279Faraut, J.; Koranyi, A. (1994), Analysis on symmetric cones, Oxford Mathematical Monographs, Oxford University Press, ISBN\u00a0978-0-19-853477-8Faulkner, J. R. (1972), “A geometry for E7“, Trans. Amer. Math. Soc., 167: 49\u201358, doi:10.1090\/s0002-9947-1972-0295205-4Faulkner, J. R. (1983), “Stable range and linear groups for alternative rings”, Geom. Dedicata, 14 (2): 177\u2013188, doi:10.1007\/bf00181623, S2CID\u00a0122923381Helgason, Sigurdur (1978), Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, ISBN\u00a0978-0-12-338460-7Jacobson, Nathan (1968), Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, vol.\u00a039, American Mathematical Society, Zbl\u00a00218.17010Jacobson, Nathan (1969), Lectures on quadratic Jordan algebras (PDF), Tata Institute of Fundamental Research Lectures on Mathematics, vol.\u00a045, Bombay: Tata Institute of Fundamental Research, MR\u00a00325715, Zbl\u00a00253.17013Jacobson, Nathan (1996), Finite-dimensional division algebras over fields, Berlin: Springer-Verlag, ISBN\u00a0978-3-540-57029-5, Zbl\u00a00874.16002Koecher, Max (1967), “\u00dcber eine Gruppe von rationalen Abbildungen”, Invent. Math., 3 (2): 136\u2013171, doi:10.1007\/BF01389742, S2CID\u00a0120969584, Zbl\u00a00163.03002Koecher, Max (1969a), “Gruppen und Lie-Algebren von rationalen Funktionen”, Math. Z., 109 (5): 349\u2013392, doi:10.1007\/bf01110558, S2CID\u00a0119934963Koecher, Max (1969b), An elementary approach to bounded symmetric domains, Lecture Notes, Rice UniversityKoecher, Max (1999) [1962], Krieg, Aloys; Walcher, Sebastian (eds.), The Minnesota Notes on Jordan Algebras and Their Applications, Lecture Notes in Mathematics, vol.\u00a01710, Berlin: Springer-Verlag, ISBN\u00a0978-3-540-66360-7, Zbl\u00a01072.17513Koecher, Max (1971), “Jordan algebras and differential geometry” (PDF), Actes du Congr\u00e8s International des Math\u00e9maticiens (Nice, 1970), Tome I, Gauthier-Villars, pp.\u00a0279\u2013283K\u00fchn, Oda (1975), “Differentialgleichungen in Jordantripelsystemen”, Manuscripta Math., 17 (4): 363\u2013381, doi:10.1007\/BF01170732, S2CID\u00a0121509094Loos, Ottmar (1975), Jordan pairs, Lecture Notes in Mathematics, vol.\u00a0460, Springer-VerlagLoos, Ottmar (1977), Bounded symmetric domains and Jordan pairs (PDF), Mathematical lectures, University of California, Irvine, archived from the original (PDF) on 2016-03-03, retrieved 2013-05-12Loos, Ottmar (1978), “Homogeneous algebraic varieties defined by Jordan pairs”, Monatsh. Math., 86 (2): 107\u2013129, doi:10.1007\/bf01320204, S2CID\u00a0121527561Loos, Ottmar (1979), “On algebraic groups defined by Jordan pairs”, Nagoya Math. J., 74: 23\u201366, doi:10.1017\/S0027763000018432Loos, Ottmar (1995), “Elementary groups and stability for Jordan pairs”, K-Theory, 9: 77\u2013116, doi:10.1007\/bf00965460McCrimmon, Kevin (1978), “Jordan algebras and their applications”, Bull. Amer. Math. Soc., 84 (4): 612\u2013627, doi:10.1090\/s0002-9904-1978-14503-0McCrimmon, Kevin (2004), A taste of Jordan algebras, Universitext, Berlin, New York: Springer-Verlag, doi:10.1007\/b97489, ISBN\u00a0978-0-387-95447-9, MR\u00a02014924, ErrataMeyberg, K. (1972), Lectures on algebras and triple systems (PDF), University of VirginiaRoos, Guy (2008), “Exceptional symmetric domains”, Symmetries in complex analysis, Contemp. Math., vol.\u00a0468, Amer. Math. Soc., pp.\u00a0157\u2013189Springer, Tonny A. (1998), Jordan algebras and algebraic groups, Classics in Mathematics, Berlin, New York: Springer-Verlag, ISBN\u00a0978-3-540-63632-8Wolf, Joseph A. (1972), “Fine structure of Hermitian symmetric spaces”, in Boothby, William; Weiss, Guido (eds.), Symmetric spaces (Short Courses, Washington University), Pure and Applied Mathematics, vol.\u00a08, Dekker, pp.\u00a0271\u2013357"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/mutation-jordan-algebra-wikipedia\/#breadcrumbitem","name":"Mutation (Jordan algebra) – Wikipedia"}}]}]