Generalized conformal realizations of Kac-Moody algebras
aa r X i v : . [ h e p - t h ] N ov AEI-2007-148
Generalized conformal realizationsof Kac-Moody algebras
Jakob Palmkvist
Albert-Einstein-InstitutMax-Planck-Institut f¨ur GravitationsphysikAm M¨uhlenberg 1, D-14476 Golm, Germany [email protected]
To the memory of Issai Kantor
Abstract
We present a construction which associates an infinite sequence of Kac-Moody al-gebras, labeled by a positive integer n , to one single Jordan algebra. For n = 1, thisreduces to the well known Kantor-Koecher-Tits construction. Our generalization uti-lizes a new relation between different generalized Jordan triple systems, together withtheir known connections to Jordan and Lie algebras. Applied to the Jordan algebraof hermitian 3 × R , C , H , O , the constructiongives the exceptional Lie algebras f , e , e , e for n = 2. Moreover, we obtain theirinfinite-dimensional extensions for n ≥
3. In the case of 2 × so ( p + n, q + n ) and the concomitant nonlinear realizationgeneralizes the conformal transformations in a spacetime of signature ( p, q ).1 Introduction
Jordan algebras are commutative but non-associtive algebras, which were originallystudied in order to understand the foundations of quantum mechanics [1, 2]. Throughtheir connection to Lie algebras, Jordan algebras play an important role in fundamentalphysics, and can be used to define generalized spacetimes [3, 4]. The origin of thisconnection lies in the observation that the triple product( x, y, z ) ( xyz ) ≡ [[ x, τ ( y )] , z ] (1.1)in the subspace g − of a 3-graded Lie algebra g − + g + g , where τ is an involution g − → g , has the same general properties as the triple product( x, y, z ) ( xyz ) ≡ ( xy ) z + x ( yz ) − y ( xz ) (1.2)formed from the multiplication in a Jordan algebra. In the Kantor-Koecher-Tits con-struction [5–7], any Jordan algebra gives rise to a 3-graded Lie algebra, such that thetwo triple products coincide. In the present paper we generalize this construction fora certain kind of Jordan algebras. We will show that any such Jordan algebra givesnot only one Lie algebra, but an infinite sequence of Lie algebras, labeled by a positiveinteger n . For n = 1, we get back the original Kantor-Koecher-Tits construction.The Kantor-Koecher-Tits construction has already been generalized by Kantorfrom Jordan algebras to Jordan triple systems, and further to generalized Jordantriple systems [8]. The generalization of Jordan algebras to Jordan triple systems isneeded for the inverse of the Kantor-Koecher-Tits construction – any 3-graded Liealgebra with an involution τ gives rise to a Jordan triple system, but not all of themcan be obtained from a Jordan algebra by (1.2). Generalized Jordan triple systemcorrespond to graded Lie algebras in general, not necessarily 3-graded. These wellknown connections between Jordan algebras, (generalized) Jordan triple systems andgraded Lie algebras are tools that we will use to derive the results in this paper.Our construction is based on a new generalization of a single generalized Jordantriple system to an infinite sequence of such triple systems. We study the case when theLie algebra associated to the first one (the original generalized Jordan triple system)is a finite Kac-Moody algebra, which means that it can be characterized by a Dynkindiagram. We find that continuing the sequence then corresponds to adding more nodesto the Dynkin diagram. Each node will be connected to the previous one by a singleline, starting from an arbitrary node in the original Dynkin diagram. For the classicalLie algebras b r ( r ≥
3) and d r ( r ≥
4) and for the exceptional Lie algebras f , e , e , e ,there is a unique node in the Dynkin diagram such that we get the affine extensionif we connect an additional node to it by a single line. In this case our constructiononly gives the current algebra extension, which means that the central element andthe derivation must be added by hand. In all other cases we get the full Kac-Moodyalgebra, whether it is finite-dimensional or not.2his work is motivated by the ’magic square’ constructions [9–12], which associatea Lie algebra M ( K , K ′ ) to any pair ( K , K ′ ) of division algebras K = R , C , H , O . Theseconstructions involve the simple Jordan algebras H ( K ) of hermitian 3 × K = R , C , H , O , where the product is the symmetrizedmatrix product. Our construction gives a Lie algebra for each simple Jordan algebraand each positive integer value of a parameter n in the following way. The Jordanalgebra first leads to a Jordan triple system by (1.2) which in turn generalizes toinfinitely many generalized Jordan triple systems. Each of them has an associated Liealgebra. We will show that when we apply the construction to the Jordan algebras H ( K ), we obtain the third row in the magic square for n = 1 (since this is the ordinaryKantor-Koecher-Tits construction) and the fourth row for n = 2. Moreover, we getthe current algebra extension of the algebras in the fourth row for n = 3 (since thenode that we start from is the ’affine’ one), and their hyperbolic extensions for n = 4.Thus our construction not only unifies the third and the fourth row, but also extendsthe magic square with infinitely many new rows. In particular, for K = O , we geta unified construction of e , e , e , and further extensions. When we instead applyour construction to the Jordan algebras H ( K ) of hermitian 2 × K = R , C , H , O , then the associated Lie algebras will always befinite-dimensional, and we consider here not only the complex Lie algebras but alsotheir real forms. In particular, for K = O , we get the pseudo-orthogonal algebras so (1 + n, n ), with the conformal algebra in a ten-dimensional Minkowski spacetimeas the well known case n = 1.Our method is useful for studying nonlinear realizations of Lie algebras [13,14]. Anygraded Lie algebra can be realized nonlinearly on its subspaces of negative (or positive)degree [15]. This nonlinear realization can be expressed in terms of the correspondinggeneralized Jordan triple system. When this in turn is obtained from an original onefor some n in the way that we will describe, then the nonlinear realization can beexpressed in terms of the original generalized Jordan triple system. We will illustratethis for so ( p + n, q + n ).The paper is organized as follows. In Section 2 we show that any generalizedJordan triple system corresponding to a finite Kac-Moody algebra generalizes to aninfinite sequence of such triple systems, labeled by a positive integer n , and that thiscorresponds to adding nodes to the original associated Dynkin diagram. In Section 3,we review the relation between Jordan algebras and the magic square of Lie algebras.Then we show that the associated Lie algebras in the H ( K ) case are the exceptionalLie algebras and their extensions. The nonlinear realization of so ( p + n, q + n ), withthe linearly realized subalgebra so ( p, q ) is given in Section 4. In the appendix wereview in detail how any generalized Jordan triple system gives rise to a graded Liealgebra and how the graded Lie algebra can be nonlinearly realized.3 Kac-Moody algebras
In this section we will prove our main result (Theorem 2.1) about Kac-Moody alge-bras and generalized Jordan triple systems. First we will briefly recall how a complexKac-Moody algebra can be constructed from its (generalized) Cartan matrix, or equiv-alently, from its Dynkin diagram (for details, see [16]), and then how it can be given agrading. We will assume that the determinant of the Cartan matrix is non-zero, whichin particular means that we leave the affine case for now. The Kac-Moody algebrawill then be finite-dimensional (or simply finite ) if and only if the Cartan matrix ispositive-definite.The Cartan matrix is of type r × r , where r is the rank of the Lie algebra. Itsentries are integers satisfying A ii = 2 (no summation) and i = j ⇒ A ij ≤ , A ij = 0 ⇔ A ji = 0 (2.1)for i, j = 1 , , . . . , r . The Dynkin diagram consists of r nodes, and two nodes i, j areconnected by a line if A ij = A ji = −
1, but disconnected if A ij = A ji = 0 (these arethe only two cases that we will consider).In the construction of a Lie algebra from its Cartan matrix, one starts with 3 r generators e i , f i , h i satisfying the Chevalley relations (no summation)[ e i , f j ] = δ ij h j , [ h i , h j ] = 0 , [ h i , e j ] = A ij e j , [ h i , f j ] = − A ij f j . (2.2)The elements h i span the abelian Cartan subalgebra g . Further basis elements of g will then be multiple commutators of either e i or f i , generated by these elementsmodulo the Serre relations (no summation)(ad e i ) − A ji e j = 0 , (ad f i ) − A ji f j = 0 . (2.3)It follows from (2.2) that these multiple commutators (as well as the elements e i and f i themselves) are eigenvectors of ad h for any h ∈ g , and thus each of them definesan element µ in the dual space of g , such that µ ( h ) is the corresponding eigenvalue.These elements µ are the roots of g and the eigenvectors are called root vectors . Inparticular, e i are root vectors of the simple roots α i , which form a basis of the dualspace of g . In this basis, an arbitrary root µ = µ i α i has integer components µ i , eitherall non-negative (if µ is a positive root ) or all non-positive (if µ is a negative root ).For finite Kac-Moody algebras, the space of root vectors corresponding to any rootis one-dimensional. Furthermore, if µ is a root, then − µ is a root as well, but no othermultiples of µ . For any positive root µ of a finite Kac-Moody algebra g , we let e µ and f µ be root vectors corresponding to µ and − µ , respectively, such that they aremultiple commutators of e i or f i . (This requirement fixes the normalization up to asign.) Thus a basis of g is formed by these root vectors e µ , f µ for all positive roots µ ,and by the Cartan elements h i for all i = 1 , , . . . , r .4 .1 Graded Lie algebras A Lie algebra g is graded , or has a grading , if it is the direct sum of subspaces g k ⊂ g for all integers k , such that [ g m , g n ] ⊂ g m + n for all integers m, n . If there is a positiveinteger m such that g ± m = 0 but g ± k = 0 for all k > m , then the Lie algebra g is(2 m + 1)- graded . We will occasionally use the notation g ± = g ± + g ± + · · · .Any simple root α i of a Kac-Moody algebra g generates a grading of g , such that for k ≤
0, the subspace g k ( g − k ) is spanned by all root vectors e µ ( f µ ) with the component µ i = − k (the minus sign is a convention) corresponding to α i in the basis of simpleroots, and, if k = 0, by the Cartan elements h j .A graded involution τ on the Lie algebra g is an automorphism such that τ ( x ) = x for any x ∈ g and τ ( g k ) = g − k for any integer k . The simplest example of a gradedinvolution in a graded Kac-Moody algebra is given by e α ↔ ± f α and h i ↔ − h i . (Withthe minus sign, this is the Chevalley involution .)On the subspace g − of a graded Lie algebra g with a graded involution τ , we candefine a triple product, that is, a trilinear map ( g − ) → g − , given by( x, y, z ) ( xyz ) = [[ x, τ ( y )] , z ] . (2.4)Then, due to the Jacobi identity and the fact that τ is an involution, this triple productwill satisfy the identity( uv ( xyz )) − ( xy ( uvz )) = (( uvx ) yz ) − ( x ( vuy ) z ) , (2.5)which means that g − is a generalized Jordan triple system . As this name suggests,and as we have already mentioned, this kind of triple systems is related to Jordanalgebras. We will explain the relation in more detail in Section 4.1. We now consider the situation when a finite Kac-Moody algebra h is extended toanother one g in the following way, for an arbitrary integer n ≥ n − n h g ✐ ✐ ✐ ②✐ The black node, which g and h have in common, generates a grading of g as well as of h . We want to investigate how the triple systems g − and h − , corresponding to these5wo gradings, are related to each other. It is clear that dim g − = n dim h − , whichmeans that g − as a vector space is isomorphic to the direct sum ( h − ) n of n vectorspaces, each isomorphic to h − . The question is if we can define a triple product on( h − ) n such that g − and ( h − ) n are isomorphic also as triple systems. To answer thisquestion, we write a general element in ( h − ) n as ( x ) + ( x ) + · · · + ( x n ) n , where x , x , . . . are elements in h − . Furthermore, we define for any graded involution τ on h a bilinear form on h − associated to τ by ( e µ , τ ( f ν )) = δ µν for root vectors e µ ∈ h − and f ν ∈ h . The answer is then given by the following theorem, which is the mainresult of this paper. Theorem 2.1.
The vector space ( h − ) n , together with the triple product given by ( x a y b z c ) = δ ab [[ x, τ ( y )] , z ] c − δ ab ( x, y ) z c + δ bc ( x, y ) z a (2.6) for a, b, . . . = 1 , , . . . , n and x, y, z ∈ h − , is a triple system isomorphic to the triplesystem g − with the triple product ( uvw ) = [[ u, τ ( v )] , w ] , where the involution τ isextended from h to g by τ ( e i ) = − f i for the simple root vectors. Proof.
A basis of h − consists of all root vectors e µ such that the component of µ corresponding to α n in the basis of simple roots is equal to one. A basis of g − consistsof all such basis elements e µ of h − together with all commutators [ e i , e µ ], where e i ,for i = 1 , , . . . , n −
1, is the root vector e i = [ . . . [[ e i , e i +1 ] , e i +2 ] , . . . , e n − ] (2.7)of the a n − subalgebra of g . We also define the root vector f i = ( − n − − i [ . . . [[ f i , f i +1 ] , f i +2 ] , . . . , f n − ] (2.8)for the corresponding negative root, and the element h i = h i + h i +1 + h i +2 + · · · + h n − (2.9)in the Cartan subalgebra of g , such that[ e i , f i ] = h i , [ h i , e i ] = 2 e i , [ h i , f i ] = − f i , (2.10)(no summation). If i = j , then[ h i , e j ] = e j , [ h i , f j ] = − f j , (2.11)while [ e i , e j ] is either zero or a root vector of g that does not belong to g − . (We stressthe difference between having the indices i, j, . . . on e, f, h upstairs and downstairs.The root vectors e i correspond to the simple roots of the a n − subalgebra, while theroot vectors e i correspond to roots of the a n − subalgebra for which the component6orresponding to the simple root α n − is equal to one, and these roots are not simple,except for α n − itself.) Using the relations (2.10) − (2.11) we get[[ e i , f j ] , e µ ] = − δ ij e µ , [[ e i , f j ] , f ν ] = δ ij f ν , [[ e µ , f ν ] , e i ] = − δ µν e i , [[ e µ , f ν ] , f j ] = δ µν f j , (2.12)and then [[ e i , e µ ] , [ f j , f ν ]] = − δ ij [ e µ , f ν ] − δ µν [ e i , f j ] , [[ e i , e µ ] , f ν ] = δ µν e i , [ e µ , [ f j , f ν ]] = − δ µν f j . (2.13)Finally we have [[ e i , f j ] , e k ] = δ jk e i + δ ji e k . (2.14)We introduce the bilinear form on h − associated to τ , defined by ( e µ , τ ( f ν )) = δ µν .Then, from (2.12) − (2.13) we get[[ e µ , τ ( e ν )] , e i ] = − ( e µ , e ν ) e i , [[ e µ , τ ( e ν )] , f j ] = ( e µ , e ν ) f j , [[ e i , e µ ] , [ f j , τ ( e ν )]] = − δ ij [ e µ , τ ( e ν )] − ( e µ , e ν )[ e i , f j ] , [[ e i , e µ ] , τ ( e ν )] = ( e µ , e ν ) e i , [ e µ , [ f j , τ ( e ν )]] = − ( e µ , e ν ) f j . (2.15)Consider now the direct sum ( h − ) n of n vector spaces, each isomorphic to h − , andwrite a general element in ( h − ) n as ( x ) + ( x ) + · · · + ( x n +1 ) n , where x , x , . . . are elements in h − . It is easy to see that the map ψ , defined by e µ e µ and( e µ ) i +1 [ e i , e µ ] for i = 1 , , . . . , n − ψ (( uvw )) = ( ψ ( u ) ψ ( v ) ψ ( w )) for all u, v, w ∈ ( h − ) n . (cid:3) In the Kantor-Koecher-Tits construction, any Jordan algebra J is associated to a3-graded Lie algebra g − + g + g , spanned by the operators u ∈ g − : x u (constant)[ u, τ ( v )] ∈ g : x ( uvx ) (linear) τ ( u ) ∈ g : x
7→ − ( xux ) (quadratic) (3.1)7cting on the Jordan algebra, where( xyz ) = ( xy ) z + x ( yz ) − y ( xz ) . (3.2)The associated Lie algebra g − + g + g is the conformal algebra con J , and g is the structure algebra str J . If J has an identity element, then all scalar multiplicationsform a one-dimensional ideal of str J . Factoring out this ideal, we obtain the reduced structure algebra str ′ J , which in turn contains the Lie algebra der J of all derivations of J . Thus, g = str J ⊃ str ′ J ⊃ der J . To see why the resulting Lie algebra is called’conformal’ we consider the Jordan algebras H ( K ) of hermitian 2 × K = R , C , H , O . We have der H ( K ) = so ( d − str ′ H ( K ) = so (1 , d − con H ( K ) = so (2 , d ) (3.3)for d = 3 , , ,
10, respectively [17]. It is well known that con H ( K ) is the algebrathat generates conformal transformations in a d -dimensional Minkowski spacetime.Furthermore, str ′ H ( K ) is the Lorentz algebra and der H ( K ) its spatial part.The Kantor-Koecher-Tits construction can be applied also to the Jordan algebras H ( K ) of hermitian 3 × K . Then we obtain the first three rows in a’magic square’ of Lie algebras [9–12, 17]. The magic square construction associates aLie algebra M ( K , K ′ ) to any pair ( K , K ′ ) of division algebras R , C , H , O , in a naturalway that leads the following symmetric 4 × K ′ \ K R C H OR a a c f C a a ⊕ a a e H c a d e O f e e e For simplicity, and we only specify the complex Lie algebras here. In this magicsquare, the real Lie algebras would actually be the compact forms of the complex Liealgebras that we have specified, but we also get other magic squares of real Lie algebrasif we replace K or K ′ by the corresponding ’split’ algebra C s , H s , O s [18]. When K ′ issplit and K non-split, we get the derivation, reduced structure and conformal algebrasof H ( K ) as the first three rows. When K and K ′ are both split, we get the split realforms of the complex Lie algebras above. We focus on the 3 × ′ \ K C H OC ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞❞ ❞ ❞ ❞ ❞ ❞❞ H ❞ ❞ ❞ ❞ ❞t ❞ ❞ ❞ ❞ ❞❞ t ❞ ❞ ❞ ❞ ❞ ❞❞t O ❞ ❞ ❞ ❞ ❞❞t ❞ ❞ ❞ ❞ ❞ ❞❞ t ❞ ❞ ❞ ❞ ❞ ❞ ❞❞t In the middle row of the 3 × g ± are one-dimensional. With this 5-grading, the algebras in the lastrow are called ’quasiconformal’, associated to Freudenthal triple systems [13, 19, 20].This is usually the way e is included in the context of Jordan algebras and octonions.Here we are more interested in the grading generated by the black node itself. Thenwe have the same situation in the last row as in Section 2.2, for n = 2. With thenotation used in Section 2.2 we thus have g in the last row and h in the second lastrow. Theorem 2.1 now implies that the algebras in the last row follow after those inthe second last row in the sequence of Lie algebras that is obtained from H ( K ) viathe (generalized) Jordan triple systems. (This holds even for K = R although we havenot included it in the illustration above.) However, this sequence does not end with n = 2 but can be continued to infinity, with one Kac-Moody algebra for each positiveinteger value of n . Since the black node in the last row is the ’affine’ one, we will getthe corresponding current algebra for n = 3, and the hyperbolic extension for n = 4.If we apply Theorem 2.1 to the Jordan algebras H ( K ) instead of H ( K ), then weget d r + n where r = 2 , , K = C , H , O , respectively. We will show this in detailin the next section. In this section we first give the nonlinear realization of so ( p + n, q + n ) with a lin-early realized subalgebra so ( p, q ). Then we show that the special case n = 1, whichcorresponds to conformal transformations in a spacetime of signature ( p, q ), is relatedto the general case in the way it should, according to Theorem 2.1. Finally, we re-late the generalized conformal realization to the Jordan algebras H ( K ) for Minkowskispacetimes in 3 , , ,
10 dimensions. 9e start with some basic facts. Let V be a real vector space with an inner product.The real Lie group SO ( V ) consists of all endomorphisms F of V which preserve theinner product, ( F ( u ) , F ( v )) = ( u, v ) (4.1)for all u, v ∈ V . The corresponding real Lie algebra so ( V ) consists of all endomor-phisms f of V which are antisymmetric with respect to the inner product,( f ( u ) , v ) + ( u, f ( v )) = 0 (4.2)for all u, v ∈ V . If V is non-degenerate and finite-dimensional with signature ( p, q ),then we can identify SO ( V ) with the real Lie group SO ( p, q ) consisting of all real( p + q ) × ( p + q ) matrices X such that X t ηX = η, det X = 1 , (4.3)where η is the diagonal matrix associated to the inner product. Correspondingly, wecan identify so ( V ) with the real Lie algebra so ( p, q ) consisting of all ( p + q ) × ( p + q )matrices x such that x t η + ηx = 0 . (4.4)In other words, so ( p, q ) consists of all real matrices of the form x = (cid:18) a c t c b (cid:19) , (4.5)where a and b are orthogonal p × p and q × q matrices respectively. These groupsand algebras are said to be pseudo-orthogonal or, if p = 0, orthogonal , written simply SO ( q ) and so ( q ).We consider now the pseudo-orthogonal algebra so ( p + n, q + n ), with the innerproduct given by η = diag( − , . . . , − | {z } p , +1 , . . . , +1 | {z } q , − , . . . , − | {z } n , +1 , . . . , +1 | {z } n ) , (4.6)for some arbitrary positive integers n, p, q . It is spanned by all matrices G I J , wherethe entry in row L , column K is given by( G I J ) K L = δ I L δ K J − η IK η JL , (4.7)and I, J, . . . = 0 , , . . . , p + q + 2 n −
1. It follows that ( G I J ) t = G J I . If I = J ,then the entry of G I J in row I , column J is 1 while the entry in row J , column I is ± I = J , then G I J = 0. These matrices satisfy thecommutation relations[ G I J , G
K L ] = δ I L G K J − δ K J G I J + η IK η JM G M L − η JL η IM G K M , (4.8)10nd all those G I J with
I < J (say) form a basis of so ( p + n, q + n ).For µ, ν, . . . = 0 , , . . . , p + q − a, b, . . . = 1 , , . . . , n , with µ < ν and a < b , we take the linear combinations K ab = ( − G a + m + nb + m + n + G a + mb + m + n − G a + m + nb + m + G a + mb + m ) ,K µa = − G µa + m + n − G µa + m ,D ab = ( G a + m + nb + m + n + G a + mb + m + n + G a + m + nb + m + G a + mb + m ) ,P µa = − G µa + m + n − G µa + m ,P ab = ( − G a + m + nb + m + n − G a + mb + m + n + G a + m + nb + m + G a + mb + m ) , (4.9)as a new basis, where we have set m = p + q − K ab and P ab vanish when n = 1, since they are antisymmetric in the indices a, b . Thebasis elements (4.9) satisfy the commutation relations[ G µ ν, D ab ] = [ G µν , P ab ] = [ G µν , K ab ] = 0 , [ G µν , G ρσ ] = δ µσ G ρν − δ ρν G µσ + η µρ η νλ G λσ − η νσ η µλ G ρλ , [ D ab , D cd ] = δ ad D cb − δ cb D ad , [ P µa , K νb ] = 2( δ ab G νµ − δ νµ D ab ) , [ G µν , P ρa ] = δ µρ P νa − η νρ η µλ P λa , [ G µν , K ρa ] = − δ ρν K µa + η µρ η νλ K λa , [ D ab , P µc ] = − δ cb P µa , [ D ab , K µc ] = δ ac K µb , [ P µa , P νb ] = − η µν P ab , [ K µa , K νb ] = − η µν K ab , [ P ab , P µc ] = 0 , [ K ab , K µc ] = 0 , [ D ab , P cd ] = δ db P ac − δ cb P ad , [ D ab , K cd ] = δ ac K bd − δ ad K bc , [ K µa , P bc ] = δ ca η µλ P λb − δ ba η µλ P λc , [ P µa , K bc ] = δ ac η µλ K λb − δ ab η µλ K λc , [ P ab , P cd ] = 0 , [ K ab , K cd ] = 0 , [ P ab , K cd ] = δ ac D bd − δ bc D ad − δ ad D bc + δ bd D ac . (4.10)We see that so ( p + n, q + n ) has the following 5-grading, which reduces to a 3-gradingwhen n = 1. 11ubspace g − g − g g g basis P ab P µa G µν , D ab K µa K ab Furthermore, we see that D ab satisfy the commutation relations for gl ( n, R ). Sincethey also commute with G µν , we have g = so ( p, q ) ⊕ gl ( n, R ) as a direct sum ofsubalgebras. Finally, a graded involution τ is given by τ ( P µa ) = η µν K νa = − G µa + m + n + G µa + m ,τ ( K µa ) = η µν P νa = − G µa + m + n + G µa + m ,τ ( D ab ) = − D ba , τ ( K ab ) = P ab ,τ ( G µν ) = G µν τ ( P ab ) = K ab . (4.11)Thus g − is a generalized Jordan triple system with the triple product( P µa P νb P ρc ) = [[ P µa , τ ( P νb )] , P ρc ] = [[ P µa , η νλ K λb ] , P ρc ]= − δ ab η µλ ( δ λρ P νc − η νρ η λκ P κc ) + 2 δ bc η µν P ρa = 2 δ ab ( η νρ P µc − η µρ P νc ) + 2 δ bc η µν P ρa . (4.12)If we now insert (4.12) in (A.29) (but rescale the elements in g − according to [14]),and use the isomorphism (A.20), so that we identify any operator f with the vectorfield − f µa ∂ µa − f ab ∂ ab , then we get the realization P ab = − ∂ ab ,P µa = ∂ µa − x µb ∂ ab ,G µν = x νa ∂ µa − x µa ∂ νa ,D ab = x µb ∂ µa + 2 x bc ∂ ac ,K µa = − x ν a x µb ∂ ν b + x ν a x νb ∂ µb − x ab ∂ µb − x ν a x µb x νc ∂ bc + 2 x ab x µc ∂ bc ,K ab = x µa x νb x µc ∂ ν c − x µb x νa x µc ∂ νc − x ac x µb ∂ µc + x bc x µa ∂ µc + 2 x µa x ν b x µc x νd ∂ cd − x ac x bd ∂ cd . (4.13)Straightforward calculations show that these generators indeed satisfy the commuta-tion relations (4.10). When n = 1, the gl ( n, R ) indices a, b, . . . take only one value,12o we can suppress them, and everything antisymmetric in these indices vanishes. Weare then left with the conformal realization: P µ = ∂ µ (translations) G µν = x ν ∂ µ − x µ ∂ ν (Lorentz transformations) D = x µ ∂ µ (dilatations) K µ = − x ν x µ ∂ ν + x ν x ν ∂ µ (special conformal transformations) (4.14)We will now show that the 5-grading of so ( p + n, q + n ) in this section is generated(as described in Section 2.1) by the simple root corresponding to node n in the Dynkindiagram below of d r for p + q = 2 r . We will then show that the cases n = 1 and n > n ✐ ✐ ✐ ✐✐ ✐② For this we must relate the P µ basis of g − used in this section to the basis consistingof root vectors. The relation will of course be different for different Lie algebras so ( p + n, q + n ). We consider first the case n = 1. Below we give explicitly the relations(with a suitable choice of Cartan-Weyl generators) for two examples, ( p, q ) = (5 , p, q ) = (1 ,
9) in the right table. µ e µ f µ ( P + P ) ( K + K ) ( P + P ) ( K + K ) ( P + P ) ( K + K ) ( P + P ) ( K + K ) ( P + P ) ( K + K ) ( P − P ) ( K − K ) ( P − P ) ( K − K ) ( P − P ) ( K − K ) ( P − P ) ( K − K ) ( P − P ) ( K − K ) µ e µ f µ ( P + P ) ( K + K ) ( P − i P ) ( K + i K ) ( P − i P ) ( K + i K ) ( P − i P ) ( K − i K ) ( P − i P ) ( K + i K ) ( P + i P ) ( K − i K ) (i P + P ) (i K − K ) ( P + i P ) ( K − i K ) (i P + P ) (i K − K ) ( P − P ) ( K − K )13e have indicated the roots by their coefficients in the basis of simple roots, corre-sponding to the nodes in the Dynkin diagram above. (For example, e µ = [ e , e ] inthe second row, and e µ = [[ e , e ] , e ] in the third.) It is evident from these tableshow to generalize them to arbitrary values of p, q (with p + q even). It follows thatthe bilinear form associated to the involution τ ( P µ ) = η µν K ν , (4.15)is given in the P µ basis by ( P µ , P ν ) = 2 η µν .When we extend so ( p +1 , q +1) to g = so ( p + n, q + n ) for n >
1, we put a superscript1 on P µ in the expressions for the root vectors e µ of the subalgebra h = so ( p + 1 , q + 1)in h − (and a subscript 1 on K µ ). Then the graded involution (4.11) on g is indeedthe extension of the original involution (4.15) on h that we described in Theorem 2.1and we get( P µa P ν b P ρc ) = 2 δ ab η νρ P µc − δ ab η µρ P νc + 2 δ bc η µν P ρa = 2 δ ab η νρ P µc − δ ab η µρ P νc + 2 δ ab η µν P ρc − δ ab η µν P ρc + 2 δ bc η µν P ρa = δ ab ( P µ P ν P ρ ) c − δ ab η µν P ρc + 2 δ bc η µν P ρa = δ ab [[ P µ , τ ( P ν )] , P ρ ] c − δ ab ( P µ , P ν ) P ρc + δ bc ( P µ , P ν ) P ρa (4.16)as we should, according to Theorem 2.1. When n = 1, the triple product (4.12) becomes( P µ P ν P ρ ) = 2 η νρ P µ − η µρ P ν + 2 η µν P ρ . (4.17)If we introduce an inner product in the vector space g − by P µ · P ν = η µν , then thiscan be written ( xyz ) = 2( z · y ) x − z · x ) y + 2( x · y ) z. (4.18)Let U be the subspace of g − spanned by P i for i = 1 , , . . . , p + q −
1. Then we canconsider g − as the Jordan algebra J ( U ), with the product P i ◦ P j = ( P i · P j ) P (4.19)for i, j = 1 , , . . . , p + q −
1, and P as identity element. If we introduce a linear map J ( U ) → J ( U ) , z ˜ z, (4.20)14hich changes sign on P , but otherwise leaves the basis elements P µ unchanged, thenwe can write the inner product as2( u · v ) = u ◦ ˜ v + v ◦ ˜ u. (4.21)Inserting (4.21) in (4.18), we get( xyz ) = (˜ z ◦ y ) ◦ x − (˜ z ◦ x ) ◦ y + (˜ x ◦ y ) ◦ z + ( z ◦ ˜ y ) ◦ x − ( z ◦ ˜ x ) ◦ y + ( x ◦ ˜ y ) ◦ z = [ y, ˜ z, x ] + [ y, ˜ x, z ] + ( z ◦ ˜ y ) ◦ x + ( x ◦ ˜ y ) ◦ z. (4.22)It is easy to see that the associators in the last line remain unchanged if we move thetilde from one element to another. Thus we get( xyz ) = [ y, ˜ z, x ] + [ y, ˜ x, z ] + ( z ◦ ˜ y ) ◦ x + ( x ◦ ˜ y ) ◦ z = [˜ y, z, x ] + [˜ y, x, z ] + ( z ◦ ˜ y ) ◦ x + ( x ◦ ˜ y ) ◦ z = 2( z ◦ ˜ y ) ◦ x − z ◦ x ) ◦ ˜ y + 2( x ◦ ˜ y ) ◦ z. (4.23)If we instead use the involution given by τ ( ˜ P µ ) = η µν K ν , (4.24)then we can remove the tilde,( xyz ) = 2( z ◦ y ) ◦ x − z ◦ x ) ◦ y + 2( x ◦ y ) ◦ z. (4.25)Any Jordan algebra J is also a Jordan triple system with this triple product. Theassociated Lie algebra, defined by the construction in the appendix or in (3.1), is itsconformal algebra con J .Consider now the case p = 1 (and still n = 1). Then U is a Euclidean space, and J ( U ) is a formally real Jordan algebra. For q = 2 , , ,
9, there is an isomorphismfrom J ( U ) to H ( K ), where K = R , C , H , O , respectively, given by P (cid:18) (cid:19) , P (cid:18) (cid:19) , P i +1 (cid:18) − e i e i (cid:19) , P p + q − (cid:18) − (cid:19) (4.26)for i = 1 , , . . . , p + q −
3. (Here, e i are the ’imaginary units’ that anticommuteand square to − τ ( P µ ) = K µ and we see from thetables that the associated bilinear form on g − has the simple form ( x, y ) = tr ( x ◦ y ).If we instead consider the split form ( p = q ) for p = 3 , ,
9, then (4.26) is still anisomorphism, if we replace K = C , H , O by the ’split’ algebra K s which is obtainedby changing the square of, respectively, 1 , , − g − , associated to the graded involution (4.24) still has the form ( x, y ) = tr ( x ◦ y ).15o sum up, we have in this section given the 3-grading of so ( p + 1 , q + 1) andshown that it is generated by the simple root corresponding to the leftmost node inthe Dynkin diagram. If we add n − ≥ so ( p + n, q + n ). Theorem 2.1 tellsus how the two triple systems, associated to the 5-graded Lie algebra so ( p + n, q + n )and its 3-graded subalgebra so ( p + 1 , q + 1), respectively, are related to each other. Itfollows that so ( p + n, q + n ) for ( p, q ) = (1 , , (1 , , (1 , , (1 , H ( K ) n , for K = R , C , H , O , withthe triple product( x a y b z c ) = 2 δ ab (( z ◦ y ) ◦ x ) c − δ ab (( z ◦ x ) ◦ y ) c + 2 δ ab (( x ◦ y ) ◦ z ) c − δ ab ( x, y ) z c + δ bc ( x, y ) z a , (4.27)where a, b, c = 1 , , . . . , n and ( x, y ) = tr ( x ◦ y ). The same holds if we replacethese pseudo-orthogonal algebras by the split forms of the corresponding complex Liealgebras, and C , H , O by C s , H s , O s . In this paper we have shown that any Jordan algebra H ( K ) or H ( K ), via (generalized)Jordan triple systems, leads to an infinite sequence of Kac-Moody algebras, labeledby a positive integer n . The generalized Jordan triple product is given by (4.27)where a, b, c = 1 , , . . . , n , and ( x, y ) is the bilinear form associated to the gradedinvolution on the Lie algebra in the n = 1 case. We have shown that it is given by( x, y ) = tr ( x ◦ y ) in the H ( K ) case, but not checked it for H ( K ). However, thebilinear form is well defined by ( e µ , τ ( f ν )) = δ µν , and possible to determine if onewould like to study the H ( K ) case in detail, as we have done for H ( K ). This wouldbe an interesting subject of future research.An important difference between the H ( K ) and H ( K ) cases is that the Lie algebraassociated to H ( K ) n is 3-graded for n = 1 and then 5-graded for all n ≥
2, while theLie algebra associated to H ( K ) n is 3-graded for n = 1 but 7-graded for n = 2, andfor n = 3 , , , . . . , we get infinitely many subspaces in the grading, since these Liealgebras are infinite-dimensional. In the affine case, we only get the correspondingcurrent algebra directly in this construction, which means that the central elementand the derivation must be added by hand. The construction might be related tothe ’affinization’ of generalized Jordan triple systems used in [21, 22] (see also [23]).Finally, concerning the hyperbolic case and further extensions, we hope that our newconstruction can give more information about these indefinite Kac-Moody algebras,for example e and e , which both (but in different approaches) are conjectured tobe symmetries underlying M-theory [24, 25]. In spite of a great interest from bothmathematicians and physicists, these algebras are not yet fully understood.16 cknowledgments: I would like to thank Martin Cederwall, Axel Kleinschmidt,Daniel Mondoc, Hermann Nicolai, Bengt E.W. Nilsson, Daniel Persson, ChristofferPetersson and Hidehiko Shimada for discussions.
A The Lie algebra associated to a generalizedJordan triple system
In the end of Section 2.1 we saw that any graded Lie algebra with a graded involutiongives rise to a generalized Jordan triple system. In this section, we will show theconverse, that any generalized Jordan triple system gives rise to a graded Lie algebrawith a graded involution. The associated Lie algebra has been defined in different (butequivalent) way by Kantor [26] and called the
Kantor algebra [27].We recall from Section 2.1 that a generalized Jordan triple system is a triple systemthat satisfies the identity( uv ( xyz )) − ( xy ( uvz )) = (( uvx ) yz ) − ( x ( vuy ) z ) . (A.1)For any pair of elements x, y in a generalized Jordan triple system T , we define thelinear map s xy : T → T, s xy ( z ) = ( xyz ) . (A.2)Thus (A.1) (for all z ) can be written[ s uv , s xy ] = s ( uvx ) y − s x ( vuy ) . (A.3)For any x ∈ T , we also define the linear map v x : T → End
T, v x ( y ) = s xy , (A.4)which we will use in the following subsection. A.1 Construction
Let T be a vector space and set ˜ U = End T . For k <
0, define ˜ U k recursively as thevector space of all linear maps from T to ˜ U k +1 . Let ˜ U − be the direct sum of all thesevector spaces, ˜ U − = ˜ U − ⊕ ˜ U − ⊕ · · · , (A.5)and define a graded Lie algebra structure on ˜ U − recursively by the relations[ u, v ] = (ad u ) ◦ v − (ad v ) ◦ u. (A.6)17ssume now that T is a generalized Jordan triple system. Let U be the subspace of˜ U spanned by s uv for all u, v ∈ T , and let U − be the subspace of ˜ U − generated by v x for all x ∈ T . Furthermore, let U + be a Lie algebra isomorphic to U − , with theisomorphism denoted by ∗ : U − → U + , u u ∗ . (A.7)Thus U + is generated by v x ∗ for all x ∈ T . Consider the vector space L ( T ) = U − ⊕ U ⊕ U + . (A.8)We can extend the Lie algebra structures on each of these subspaces to a Lie algebrastructure on the whole of L ( T ), by the relations[ s xy , v z ] = v ( xyz ) , [ v x , v y ∗ ] = s xy , [ s xy , v z ∗ ] = − v ( yxz ) ∗ . (A.9)Furthermore, we can extend the isomorphism ∗ between the subalgebras U − and U + to a graded involution on the Lie algebra L ( T ). On U + , it is given by the inverse ofthe original isomorphism, ( u ∗ ) ∗ = u , and on U by s xy ∗ = − s yx . Theorem A.1.
Let g be a graded simple Lie algebra, generated by its subspaces g ± ,with a graded involution τ . Let g − be the generalized Jordan triple system derivedfrom g by ( uvw ) = [[ u, τ ( v )] , w ] . (A.10) Then the Lie algebra L ( g − ) is isomorphic to g . Proof.
Define the linear map ϕ : g → L ( g − ), with g k → U k for all integers k ,recursively by u ∈ g − : ϕ ( u )( x ) = ϕ ([ u, τ ( x )]) ,s ∈ g : ϕ ( s )( x ) = [ s, x ] ,τ ( u ) ∈ g + : ϕ ( τ ( u )) = ϕ ( u ) ∗ , (A.11)where x ∈ g − . We will show that ϕ is an isomorphism. • ϕ is injective Suppose that r and s are elements in g such that ϕ ( r ) = ϕ ( s ). Then [ r − s, x ] = 0for all x ∈ g − , which means that [ r − s, g − ] = 0 since g − is generated by g − . Butthen the proper subspace X k ∈ N (ad ( g + g + )) k ( r − s ) ⊂ g + + g (A.12)18f g is an ideal. Since g is simple, it must be zero, but r − s is an element of thissubspace, so r = s . Suppose now that u and v are elements in g − with ϕ ( u ) = ϕ ( v ).Then ϕ ([ u, τ ( x )]) = ϕ ([ v, τ ( x )]) for all x ∈ g − , and by induction we can show thatthis implies [ u − v, τ ( x )] = 0 for all x ∈ g − . Now we can use the same argument asbefore (but with g + replaced by g − ) to show that u and v must be equal. The case u, v ∈ g + then easily follows by ϕ ( τ ( u )) − ϕ ( τ ( v )) = ϕ ( u ) ∗ − ϕ ( v ) ∗ = ( ϕ ( u ) − ϕ ( v )) ∗ . (A.13) • ϕ is a homomorphism It is sufficient to show this when u ∈ g i and v ∈ g j for all integers i, j and we willdo it by induction over | i | + | j | . One easily checks that ϕ ([ u, v ]) = [ ϕ ( u ) , ϕ ( v )] when | i | + | j | ≤
1. Thus suppose that this is true if | i | + | j | = p for some integer p ≥
1. For i, j < ϕ ( u ) , ϕ ( v )]( x ) = [ ϕ ( u ) , ϕ ( v )( x )] − [ ϕ ( v ) , ϕ ( u )( x )]= [ ϕ ( u ) , ϕ ([ v, τ ( x )])] − [ ϕ ( v ) , ϕ ([ u, τ ( x )])]= ϕ ([ u, [ v, τ ( x )]] − [ v, [ u, τ ( x )]]) = ϕ ([ u, v ])( x ) (A.14)by the assumption of induction in the third step, and by the Jacobi identity in thelast one. We use this for the case i, j >
0, where we have ϕ ([ τ ( u ) , τ ( v )]) = ϕ ( τ ([ u, v ])) = ϕ ([ u, v ]) ∗ = [ ϕ ( u ) , ϕ ( v )] ∗ = [ ϕ ( u ) ∗ , ϕ ( v ) ∗ ] = [ ϕ ( τ ( u )) , ϕ ( τ ( v ))] . (A.15)Finally, we consider the case where i ≥ j ≤
0. Again, we show it by inductionover | i | + | j | , which means i − j in this case. One easily checks that it is true when i = 1 and j = −
1, so we can assume that j ≤ − i ≥ v canbe written as a sum of elements [ x, y ] where x ∈ g m and y ∈ g n for j < m, n <
0. Weconsider one such term and, using what we have already proven, we get[ ϕ ( u ) , ϕ ([ x, y ])] = [ ϕ ( u ) , [ ϕ ( x ) , ϕ ( y )]]= [[ ϕ ( u ) , ϕ ( x )] , ϕ ( y )] − [[ ϕ ( u ) , ϕ ( y )] , ϕ ( x )]= [ ϕ ([ u, x ]) , ϕ ( y )] − [ ϕ ([ u, y ]) , ϕ ( x )]= ϕ ([[ u, x ] , y ] − [[ u, y ] , x ]) = ϕ ([ u, [ x, y ]]) (A.16)by the assumption of induction in the third and fourth steps. • ϕ is surjective Since ϕ is a homomorphism, this follows from the fact that g and L ( g − ) are generatedby g ± and U ± , respectively. The proof is complete. (cid:3) As we have seen, the theorem is useful when a generalized Jordan triple system T happens to be isomorphic to g − , because it then tells us how to construct g from T .19 .2 Realization The construction of the Lie algebra in the previous subsection may seem rather ab-stract, where the elements are linear operators acting on vector spaces of other linearoperators, which in turn act on other vector spaces, and so on. However, once the Liealgebra is constructed, it can also be realized in a way such that the elements act onthe same vector space, but in general non-linearly, and there is a very simple formulafor this, as we will see in this subsection.Let V be the direct sum of (infinitely many) vector spaces V , V , . . . . We write anelement v ∈ V as v = v + v + · · · , where v k ∈ V k , for k = 1 , , . . . . With an operatoron V of grade p we mean a map f : V → V such that for any i = 1 , , . . . , there is asymmetric ( p + p + · · · )-linear map F i : ( V ) p × ( V ) p × · · · → V i , (A.17)where p + 2 p + 3 p + · · · = i + p , that satisfies f ( v ) i = F i ( v , v , . . . , v ; v , v , . . . , v ; . . . ) . (A.18)We define the composition f ◦ g of such an operator f and another operator g , of grade q , as the operator of grade p + q given by( f ◦ g ) i ( v ) = p F i ( g ( v ) , v , . . . , v ; v , v , . . . , v ; . . . )+ p F i ( v , v , . . . , v ; g ( v ) , v , . . . , v ; . . . ) + · · · (A.19)for all i = 1 , , . . . , and a Lie bracket as usual by [ f, g ] = f ◦ g − g ◦ f . Let M p ( V )be the vector space of all operators on V of grade p , and let M ( V ) be the direct sumof all M p ( V ) for all integers p (note that they can also be negative). It follows that M ( V ) is a graded Lie algebra. It is isomorphic to the Lie algebra of all vector fields f i ∂ i on V , where f ∈ M ( V ), with an isomorphism given by f
7→ − f i ∂ i . (A.20)Any graded Lie algebra g = g − + g + g + is isomorphic to a subalgebra of M ( g − ). Itcan be shown [8, 15] that an injective homomorphism χ : g → M ( g − ) is given by χ ( u ) : x (cid:18) ad x − e − ad x P e − ad x (cid:19) ( u ) , (A.21)where P is the projection onto U − along U + U + , and the ratio should be consideredas the power series ad x − e − ad x = 1 + ad x x ) − (ad x )
720 + · · · . (A.22)20 .3 Examples We will now illustrate the ideas in two cases, where the generalized Jordan triplesystem satisfies further conditions.First of all, we assume that the triple systems are such that if ( xyz ) = 0 for all y, z , then x = 0. This allows us to identify x with v x for all x in the triple system,that is, we can identify U − with T , and we can consider any element [ v x , v y ] ∈ U − as a linear map on T , which we denote by h x, y i . Since[ v x , v y ]( z ) = [ v x , v y ( z )] − [ v y , v x ( z )]= [ v x , s yz ] − [ v y , s xz ] = v ( xzy ) − v ( yzx ) , (A.23)this linear map is given by h x, y i ( z ) = ( xzy ) − ( yzx ) . (A.24)A generalized Jordan triple system is generalized in the sense that this linear map doesnot have to be zero − in a Jordan triple system [28], the triple product ( xyz ) is bydefinition symmetric in x and z . Accordingly, the Lie algebra associated to a Jordantriple system is 3-graded, g = g − + g + g , and it can be realized on its subspace g − by applying the formula (A.21). Everything that is left from the power seriesexpansion (A.22) is then the identity map,ad x − e − ad x = 1 , (A.25)and we get u ∈ g − : x P u = u [ u, τ ( v )] ∈ g : x P ([ u, τ ( v )] − [ x, [ u, τ ( v )]]) = − [ x, [ τ ( u ) , v ]]= [ s uv , x ] = ( uvx ) ,τ ( u ) ∈ g : x P ( τ ( u ) − [ x, τ ( u )] + [ x, [ x, τ ( u )]]) = [ x, [ x, τ ( u )]]= − [ s xu , x ] = − ( xux ) . (A.26)This is the conformal realization of g on g − .We now turn to Kantor triple systems [29] (or generalized Jordan triple systemsof second order [26]). These are generalized Jordan triple systems that in addition tothe condition (A.1) satisfy the identity hh u, v i ( x ) , y i = h ( yxu ) , v i − h ( yxv ) , u i . (A.27)It follows that the Lie algebra associated to a Kantor triple system is 5-graded, andthe only part of (A.22) that we have to keep isad x − e − ad x = 1 + ad x . (A.28)21hen we get[ u, v ] ∈ g − : z + Z
7→ h u, v i ,u ∈ g − : z + Z u + h u, z i , [ u, τ ( v )] ∈ g : z + Z ( uvz ) − h u, Z ( v ) i ,τ ( u ) ∈ g : z + Z
7→ − ( zuz ) − Z ( u )+ h ( zuz ) , z i − h Z ( u ) , z i ,τ ( u ) , τ ( v )] ∈ g : z + Z
7→ − ( z h u, v i ( z ) z ) − Z ( h u, v i ( z ))+ h ( z h u, v i ( z ) z ) , z i + h Z ( u ) , Z ( v ) i , (A.29)where z ∈ g − and Z ∈ g − , which is the same realization as in [14], apart from arescaling of the elements in g − by a factor of two. References [1] P. Jordan,
Uber eine Klasse nichtassociativer hyperkomplexer Algebren , Nachr.Ges. Wiss. G¨ottingen 569–575 (1932).[2] P. Jordan, J. von Neumann and E. Wigner,
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