Albert Einstein Institute, Am M¨uhlenberg 1, DE-14476 Golm, Germany E-mail: [email protected]
Received date: December 16, 2007 Revised date: March 31, 2008
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The Kantor-Koecher-Tits construction associates a Lie algebra to any Jordan algebra. We generalize this construction to include also extensions of the associated Lie algebra. In particular, the conformal realization of so(p + 1, q + 1) generalizes to so(p + n, q + n), for arbitrary n, with a linearly realized subalgebra so(p, q). We also show that the construction applied to 3 × 3 matrices over the division algebras R, C, H, O gives rise to the exceptional Lie algebras f4, e6, e7, e8, as well as to their affine, hyperbolic and further extensions.
This article aims to give a brief overview of results that have already been shown by the author in , where details and a more comprehensive list of references are provided. We will mostly consider algebras over the real numbers, even though we will complexify real Lie algebras in order to study properties of the corresponding Dynkin diagrams. However, algebras are assumed to be real if nothing else is stated.
A Jordan algebra is a commutative algebra that satisfies the Jordan identity
The symmetric part of the product in an associative but noncommutative algebra, leads to a Jordan algebra in the same way as the antisymmetric part leads to a Lie algebra. A deeper relationship between these two important kinds of algebras emerges in the generalization of a Jordan algebra to a Jordan triple system J with a triple product such that (xyz) = (zyx) and (2)
This is indeed a generalization since any Jordan algebra, with triple product satisfies the conditions for a Jordan triple system. The same structure arises also in a 3-graded Lie algebra Let be a graded involution on g, which means that Then the subspace g−1 is a Jordan triple system with the triple product Conversely, any Jordan triple system gives rise to a 3-graded Lie algebra g−1 + g0 + g1, spanned by the operators (3)
where x is an element in the Jordan triple system, which can be identified with g−1.If the triple system is derived from a Jordan algebra J by (2), then the associated Lie algebra g−1+g0+g1 is the conformal algebra con J, and g0 is the structure algebra str J. If J has an identity element, then all scalar multiplications form a one-dimensional ideal of str J. Factoring out this ideal, we obtain the reduced structure algebra str ´J. Finally, all derivations of J form a subalgebra der J of str ´J. Thus, g0 = str Jstr ´J der J.This construction of a 3-graded Lie algebra from a given Jordan algebra is called the Kantor-Koecher-Tits construction [3, 5, 9]. To see why the resulting Lie algebra is called ’conformal’ we consider the Jordan algebras H2(K) of hermitian 2 × 2 matrices over the division algebras K = R, C, H,O. We have
for d = 3, 4, 6, 10, respectively . It is well known that conH2(K) is the algebra that generates conformal transformations in a d-dimensional Minkowski spacetime. Furthermore, str ´H2(K) is the Lorentz algebra and der H2(K) its spatial part. With a d-dimensional Minkowski spacetime we mean a vector space with a basis Pμ for μ = 0, 1, . . . , d − 1 and an inner product such that (Pμ, Pv) = , where = diag(−1, 1, . . . , 1). We let a vector x have components xμ in this basis, x = , and let be the corresponding partial derivative. Then we can identify g−1 with this vector space and the operators (3) with the following vector fields:
where the indices are raised and lowered by . The fact that these operators satisfy the commutation relations for so(2, d) does not depend on the signature (1, d − 1) of so we have the same conformal realization of so(p + 1, q + 1) for any signature (p, q).
The Kantor-Koecher-Tits construction can be applied also to the Jordan algebras H3(K) of hermitian 3 × 3 matrices over K and then we obtain the first three rows in the ’magic square’, which is a symmetric 4 × 4 array of complex Lie algebras  (see also references therein). In particular, the exceptional Jordan algebra H3(O) gives rise to exceptional Lie algebras:
(For simplicity, we do not specify the real forms of these complex Lie algebras.) We focus on the 3×3 subarray in the lower right corner of the magic square, consisting of simply laced algebras:
It is easily seen that any simple root α of a complex Lie algebra g (or the corresponding node in the Dynkin diagram) ’generates’ a grading of g, where the subspace gk is spanned by all root vectors eμ or f−μ, such that the root μ has the coefficient −k for α in the basis of simple roots . In the middle row above, the black node generates the 3-grading of the conformal algebra. (Here and below, this meaning of a black node in a Dynkin diagram should not be confused with any different meaning used elsewhere.) The outermost node next to it in the last row generates the unique 5-grading where the subspaces g±2 are one-dimensional. With this 5-grading, the algebras in the last row are called ’quasiconformal’, associated to Freudenthal triple systems . This is usually the way e8 is included in the context of Jordan algebras and octonions. The approach in this contribution is different: we want to generalize the conformal realization but keep the linear realization of the reduced structure algebra, and therefore we consider in the last row the grading generated by the black node itself. This grading seems better suited for a further generalization to extensions of these exceptional Lie algebras. We thus consider the case when a finite Kac-Moody algebra h is extended to another Kac-Moody algebra g in the following way, for an arbitrary integer n≥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−1 and h−1, corresponding to these two gradings, are related to each other. It is clear that dim g−1 = n dim h−1, which means that g−1 as a vector space is isomorphic to the direct sum (h−1)n of n vector spaces, each isomorphic to h−1. The question is if we can define a triple product on (h−1)n such that g−1 and (h−1)n are isomorphic also as triple systems. To answer this question, we write a general element in (h−1)n as (x1)1 +(x2)2 +· · ·+(xn)n, where x1, x2, . . . are elements in h−1. Furthermore, we define for any graded involution on h a bilinear form on h−1 associated to by (eμ, (fv)) = for root vectors eμ h−1 and fv h1. The answer is then given by the following theorem (for a proof, see ).
Theorem 1. The vector space (h−1)n, together with the triple product given by
for a, b, . . . = 1, 2, . . . , n and x, y, z h−1, is a triple system isomorphic to the triple system g−1 with the triple product (uvw) = [[u, (v)], w], where the involution is extended from h to g by (ei) = −fi for the simple root vectors.
Even though the grading of h generated by the black node is a 3-grading, the grading of the extended algebra g generated by the same black node is an m-grading, where m can be any odd positive integer, or even infinity. Equivalently, even though the triple system h−1 is a Jordan triple system, this is in general not the case for the triple system g−1. However, g−1 is always a generalized Jordan triple system, which means that (2) holds, but the triple product (xyz) does not need to be symmetric in x and z. The construction of the associated Lie algebra can be extended to any generalized Jordan triple system, as we will see an example of next. .
When h = so(p, q) for any positive integers p, q, the black node always generates a 5-grading of g = so(p + n, q + n). Equivalently, g−1 is a generalized Jordan triple system of second order, or a Kantor triple system [1, 4]. We get back the associated Lie algebra so(p + n, q + n) as the one spanned by the operators
where z is an element in the Kantor triple system g−1, while Z and the h , i expressions belong to a certain subspace of End g−1, which can be identified with g−2 . In the same way as (3), we can write the operators (6) as the following vector fields:
When (p, q) = (1, 2), (1, 3), (1, 5), (1, 9), we can thus construct so(p + n, q + n) starting from the Jordan algebra H2(K) and using the theorem. In turns out that the bilinear form associated to the graded involution in this case is given by the trace: (x, y) = tr (xy). Then H2(K)n will be a Kantor triple system with the triple product
where a, b, c = 1, 2, . . . , n, and the Lie algebra associated to this Kantor triple system is thus so(p + n, q + n). When we apply the same idea to the Jordan algebras H3(K), we find that the exceptional algebras f4, e6, e7, e8 are the Lie algebras associated to H3(K)2, with the triple product (7) for K = R, C, H, O, respectively. Their affine and hyperbolic extensions are those associated to H3(K)3 and H3(K)4, respectively, while further extensions correspond to H3(K)n for n = 5, 6, . . .. It remains to show that the bilinear form associated to the graded involution really is given by (x, y) = tr (x y), also in the case of 3 × 3 matrices. However, one can show that the triple product (7) indeed satisfies the definition of a generalized Jordan triple system when x, y, z are elements inH3(K) and (x, y) = tr (x y).
An important and interesting difference between the H2(K)n and H3(K)n cases is that the Lie algebra associated to H2(K)n is 3-graded for n = 1 and then 5-graded for all n ¸ 2, while the Lie algebra associated to H3(K)n is 3-graded for n = 1 but 7-graded for n = 2, and for n = 3, 4, 5, . . ., we get infinitely many subspaces in the grading, since these Lie algebras are infinite-dimensional. In the affine case, we only get the corresponding current algebra directly in this construction, which means that the central element and the derivation must be added by hand. It would be interesting to find an interpretation of these elements in the Jordan algebra approach. Finally, concerning the hyperbolic case and further extensions, we hope that our new construction can give more information about these indefinite Kac-Moody algebras, which, in spite of a great interest from both mathematicians and physicists, are not yet fully understood.
The author would like to thank the organizers of the AGMF workshop for the opportunity to present a talk at this nice conference.
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