Medical, Pharma, Engineering, Science, Technology and Business

**Jakob PALMKVIST**

Albert Einstein Institute, Am M¨uhlenberg 1, DE-14476 Golm, Germany E-mail: [email protected]

- *Corresponding Author:
- Jakob PALMKVIST

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

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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 [7], 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

(1)

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} + g_{0} + g_{1}, 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}+g_{0}+g_{1} is
the conformal algebra con *J*, and g_{0} 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, g_{0} = str *J*str ´*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 H_{2}(K) of hermitian
2 × 2 matrices over the division algebras K = R, C, H,O. We have

for d = 3, 4, 6, 10, respectively [8]. It is well known that conH_{2}(K) is the algebra that generates
conformal transformations in a d-dimensional Minkowski spacetime. Furthermore, str ´H_{2}(K) is
the Lorentz algebra and der H_{2}(K) its spatial part. With a d-dimensional Minkowski spacetime
we mean a vector space with a basis P_{μ} for _{μ} = 0, 1, . . . , *d − 1* and an inner product such that
(P*μ*, P*v*) = , where = diag(−1, 1, . . . , 1). We let a vector x have components x^{μ} 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:

(4)

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 *H*_{3}(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 [8] (see also references therein). In
particular, the exceptional Jordan algebra H_{3}(O) gives rise to exceptional Lie algebras:

(5)

(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 g_{k} is spanned by all root vectors e_{μ} or f_{−μ}, such that the root *μ* has the coefficient *−k *for α in the basis of simple roots
[4]. 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 [2].
This is usually the way e_{8} 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 (x_{1})^{1} +(x_{2})^{2} +· · ·+(x_{n})^{n}, where x_{1}, x_{2}, . . . are elements in h_{−1}. Furthermore, we define for
any graded involution on h a bilinear form on h_{−1} associated to by (e_{μ}, (f_{v})) = for root
vectors e_{μ} h_{−1} and f_{v} h_{1}. The answer is then given by the following theorem (for a proof,
see [7]).

**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 (e^{i}) = −f^{i} 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

(6)

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} [6]. 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 *H*_{2}(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 (*x**y*). Then H_{2}(K)^{n} will
be a Kantor triple system with the triple product

(7)

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 H_{3}(K), we find that
the exceptional algebras f_{4}, e_{6}, e_{7}, e_{8} are the Lie algebras associated to H_{3}(K)^{2}, with the triple
product (7) for K = R, C, H, O, respectively. Their affine and hyperbolic extensions are those
associated to H_{3}(K)^{3} and H_{3}(K)^{4}, respectively, while further extensions correspond to H_{3}(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 inH_{3}(K) and (x, y) = tr (x y).

An important and interesting difference between the H_{2}(K)^{n} and H_{3}(K)^{n} cases is that the
Lie algebra associated to H_{2}(K)^{n} is 3-graded for *n* = 1 and then 5-graded for all *n* ¸ 2, while
the Lie algebra associated to H_{3}(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.

- AsanoH,KaneyukiS (1988)On compact generalized Jordan triple systems of the second kind. Tokyo J Math 11: 105–118.
- G ̈unaydinM, KoepsellK, NicolaiH (2001) Conformal and quasiconformal realizations of exceptional Lie groups Comm Math Phys221: 57–76.
- KantorIL (1964) Classification of irreducible transitively differential groups. Soviet Math Dokl 5: 1404–1407.
- KantorIL (1972)Some generalizations of Jordan algebras. Trudy SemVektorTenzor Anal 16: 407–499.
- KoecherM,Imbedding of Jordan algebras into Lie algebras I. Amer J Math 89: 787–816.
- PalmkvistJ (2006)A realization of the Lie algebra associated to a Kantor triple system. J Math Phys47: 023505.
- PalmkvistJ, Generalized conformal realizations of Kac-Moody algebras. PreprintarXiv:0711.0441(hep-th).
- SudberyA (1984) Division algebras, (pseudo)orthogonal groups and spinors. J Phys A: Math Gen17: 939–955.
- TitsJ (1962)Uneclassed’alg`ebres de Lie en relation avec les alg`ebres de Jordan. Indag Math 24: 530–534.

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