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Journal of Generalized Lie Theory and Applications | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Journal of Generalized Lie Theory and Applications

Efim ZELMANOV*

Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA

*Corresponding Author:
Efim ZELMANOV
Department of Mathematics
University of California San Diego 9500 Gilman Drive
La Jolla, CA 92093-0112, USA
E-mail: [email protected]

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Abstract

Leo Landau once said that all physicists can be divided into physicists-composers and physicistsperformers. Issai Kantor is, in my opinion, a mathematician-composer. Here are two his compositions on the theme of the Jordan algebras.

Leo Landau once said that all physicists can be divided into physicists-composers and physicistsperformers. Issai Kantor is, in my opinion, a mathematician-composer. Here are two his compositions on the theme of the Jordan algebras.

Jacques Tits in his study of models for exceptional Lie algebras made the following observation. Let L be a Lie algebra over a field image, which contains the Lie algebra

image

image

If the adjoint operator ad(h) acting on L has only the eigenvalues −2, 2, 2, then the 2-eigenspace L(2) with the operation image is a Jordan algebra.

Issai Kantor generalized it in the following way. Let L be a image graded Lie algebra

image

Then for an arbitrary element a from L(−1) the operation image defines a structure of a Jordan algebra on L(1). This lead to the notions of the Jordan triple systems and pairs (M. Koecher came to the same structure from the side of Hermitian symmetric spaces).

Moreover, Kantor and Koecher independently showed that every Jordan algebra arises in this way from a imagegraded Lie algebra via the Tits-Kantor-Koecher construction

Kantor’s observation about the operation [[x, a], y] played a crucial role in my proof of the Restricted Burnside problem.

Another brilliant example of Kantor’s insight is the discovery that every Poisson bracket leads to a Jordan superalgebra. Let R be an associative commutative algebra with a bilinear bracket image The bracket is said to be a Poisson bracket if it satisfies the product rule and image is a Lie algebra. Kantor noticed that the superalgebra image where

image

is a Jordan superalgebra, which is now called the Kantor double of R. The application of this construction to the supercommutative Grassmann algebras yielded the first examples of finite dimensional Jordan superalgebras with nonsemisimple even parts.

Kantor had fantastic intuition and a sense of what is important.

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