##### Department of Mathematics,

University of California San Diego

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### Algebra Seminar

## Iryna Kashuba

#### University of San Paulo, Brasil

## Jordan Algebras and Their One-sided Representations

##### Abstract:

\indent This talk is a survey on the joint results with S.Ovsienko, V. Serganova and I. Shestakov. It is devoted to the problem of classification of indecomposable Jordan bimodules over finite dimensional Jordan algebras when squared radical is zero. \indent Recall, that for a Jordan algebra $J$ the category $J$-bimod of $k$-finite dimensional $J$-bimodules is equivalent to the category $U$-mod of (left) finitely dimensional modules over an associative algebra $U = U(J)$, which is called the universal multiplication envelope of $J$. If $J$ has finite dimension the algebra $U$ is finite dimensional as well. In particular, in accordance with the representation type of the algebra $U$ one can define Jordan algebras of the finite, tame and wild representation types. \indent From the other side to each Jordan algebra corresponds a Lie algera $TKK(J)$. Moreover there is a correspondence between the finite dimensional Jordan modules over $J$ and finite dimensional Lie modules over $TKK(J)$. \indent This allows us to apply to the category $J$-bimod all the machinery developed in the representation theory of finite dimensional algebras as well as the representation theory of Lie algebras.

### January 10, 2011

### 2:00 PM

### AP&M 7321

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