##### Department of Mathematics,

University of California San Diego

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### Algebra Seminar

## W.A. Moens

#### Univ. of Vienna and UCSD

## Faithful Lie Algebra Representations of Minimal Degree

##### Abstract:

In the seminar I would like to talk about two general problems in the theory of finite-dimensional Lie algebras: â€¢ Finding a faithful representation of a Lie algebra (of lowest possible degree). E.g.: Can the $(2n + 1)$-dimensional Heisenberg Lie algebra $h_n$ be em- bedded into $gl_{n+1}(C)$ Into $gl_{n+2}(C)$ â€¢ Finding properties of Lie algebras that admit a regular transformation. E.g.: Is a Lie algebra that admits a regular periodic derivation neces- sarily abelian? These two problems appear to be unrelated, but this is not the case. They follow naturally from some problems on the existence of affine structures on Lie algebras (or groups). I plan to briefly mention the historical background material, then formulate both problems more explicitly, illustrate which results are already known, and finally, how I hope to obtain some more (partial) answers. For the first problem, it will be interesting to find good bounds on the minimal degree of a faithful representation. For the second problem, we would like to find good bounds on the nilpotency or solvability class of the Lie algebra.

Host: Efim Zelmanov

### February 19, 2013

### 2:00 PM

### AP&M 7218

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