##### Department of Mathematics,

University of California San Diego

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### Algebra Colloquium

## Alexander Mikhalev

#### Moscow State University

## Lie algebras with one defining relation

##### Abstract:

In the talk we consider Lie algebras with one defining relation. Starting with an analog of Freiheitssatz (Shirshov's theorem), we give an example of a Lie algebra over a field of prime characteristic with cohomological dimension one which is not a free Lie algebra (this gives a counterexample to a hypothesis that the analog of Stallings-Swan theorem takes place for Lie algebras; the problem in the case of field of zero characteristic is still open, we formulate some related conjectures). For a finitely generated free Lie algebra L we construct an example of two elements $u$ and $v$ of $L$ such that the factor algebras $L/(u)$ and $L/(v)$ are isomorphic, where $(u)$ and $(v)$ are ideals of L generated by $u$ and $v$, respectively, but there is no automorphism $f$ of $L$ such that $f(u)=v$. We consider also the situation where a Lie algebra with one defining relation is a free Lie algebra.

Host: Efim Zelmanov

### October 16, 2006

### 3:00 PM

### AP&M 7218

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