##### Department of Mathematics,

University of California San Diego

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### Algebra Seminar

## Jason Bell

#### Simon Fraser University

## Free subalgebras of division algebras

##### Abstract:

In 1983, Makar-Limanov showed that the quotient division algebra of the complex Weyl algebra contains a copy of the free algebra on two generators. This results shows that, unlike in the commutative case, noncommutative localization can behave very pathologically. Stafford and Makar-Limanov conjectured that the following general dichotomy should hold: if a division ring is not finite-dimensional over its center (essentially commutative) then it must contain a free algebra on two generators. We show that for division algebras with uncountable centers a weaker dichotomy holds: such a division ring must either contain a free algebra on two generators or it must be in some sense algebraic over certain division subalgebras. We use this to show that if $A$ is a finitely generated complex domain of Gelfand-Kirillov dimension two then the conjectured dichotomy of Stafford and Makar-Limanov holds for the quotient division ring of $A$; that is, it is either finite-dimensional over its center or it contains a free algebra on two generators. This is joint work with Dan Rogalski.

Host: Dan Rogalski

### February 13, 2012

### 2:00 PM

### AP&M 7218

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