Centralizers in associative algebras

This thesis is divided into two parts. The subject of the first part is the structure of centralizers in associative algebras. We prove that over an algebraically closed field of characteristic zero, the centralizer of a nonconstant element in the second Weyl algebra has Gelfand-Kirillov (GK for short) dimension one, two or three. Those centralizers of GK dimension one or two are commutative and those of GK dimension three contain a finitely generated subalgebra which does not satisfy a polynomial identity. We show that for each $n \in \{1,2,3\}$ there exists a centralizer of GK dimension $n$. We also give explicit forms of centralizers for some elements of the second Weyl algebra and a connection between the problem of finite generation of centralizers in the second Weyl algebra and Dixmier's Fourth Problem. Some algebras such as the first Weyl algebra, quantum planes and finitely generated graded algebras of GK dimension two can be viewed as subalgebras of some skew Laurent polynomial algebra over a field. We prove that if $K$ is a field, $\sigma \in {\rm{Aut}}(K)$ and the fixed field of $\sigma$ is algebraically closed, then the centralizer of a nonconstant element of a subalgebra of $K[x,x^{-1}; \sigma]$ is commutative and a free module of finite rank over some polynomial algebra in one variable. In the last chapter, which is the second part of this thesis, we first prove a new version of Shirshov's theorem. We then use this theorem to prove an analogue of Kaplansky's theorem, i.e. if $D$ is a central division $k$-algebra which is left algebraic of bounded degree $d$ over some subfield, which is not necessarily central, then $[D:k] \leq d^2$.