Zariski cancellation problem for noncommutative algebras

noncommutative algebras

In this lecture 1 I would like to address the following question: given an associative algebra A 0 , what are the possible ways to deform it? Consideration of this question for concrete algebras often leads to interesting mathematical discoveries. I will discuss several approaches to this question, and examples of applying them. 1. Deformation theory 1.1. Formal deformations. The most general a...

Geometric Models for Noncommutative Algebras

Noncommutative Disc Algebras for Semigroups

We study noncommutative disc algebras associated to the free product of discrete subsemigroups of R +. These algebras are associated to generalized Cuntz algebras, which are shown to be simple and purely innnite. The nonself-adjoint subalgebras determine the semigroup up to isomorphism. Moreover, we establish a dilation theorem for contractive representations of these semigroups which yields a ...

Some Noncommutative Matrix Algebras Arising in the Bispectral Problem

I revisit the so called “bispectral problem” introduced in a joint paper with Hans Duistermaat a long time ago, allowing now for the differential operators to have matrix coefficients and for the eigenfunctions, and one of the eigenvalues, to be matrix valued too. In the last example we go beyond this and allow both eigenvalues to be matrix valued.

Actions of Hopf algebras on noncommutative algebras

Often A is called H-module algebra. We refer reader to [11, 6] for the basic information concerning Hopf algebras and their actions on associative algebras. Definition 1.2 The invariants of H in A is the set AH of those a ∈ A, that ha = ε(h)a for each h ∈ H. Straightforward computations show, that AH is subalgebra of A. The notion of action of Hopf algebra on associative algebra generalize the ...