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On the derivations and automorphisms of the algebra $k\langle x, y\rangle/(yx-xy-x^N)$

Mariano Suárez-Álvarez

Abstract

We consider the algebra $A_N=k\langle x, y\rangle/(yx-xy-x^N)$, with $k$ a field of characteristic zero and $N$ a positive integer. Our main result is a complete description of the first Hochschild cohomology $\operatorname{HH}^1(A_N)$ of $A_N$ that consists both of explicit derivations of $A_N$ whose cohomology classes span it and a description of its Lie algebra structure. As we do this, we compute the automorphism group of the algebra, as well as certain characteristic subgroups thereof related to locally nilpotent derivations, classify the finite groups that act on $A_N$ and, finally, show that there are no inner-faithful actions of generalized Taft Hopf algebras on $A_N$. We establish this last result thanks to another calculation of Hochschild cohomology, now with twisted coefficients.

On the derivations and automorphisms of the algebra $k\langle x, y\rangle/(yx-xy-x^N)$

Abstract

We consider the algebra , with a field of characteristic zero and a positive integer. Our main result is a complete description of the first Hochschild cohomology of that consists both of explicit derivations of whose cohomology classes span it and a description of its Lie algebra structure. As we do this, we compute the automorphism group of the algebra, as well as certain characteristic subgroups thereof related to locally nilpotent derivations, classify the finite groups that act on and, finally, show that there are no inner-faithful actions of generalized Taft Hopf algebras on . We establish this last result thanks to another calculation of Hochschild cohomology, now with twisted coefficients.
Paper Structure (8 sections, 52 theorems, 177 equations, 2 tables)

This paper contains 8 sections, 52 theorems, 177 equations, 2 tables.

Table of Contents

  1. Preliminaries
  2. The automorphism group
  3. Hochschild cohomology
  4. A sequence of special elements in our algebra
  5. Explicit representatives for classes in HH1(A)
  6. The Lie algebra structure on HH1(A)
  7. Twisted cohomology
  8. Actions of Taft algebras

Key Result

Theorem 1

Suppose that $N\geq2$, let $q$ be a variable, and let $(c_i(q))_{i\geq0}$ be the sequence of polynomials in $\mathbb{Q}[q]$ such that

Theorems & Definitions (97)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 1.1
  • proof
  • Remark 1.2
  • Lemma 1.3
  • ...and 87 more