On the derivations and automorphisms of the algebra k⟨ x, y⟩/(yx-xy-x^N)

2024/02/19 by Mariano Suárez-Álvarez, Suárez-Álvarez, Mariano
Mathematics · Physics and Astronomy · #16E40 #16S36 #16T05 #16W20 #16W25 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #K-Theory and Homology (math.KT) #Nonlinear Waves and Solitons #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2402.11962

openalex publication_date 2024/02/19 · openalex created_date 2024/02/21 · openalex updated_date 2026/07/28

Abstract

We consider the algebra A_N=k⟨ x, y⟩/(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 HH¹(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.

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