Journal of Algebra and Its Applications

We study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra Λ, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of Λ in characteristic zero. Criteria for the (semi-)simplicity, the solvability, the reductivity, the commutativity and the nilpotency are given.

We study isomorphisms between generalized Weyl algebras, giving a complete answer to the quantum case of this problem for R = k[h].

Necessary and sufficient conditions for an Ore extension S = R[x;σ,δ] to be a PI ring are given in the case σ is an injective endomorphism of a semiprime ring R satisfying the ACC on annihilators. Also, for an arbitrary endomorphism τ of R, a characterization of Ore extensions R[x;τ] which are PI rings is given, provided the coefficient ring R is noetherian.

Based on Kostant's cohomological interpretation of the Amitsur–Levitzki theorem, we prove a super version of this theorem for the Lie superalgebras 𝔬𝔰𝔭(1, 2n). We conjecture that no other classical Lie superalgebra can satisfy an Amitsur–Levitzki type super identity. We show several (super) identities for the standard super polynomials. Finally, a combinatorial conjecture on the standard skew supersymmetric polynomials is stated.

We show that the superanalogue of the Iwasawa decomposition exists for supergroup SL(n,m,ℂ). The first component of the decomposition is the compact real form SU(n,m), which was constructed following the idea of our article [3]. The second component is the super version of the AN group, that we define in this article.

We give a construction of algebras, called articulation, which is a specific gluing of two non-simple algebras. Then, we describe the Auslander–Reiten theory of an algebra obtained in this way. This allows us to characterize the articulated algebras which are laura, left (or right) glued, weakly shod, shod, quasi-tilted or tilted.

We classify the twisted tensor products of a finite set algebra with a two elements set algebra using colored quivers obtained through considerations analogous to Ore extensions. This provides also a classification of entwining structures between a finite set algebra and the grouplike coalgebra on two elements. The resulting 2-nilpotent algebras have particular features with respect to Hochschild (co)homology and cyclic homology.