Several results about the multiplicities of indecomposable injectives in the minimal injective resolution of a ring exist in the literature. Mostly these apply to universal enveloping algebras of finite dimensional solvable Lie algebras, and to Gorenstein noetherian PI local rings. We unify these results and extend them to the much wider class of rings with Auslander dualizing complexes.

We compute the Hochschild and Gerstenhaber-Schack cohomological dimensions of the universal cosovereign Hopf algebras, when the matrix of parameters is a generic asymmetry. Our main tools are considerations on the cohomologies of free product of Hopf algebras, and on the invariance of the cohomological dimensions under graded twisting by a finite abelian group.

We prove a Schur-Weyl duality between the quantum enveloping algebra of $\mathfrak{gl}_m$ and certain quotient algebras Ariki-Koike algebras, which we give explicitly. The involves several algebraically independent parameters is realized through tensor product parabolic universal Verma module power natural representation $\mathfrak{gl}_m$. also new presentation by generators relations generaliz...

Let X be a finite, n-dimensional, r-connected CW complex. We prove the following theorem: If p ≥ n/r is an odd prime, then the loop space homology Bockstein spectral sequence modulo p is a spectral sequence of universal enveloping algebras over differential graded Lie algebras.

This research was motivated by universal algebraic geometry. One of the central questions of universal algebraic geometry is: when two algebras have the same algebraic geometry? For answer of this question (see [8],[10]) we must consider the variety Θ, to which our algebras belongs, the category Θ of all finitely generated free algebras of Θ and research how the group AutΘ of all the automorphi...

We generalize to universal algebra concepts originating from λ-calculus and programming to prove a new result on the lattice λT of λ-theories, and a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation Theorem. In this paper we introduce the class of Church algebras (which includes all Boolean algebras, combinatory algebras, rings with unit a...

This paper is about Toroidal Lie algebras which generalize the notion of an Affine Lie algebra. We study Verma type modules for these Toroidal algebras and prove an irreducibility criterion when the number of variables is two. We use the fact that the universal enveloping algebra is an Ore domain to obtain facts about the Verma type modules. Moreover, we are able to characterize the Affine Kac-...

Independence algebras were introduced in the early 1990s by specialists in semigroup theory, as a tool to explain similarities between the transformation monoid on a set and the endomorphism monoid of a vector space. It turned out that these algebras had already been defined and studied in the 1960s, under the name of v∗-algebras, by specialists in universal algebra (and statistics). Our goal i...

We generalize the notion of a Rota-Baxter operator on groups and weight 1 Lie algebras define study cocommutative Hopf algebra $H$. If $H=F[G]$ is group $G$ or $H=U(\mathfrak{g})$ universal enveloping $\mathfrak{g}$, then we prove that operators $H$ are in one to correspondence with corresponding algebras.

We propose a notion of a quantum universal enveloping algebra for any Lie algebra defined by generators and relations which is based on the quantum Lie operation concept. This enveloping algebra has a PBW basis that admits a monomial crystallization by means of the Kashiwara idea. We describe all skew primitive elements of the quantum universal enveloping algebras for the classical nilpotent al...