With the help of Galois coverings, we describe the tame tensor products $A{\otimes }_{K}B$ of basic, connected, nonsimple, finite-dimensional algebras A and B over an algebraically closed field K. In particular, the description of all tame group algebras AG of finite groups G over finite-dimensional algebras A is completed.

Assume that K is an algebraically closed field. Let D be a complete discrete valuation domain with a unique maximal ideal p and residue field D/p ≌ K. We also assume that D is an algebra over the field K . We study subamalgam D-suborders ${\Lambda }^{•}$ (1.2) of tiled D-orders Λ (1.1). A simple criterion for a tame lattice type subamalgam D-order ${\Lambda }^{•}$ to be of polynomial growth is given in Theorem 1.5. Tame lattice type subamalgam D-orders ${\Lambda }^{•}$ of non-polynomial growth are completely described in Theorem 6.2 and Corollary...

We describe all finite-dimensional algebras A over an algebraically closed field for which the algebra $T_2\left(A\right)$ of 2×2 upper triangular matrices over A is of tame representation type. Moreover, the algebras A for which $T_2\left(A\right)$ is of polynomial growth (respectively, domestic, of finite representation type) are also characterized.

A criterion for tame prinjective type for a class of posets with zero-relations is given in terms of the associated prinjective Tits quadratic form and a list of hypercritical posets. A consequence of this result is that if ${\Lambda }^{•}$ is a three-partite subamalgam of a tiled order then it is of tame lattice type if and only if the reduced Tits quadratic form $q_{{\Lambda }^{•}}$ associated with ${\Lambda }^{•}$ in [26] is weakly non-negative. The result generalizes a criterion for tameness of such orders given by Simson [28] and gives an...

We study associative ternary algebras and describe a general approach which allows us to construct various classes of ternary algebras. Applying this approach to a central bimodule with a covariant derivative we construct a ternary algebra whose ternary multiplication is closely related to the curvature of the covariant derivative. We also apply our approach to a bimodule over two associative (binary) algebras in order to construct a ternary algebra which we use to produce a large class of Lie algebras....

By abstracting the multiplication rule for ℤ₂-graded quantum stochastic integrals, we construct a ℤ₂-graded version of the Itô Hopf algebra, based on the space of tensors over a ℤ₂-graded associative algebra. Grouplike elements of the corresponding algebra of formal power series are characterised.

We prove that the first complex homology of the Johnson subgroup of the Torelli group $T_g$ is a non-trivial, unipotent $T_g$-module for all $g\ge 4$ and give an explicit presentation of it as a $Sy{m}_{.}{H}_1(T_g,C)$-module when $g\ge 6$. We do this by proving that, for a finitely generated group $G$ satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that the first homology of its Johnson kernel is a nilpotent module over the corresponding Laurent polynomial ring, isomorphic to the...

In this paper we study the Hopf adjoint action of group algebras and enveloping algebras. We are particularly concerned with determining when these representations are faithful. Delta methods allow us to reduce the problem to certain better behaved subalgebras. Nevertheless, the problem remains open in the finite group and finite-dimensional Lie algebra cases.

Quantum integrals associated to quantum Hom-Yetter-Drinfel’d modules are defined, and the affineness criterion for quantum Hom-Yetter-Drinfel’d modules is proved in the following form. Let (H,α) be a monoidal Hom-Hopf algebra, (A,β) an (H,α)-Hom-bicomodule algebra and $B=A^{coH}$. Under the assumption that there exists a total quantum integral γ: H → Hom(H,A) and the canonical map $\beta :A{\otimes }_{B}A\to A\otimes H$, $a{\otimes }_{B}b\mapsto S^{-1}\left(b_{\left[1\right]}\right)\alpha \left(b_{\left[0\right][-1]}\right)\otimes {\beta }^{-1}\left(a\right)\beta \left(b_{\left[0\right]\left[0\right]}\right)$, is surjective, we prove that the induction functor $A{\otimes }_B-:̃{{(}_k)}_B{\to }_A^H$ is an equivalence of categories.