We introduce the algebras satisfying the $(ℬ,n)$ condition. If $\Lambda$, $\Gamma$ are algebras satisfying the $(ℬ,n)$, $(ℰ,m)$ condition, respectively, we give a construction of $(m+n)$-almost split sequences in some subcategories ${(ℬ\otimes ℰ)}^{(i_0,j_0)}$ of $mod(\Lambda \otimes \Gamma )$ by tensor products and mapping cones. Moreover, we prove that the tensor product algebra $\Lambda \otimes \Gamma$ satisfies the $({(ℬ\otimes ℰ)}^{(i_0,j_0)},n+m)$ condition for some integers $i_0$, $j_0$; this construction unifies and extends the work of A. Pasquali (2017), (2019).

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.

The semigroup $S=S\left(M₂{(}_q)\right)$ of subspaces of the algebra $M₂{(}_q)$ of 2 × 2 matrices over a finite field ${}_q$ is studied. The ideal structure of S, the regular -classes of S and the structure of the complex semigroup algebra ℂ[S] are described.

A longstanding open problem in the theory of von Neumann regular rings is the question of whether every directly finite simple regular ring must be unit-regular. Recent work on this problem has been done by P. Menal, K. C. O'Meara, and the authors. To clarify some aspects of these new developments, we introduce and study the notion of almost isomorphism between finitely generated projective modules over a simple regular ring.

We compute a complete set of nonisomorphic minimal Auslander generators for the exterior algebra in two variables.