Group algebras are Hopf algebras, and their Hopf structure plays crucial roles in representation theory and cohomology of groups. A Hopf algebra is an algebra A (say over a field k) that has a comultiplication (∆ : A → A ⊗k A) generalizing the diagonal map on group elements, an augmentation (ε : A → k) generalizing the augmentation on a group algebra, and an antipode (S : A → A) generalizing th...

In this paper we present the results of computation of cohomology for some Lie (super)algebras of Hamiltonian vector fields and related algebras. At present, the full cohomology rings for these algebras are not known even for the low dimensional vector fields. The partial “experimental” results may give some hints for solution of the whole problem. The computations have been carried out with th...

Group actions are ubiquitous in mathematics. To understand a mathematical object, it is often helpful to understand its symmetries as expressed by a group. For example, a group acts on a ring by automorphisms (preserving its structure). Analogously, a Lie algebra acts on a ring by derivations. Unifying these two types of actions are Hopf algebras acting on rings. A Hopf algebra is not only an a...

In this paper, first we introduce the notion of a nonabelian embedding tensor, which is generalization an tensor. Then Leibniz-Lie algebra, underlying algebraic structure and can also be viewed as Leibniz algebra. Next using derived bracket, construct differential graded Lie whose Maurer-Cartan elements are exactly tensors. Consequently, obtain algebra that governs deformations Finally, define ...

In this paper, first we give the notion of a representation relative Rota-Baxter Lie algebra and introduce cohomologies with coefficients in representation. Then classify abelian extensions algebras using second cohomology group, skeletal 2-algebras third group as applications. At last, established general framework representations algebras, which is consistent associative literature, Applicati...

Using methods from algebraic topology and group cohomology, I pursue Grothendieck’s question on equality of geometric and cohomological Brauer groups in the context of complex-analytic spaces. The main result is that equality holds under suitable assumptions on the fundamental group and the Pontrjagin dual of the second homotopy group. I apply this to Lie groups, Hopf manifolds, and complex-ana...