We give a treatment of the Brenner-Monsky example based on polynomial algebra and linear algebra. No prior knowledge of tight closure theory, Hilbert-Kunz theory, algebraic geometry or local cohomology is assumed.

The deformation theory of an algebra is controlled by the Gerstenhaber bracket, a Lie bracket on Hochschild cohomology. We develop techniques for evaluating brackets semidirect product algebras recording actions finite groups over fields positive characteristic. cohomology and these skew group can be complicated when characteristic underlying field divides order. show how to investigate using t...

In this article, we generalize Loday and Pirashvili's [11] computation of the Ext-category Leibniz bimodules for a simple Lie algebra to case (non Lie) algebra. Most arguments easily, while main new ingredient is Feldvoss-Wagemann's cohomology vanishing theorem semi-simple algebras.

A Rota-Baxter Leibniz algebra is a $(\mathfrak{g},[~,~]_{\mathfrak{g}})$ equipped with operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. We define representation and dual of algebras. Next, we cohomology theory also study the infinitesimal formal deformation algebras show that our cohomology. Moreover, an abelian extension equivalence classes such extensions are related to groups.

In one of his last papers, Boris Weisfeiler proved that if modular semisimple Lie algebra possesses a solvable maximal subalgebra which defines in it a long filtration, then associated graded algebra is isomorphic to one constructed from the Zassenhaus algebra tensored with the divided powers algebra. We completely determine such class of algebras, calculating in process low-dimensional cohomol...

We introduce the concept of para-Hopf algebroid and define their cyclic cohomology in the spirit of Connes-Moscovici cyclic cohomology for Hopf algebras. Para-Hopf algebroids are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structures, ...

We construct the Gerstenhaber bracket on Hochschild cohomology of a twisted tensor product of algebras, and, as examples, compute Gerstenhaber brackets for some quantum complete intersections arising in work of Buchweitz, Green, Madsen, and Solberg. We prove that a subalgebra of the Hochschild cohomology ring of a twisted tensor product, on which the twisting is trivial, is isomorphic, as a Ger...

We determine the possible cohomology algebra of orbit space of any free involution on a mod-2 cohomology lens space X using the Leray spectral sequence associated to the Borel fibration X →֒ XZ2 −→ BZ2 . As an application we show that if X is a mod-2 cohomology lens space of dimension 2m − 1 where 4 ∤ m, then there does not exist any Z2-equivariant map S n → X for n ≥ 2, where S is equipped with...