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            2020

We show that for some finite group block algebras, with nontrivial defect groups, the first Hochschild cohomology is nontrivial. Along the way we obtain methods to investigate the nontriviality of the first Hochschild cohomology of some twisted group algebras.

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              2021​

Motivated by the reduction techniques involving character triples for the local-global conjectures, we show that a blockwise relation between module triples is a consequence of a derived equivalence with additional properties. Moreover, we show that this relation is compatible with wreath products.

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BD algebras (Beilinson–Drinfeld algebras) are algebraic structures which are defined similarly to BValgebras (Batalin–Vilkovisky algebras). The equation defining the BD operator has the same structure as the equation for BV algebras, but the BD operator is increasing with degree +1.We obtain methods of constructing BD algebras in the context of group cohomology.​

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Starting with group graded Morita equivalences, we obtain Morita equivalences for tensor products and wreath products.

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We show that there is an action of the symmetric group on the Hochschild cochain complex of a twisted group algebra with

coefficients in a bimodule. This allows us to define the symmetric Hochschild cohomology of twisted group algebras, similarly

to Staic's construction of symmetric group cohomology.We give explicit embeddings and connecting homomorphisms between

the symmetric cohomology spaces and symmetric Hochschild cohomology of twisted group algebras.

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  • T. Coconet, On a property of the generalized Brauer pairs, Stud. Univ. Babes-Bolyai Math 66 (2021), No. 4, 605-611 .

In this paper we give a generalization of a result of Puig and Zhou to the context of group graded algebras.We use this generalization for an alternative approach of the proof of a result involving group graded basic Morita equivalences.

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             2022

  • I.-I. Simion, C.-C. Todea, Hochschild cohomology of some  finite category algebras as simplicial cohomology, Mathematical Notes, acceptat, (2022).

By a result of Gerstenhaber and Schack, the simplicial cohomology ring H*(C; k) of a poset C is isomorphic to the Hochschild cohomology ring HH*(kC) of the category algebra kC, where the poset is viewed as a category and k is a field. Extending results of Mishchenko, under certain assumptions on a category C, we construct a category D and a graded k-linear isomorphism  between HH*(kC)  and H*(D; k). Interpreting the degree one cohomology, we also show how the k-space of derivations on kC, graded by some semigroup, corresponds to the k-space of characters on D.

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  • I. I. Simion, Covering numbers of unipotent conjugacy classes in simple algebraic groups, Transformation Groups, acceptat (2022).

For simple algebraic groups de ned over algebraically closed  fields of good characteristic, we give upper bounds on the covering numbers of unipotent conjugacy classes in terms of their (co)ranks and in terms of their dimensions.

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  • T. Coconet, C.-C. Todea, Isomorphisms of fusion subcategories on permutation algebras, in evaluare.

We prove two isomorphisms of some subcategories of fusion systems defined on N-interior p-permutation G-algebras by extending the main results of some paper of Harris.

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  • T. Coconet, The extended Brauer quotient of a complex of group graded modules, in evaluare.

We extend the main result of Section 3 of a paper of Puig and Zhou to the case of group graded D-interior algebras. As an application,

we use this result to obtain a correspondence between pairs of non-contractile points.

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  • T. Coconet , C.-C. Todea, Stable unital basis, hyperfocal subalgebras and basic Morita equivalences, in evaluare.

We investigate Conjecture 1.6 introduced by Barker and Gelvin in a recent paper, which says that any source algebra of a p-block (p is a prime) of finite group has the unit group containing a basis stabilized by left and right action of the defect group. We will reduce this conjecture to a similar statement about basis of the hyperfocal subalgebras in the source algebra. We will also show that such unital basis of source algebras of two p-blocks, stabilized by left and right action of the defect group, are transported trough basic Morita equivalences.

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  • M. W. Liebeck, I. I. Simion, Covering numbers for simple algebraic groups, in evaluare.

Let G be a simple algebraic group over an algebraically closed  field, and let C be a noncentral conjugacy class of G. The covering number cn(G;C) is defined to be the minimal k such that G  is C to the power of k. We prove that cn(G;C)   is bounded in terms of the dimension of C, dimension of G and an explicit constant c ( with is at most 120). Some consequences on the width and generation of simple algebraic groups are given.

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  • I. I. Simion, Products of conjugacy classes in simple algebraic groups in terms of diagrams, in evaluare.

For a simple algebraic group G de ned over an algebraically closed field we study products of normal subsets in terms of certain marked diagrams.

In particular, we formulate suficient conditions in terms of marked diagrams, for a product of normal subsets in G to contain regular semisimple elements.

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