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Register a new userGraded Lie structure on cohomology of some exact monoidal categories
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For some exact monoidal categories, we describe explicitly a connection between topological and algebraic definitions of the Lie bracket on the extension algebra of the unit object. The topological definition, due to Schwede and Hermann, involves loops in extension categories, and the algebraic definition involves homotopy liftings as introduced by the first author. As a consequence of our description, we prove that the topological definition indeed yields a Gerstenhaber algebra structure in the monoidal category setting, answering a question of Hermann. For use in proofs, we generalize $A_{infty}$-coderivation and homotopy lifting techniques from bimodule categories to some exact monoidal categories.
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Let $ 0rightarrow mathfrak{a} rightarrow mathfrak{e} rightarrow mathfrak{g} rightarrow 0$ be an abelian extension of the Lie superalgebra $mathfrak{g}$. In this article we consider the problems of extending endomorphisms of $mathfrak{a}$ and lifting endomorphisms of $mathfrak{g}$ to certain endomorphisms of $mathfrak{e}$. We connect these problems to the cohomology of $mathfrak{g}$ with coefficients in $mathfrak{a}$ through construction of two exact sequences, which is our main result, involving various endomorphism groups and the second cohomology. The first exact sequence is obtained using the Hochschild-Serre spectral sequence corresponding to the above extension while to prove the second we rather take a direct approach. As an application of our results we obtain descriptions of certain automorphism groups of semidirect product Lie superalgebras.
The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkovs homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.
We determine the Gerstenhaber structure on the Hochschild cohomology ring of a class of self-injective special biserial algebras. Each of these algebras is presented as a quotient of the path algebra of a certain quiver. In degree one, we show that the cohomology is isomorphic, as a Lie algebra, to a direct sum of copies of a subquotient of the Virasoro algebra. These copies share Virasoro degree 0 and commute otherwise. Finally, we describe the cohomology in degree $n$ as a module over this Lie algebra by providing its decomposition as a direct sum of indecomposable modules.
Suppose the ground field $mathbb{F}$ is an algebraically closed field of characteristic different from 2, 3. We determine the Betti numbers and make a decomposition of the associative superalgebra of the cohomology for the model filiform Lie superalgebra. We also describe the associative superalgebra structures of the (divided power) cohomology for some low-dimensional filiform Lie superalgebras.
We use a representation of a graded twisted tensor product of $K[x]$ with $K[y]$ in $L(K^{Bbb{N}_0})$ in order to obtain a nearly complete classification of these graded twisted tensor products via infinite matrices. There is one particular example and three main cases: quadratic algebras classified by Conner and Goetz, a family called $A(n,d,a)$ with the $n+1$-extension property for $nge 2$, and a third case, not fully classified, which contains a family $B(a,L)$ parameterized by quasi-balanced sequences.
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