Becker-Gottlieb transfer for Hochschild cohomology

Let G be a finite group. Over any finite G-poset \({\mathcal{P}}\) we may define a transporter category as the corresponding Grothendieck construction. Transporter categories are generalizations of subgroups of G, and we shall demonstrate the finite generation of their cohomology. We record a generalized Frobenius reciprocity and use it to examine some important quotient categories of transporter categories, customarily called local categories.

Support varieties for transporter category algebras

Let be a regular ring, and let A, B be essentially finite type -algebras. For any functor F: D(ModA) × ⋅ × D(ModA) → D(ModB) between their derived categories, we define its twist F!: D(ModA) × ⋅ × D(ModA) → D(ModB) with respect to dualizing complexes, generalizing Grothendieck's construction of f!. We show that relations between functors are preserved between their twists, and deduce that various relations hold between derived Hochschild (co)-homology and the f! functor. We also deduce that the set of isomorphism classes of dualizing complexes over a ring (or a scheme) form a group with respect to derived Hochschild cohomology, and that the twisted inverse image functor is a group homomorphism.

Hochschild cohomology and Linckelmann cohomology for blocks of finite groups

A concrete description of Hochschild cohomology is the first step toward exploring associative deformations of algebras. In this dissertation, deformation theory, geometry, combinatorics, invariant theory, representation theory, and homological algebra merge in an investigation of Hochschild cohomology of skew group algebras arising from complex reflection groups. Given a linear action of a finite group on a finite dimensional vector space, the skew group algebra under consideration is the semi-direct product of the group with a polynomial ring on the vector space. Each representation of a group defines a different skew group algebra, which may have its own interesting deformations. In this work, we explicitly describe all graded Hecke algebras arising as deformations of the skew group algebra of any finite group acting by the regular representation. We then focus on rank two exceptional complex reflection groups acting by any irreducible representation. We consider in-depth the reflection representation and a nonfaithful rotation representation. Alongside our study of cohomology for the rotation representation, we develop techniques valid for arbitrary finite groups acting by a representation with a central kernel. Additionally, we consider combinatorial questions about reflection length and codimension orderings on complex reflection groups. We give algorithms using character theory to compute reflection length, atoms, and poset relations. Using a mixture of theory, explicit examples, and calculations using the software GAP, we show that Coxeter groups and the infinite family G(m,1,n) are the only irreducible complex reflection groups for which the reflection length and codimension orders coincide. We describe the atoms in the codimension order for the groups G(m,p,n). For arbitrary finite groups, we show that the codimension atoms are contained in the support of every generating set for cohomology, thus yielding information about the degrees of generators for cohomology.

The Hochschild cohomology ring of a global quotient orbifold

We consider deformations of quantum exterior algebras extended by finite groups. Among these deformations are a class of algebras which we call truncated quantum Drinfeld Hecke algebras in view of their relation to classical Drinfeld Hecke algebras. We give the necessary and sufficient conditions for which these algebras occur, using Bergman's Diamond Lemma. We compute the relevant Hochschild cohomology to make explicit the connection between Hochschild cohomology and truncated quantum Drinfeld Hecke algebras. To demonstrate the variance of the allowed algebras, we compute both classical type examples and demonstrate an example that does not arise as a factor algebra of a quantum Drinfeld Hecke algebra.

Let k be the field with p>0 elements, and let G be a finite group. By exhibiting an E-infinity-operad action on Hom(P,k) for a complete projective resolution P of the trivial kG-module k, we obtain power operations of Dyer-Lashof type on Tate cohomology H*(G; k). Our operations agree with the usual Steenrod operations on ordinary cohomology. We show that they are compatible (in a suitable sense) with products of groups, and (in certain cases) with the Evens norm map. These theorems provide tools for explicit computations of the operations for small groups G. We also show that the operations in negative degree are non-trivial. As an application, we prove that at the prime 2 these operations can be used to determine whether a Tate cohomology class is productive (in the sense of Carlson) or not.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Let $G$ be a finite group or a compact connected Lie group and let $BG$ be its classifying space. Let $\mathcal{L}BG:=map(S^1,BG)$ be the free loop space of $BG$ i.e. the space of continuous maps from the circle $S^1$ to $BG$. The purpose of this paper is to study the singular homology $H_*(\mathcal LBG)$ of this loop space. We prove that when taken with coefficients in a field the homology of $\mathcal LBG$ is a homological conformal field theory. As a byproduct of our main theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology $H^*(\mathcal LBG)$. We also prove an algebraic version of this result by showing that the Hochschild cohomology $HH^*(S_* (G),S_*(G))$ of the singular chains of $G$ is a Batalin-Vilkovisky algebra.

Nontriviality of the first Hochschild cohomology of some block algebras of finite groups

We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig’s graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to a Drinfeld-type deformation. We fit all these deformations into a general theory, connecting Poincare-Birkhoff-Witt deformations and Hochschild cohomology when working over fields of arbitrary characteristic. We make this connection by way of a double complex adapted from Guccione, Guccione, and Valqui, formed from the Koszul resolution of a polynomial ring and the bar resolution of a group algebra.

Algebra Structure on the Hochschild Cohomology of the Ring of Invariants of a Weyl Algebra under a Finite Group

We apply new techniques to Gerstenhaber brackets on the Hochschild cohomology of a skew group algebra formed from a polynomial ring and a finite group (in characteristic 0). We show that the Gerstenhaber brackets can always be expressed in terms of Schouten brackets. We obtain as consequences some conditions under which brackets are always 0, strengthening known results.

We study extensions of p-local finite groups where the kernel is a p-group. In particular, we construct examples of saturated fusion systems \({\mathcal{F}}\) which do not come from finite groups, but which have normal p-subgroups \({A \vartriangleleft \mathcal{F}}\) such that \({\mathcal{F}/A}\) is the fusion system of a finite group. One of the tools used to do this is the concept of a “transporter system”, which is modelled on the transporter category of a finite group, and is more general than a linking system.