Schur algebras of Brauer algebras I

Schur algebras of Brauer algebras are defined as endomorphism algebras of certain direct sums of ‘permutation modules’ over Brauer algebras. Explicit combinatorial bases of these new Schur algebras are given; in particular, these Schur algebras are defined integrally. The new Schur algebras are related to the Brauer algebra by Schur–Weyl dualities on the above sums of permutation modules. Moreover, they are shown to be quasi-hereditary. Over fields of characteristic different from two or three, the new Schur algebras are quasi-hereditary 1-covers of Brauer algebras, and hence the unique ‘canonical’ Schur algebras of Brauer algebras.     

On the action of Ariki–Koike algebras on tensor space

The action of the Ariki–Koike algebra on tensor space is used to show that the Ariki–Koike algebra is isomorphic to a direct sum of matrix rings over tensor products over smaller Ariki–Koike algebras. As an application we get Schur–Weyl duality for the action of the Ariki–Koike algebra and the Levi subalgebra of the q-Schur algebra when separation conditions holds.     

Artin-Schelter regular algebras and categories

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin-Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin-Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin-Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the -category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.     

Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

The partition algebra \(\mathsf {P}_k(n)\) and the symmetric group \(\mathsf {S}_n\) are in Schur–Weyl duality on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\), so there is a surjection \(\mathsf {P}_k(n) \rightarrow \mathsf {Z}_k(n) := \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is an isomorphism when \(n \ge 2k\). We prove a dimension formula for the irreducible modules of the centralizer algebra \(\mathsf {Z}_k(n)\) in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible \(\mathsf {S}_n\)-modules in \(\mathsf {M}_n^{\otimes k}\). Our dimension expressions hold for any \(n \ge 1\) and \(k\ge 0\). Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on \(\mathsf {M}_n^{\otimes k}\) and the quasi-partition algebra corresponding to tensor powers of the reflection representation of \(\mathsf {S}_n\).     

Presenting generalized Schur algebras in types B, C, D

We give explicit presentations by generators and relations of certain generalized Schur algebras (associated with tensor powers of the natural representation) in types B, C, D. This extends previous results in type A obtained by two of the authors. The presentation is compatible with the Serre presentation of the corresponding universal enveloping algebra. In types C, D this gives a presentation of the corresponding classical Schur algebra (the image of the representation on a tensor power) since the classical Schur algebra coincides with the generalized Schur algebra in those types. This coincidence between the generalized and classical Schur algebra fails in type B, in general.     

Twistors for modules over algebras

We study the concept of module twistor for a module over an algebra. This concept provides a unifying framework for various deformed constructions of modules over algebras, such as module R-matrices, (n-factor iterated) twisted tensor products and L-R-twisted tensor products of algebras. Among the main results, we find the relations among these constructions. Furthermore, we study some properties of module twistors.     

Hochschild cohomology and quantum Drinfeld Hecke algebras

Quantum Drinfeld Hecke algebras are generalizations of Drinfeld Hecke algebras in which polynomial rings are replaced by quantum polynomial rings. We identify these algebras as deformations of skew group algebras, giving an explicit connection to Hochschild cohomology. We compute the relevant part of Hochschild cohomology for actions of many reflection groups, and we exploit computations from Naidu et al. (Proc Am Math Soc 139:1553–1567, 2011) for diagonal actions. By combining our work with recent results of Levandovskyy and Shepler (Can J Math 66:874–901, 2014) we produce examples of quantum Drinfeld Hecke algebras. These algebras generalize the braided Cherednik algebras of Bazlov and Berenstein (Selecta Math 14(3–4):325–372, 2009).     

Affine and degenerate affine BMW algebras: actions on tensor space

The affine and degenerate affine Birman–Murakami–Wenzl (BMW) algebras arise naturally in the context of Schur–Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients of the affine and degenerate affine BMW algebras. In this paper, we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce actions of the affine and degenerate affine BMW algebras. We then exploit the connection to quantum groups and Lie algebras to determine universal parameters for the affine and degenerate affine BMW algebras. Finally, we show that the universal parameters are central elements—the higher Casimir elements for orthogonal and symplectic enveloping algebras and quantum groups.     

Maximal modifications and Auslander–Reiten duality for non-isolated singularities

We first generalize classical Auslander–Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of \(\operatorname{Spec}R\) , in the main body of this paper we introduce a theory of modifying and maximal modifying modules. Under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three-dimensional Gorenstein rings are shown to be derived equivalent. We then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory. Our mutation works in arbitrary dimension, and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian.     

Bases, filtrations and module decompositions of free Lie algebras

We use the technique known as elimination to devise some new bases of the free Lie algebra which (like classical Hall bases) consist of Lie products of left normed basic Lie monomials. Our bases yield direct decompositions of the homogeneous components of the free Lie algebra with direct summands that are particularly easy to describe: they are tensor products of metabelian Lie powers. They also give rise to new filtrations and decompositions of free Lie algebras as modules for groups of graded algebra automorphisms. In particular, we obtain some new decompositions for free Lie algebras and free restricted Lie algebras over fields of positive characteristic.     

Tilting equivalences: From hereditary algebras to symmetric groups

We study Koszul duality for finite dimensional hereditary algebras, and various generalisations to trivial extension algebras, to Schur algebras, to doubles of Schur bialgebras, and to deformations of doubles of Schur bialgebras. We describe applications to the modular representation theory of symmetric groups.     

Three theorems on common splitting fields of central simple algebras

LetA ₁, …A _n be central simple algebras over a fieldF. Suppose that we possess information on the Schur indexes of some tensor products of (some tensor powers of) the algebras. What can be said (in general) about possible degrees of finite field extensions ofF splitting the algebras? In Part I, we prove a positive result of that kind. In Part II, we prove a negative result. In Part III, we develop a general approach.     

Whitney algebras and Grassmann��s regressive products

Geometric products on tensor powers ${\Lambda(V)^{\otimes m}}$ of an exterior algebra and on Whitney algebras (Crapo and Schmitt in J. Comb. Theory A 91:215?C263, 2000) provide a rigorous version of Grassmann??s regressive products of 1844 (Grassmann in Die Lineale Ausdehnungslehre, Verlag von Otto Wigand, Leipzig, 1844). We study geometric products and their relations with other classical operators on exterior algebras, such as the Hodge *?operators and the join and meet products in Cayley?CGrassmann algebras (Barnabei et?al. in J. Algebra 96:120?C160, 1985; Stewart in Nature 321:17, 1986). We establish encodings of tensor powers ${\Lambda(V)^{\otimes m}}$ and of Whitney algebras W ^m (M) in terms of letterplace algebras and of their geometric products in terms of divided powers of polarization operators. We use these encodings to provide simple proofs of the Crapo and Schmitt exchange relations in Whitney algebras and of two typical classes of identities in Cayley?CGrassmann algebras.     

Representation type of endomorphism algebras of exceptional sequences of typeA n

LetF be an algebraically closed field, be a quiver of typeA _n . In this paper we prove that the endomorphism algebras of exceptional sequences over are sums of finitely many tilted algebras of typeA _m wherem≤n by using perpendicular categories, and thus the endomorphism algebras of exceptional sequences of typeA _n are representation-finite. Supported by Chinese Postdoctorate Fund and Beijing Youth Fund     

Spectral-like duality for distributive Hilbert algebras with infimum

Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and \({\wedge}\)-semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters.     

Classification of Lie algebras of specific type in complexified Clifford algebras

We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These 16 Lie algebras are direct sums of subspaces of quaternion types. We obtain isomorphisms between these Lie algebras and classical matrix Lie algebras in the cases of arbitrary dimension and signature. We present 16 Lie groups: one Lie group for each Lie algebra associated with this Lie group. We study connection between these groups and spin groups.     

Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-Calabi-Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press]. In addition, we prove a general result about relative 3-Calabi-Yau duality over non-stable endomorphism rings. This strengthens and generalizes the Ext-group symmetries obtained in [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press] for simple modules. Finally, we generalize the results on relative Calabi-Yau duality from 2-Calabi-Yau to d-Calabi-Yau categories. We show how to produce many examples of d-cluster tilted algebras.