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## Introduction

### Algebras and Representation Theory (2000-12-01) 3: 301-302 , December 01, 2000

## Cyclotomic Birman–Wenzl–Murakami Algebras, II: Admissibility Relations and Freeness

### Algebras and Representation Theory (2011-02-01) 14: 1-39 , February 01, 2011

The cyclotomic Birman-Wenzl-Murakami algebras are quotients of the affine BMW algebras in which the affine generator satisfies a polynomial relation. We study admissibility conditions on the ground ring for these algebras, and show that the algebras defined over an admissible integral ground ring *S* are free *S*-modules and isomorphic to cyclotomic Kauffman tangle algebras. We also determine the representation theory in the generic semisimple case, obtain a recursive formula for the weights of the Markov trace, and give a sufficient condition for semisimplicity.

## Bounding Cohomology for Finite Groups and Frobenius Kernels

### Algebras and Representation Theory (2015-06-01) 18: 739-760 , June 01, 2015

Let *G* be a simple, simply connected algebraic group defined over an algebraically closed field *k* of positive characteristic *p*. Let *σ* :*G* → *G* be a strict endomorphism (i.e., the subgroup *G*(*σ*) of *σ*-fixed points is finite). Also, let *G*_{σ} be the scheme-theoretic kernel of *σ*, an infinitesimal subgroup of *G*. This paper shows that the dimension of the degree *m* cohomology group H*m*(*G*(*σ*),*L*) for any irreducible *k**G*(*σ*)-module *L* is bounded by a constant depending on the root system Φ of *G* and the integer *m*. These bounds are actually established for the degree *m* extension groups
$ Ext^{m}_{G(\sigma )}(L,L^{\prime })$
between irreducible *k**G*(*σ*)-modules
$L,L^{\prime }$
, with a similar result holding for *G*_{σ}. In these Ext*m* results, the bounds also depend on the highest weight associated to *L*, but are, nevertheless, independent of the characteristic *p*.

We also show that one can find bounds independent of the prime for the Cartan invariants of *G*(*σ*) and *G*_{σ}, and even for the lengths of the underlying PIMs.

## Monoidal Ring and Coring Structures Obtained from Wreaths and Cowreaths

### Algebras and Representation Theory (2014-08-01) 17: 1035-1082 , August 01, 2014

Let *A* be an algebra in a monoidal category
${\cal C}$
, and let *X* be an object in
${\cal C}$
. We study *A*-(co)ring structures on the left *A*-module *A* ⊗ *X*. These correspond to (co)algebra structures in
$EM({\cal C})(A)$
, the Eilenberg-Moore category associated to
${\cal C}$
and *A*. The ring structures are in bijective correspondence to wreaths in
${\cal C}$
, and their category of representations is the category of representations over the induced wreath product. The coring structures are in bijective correspondence to cowreaths in
${\cal C}$
, and their category of corepresentations is the category of generalized entwined modules. We present several examples coming from (co)actions of Hopf algebras and their generalizations. Various notions of smash products that have appeared in the literature appear as special cases of our construction.

## Filtered Multiplicative Bases of Restricted Enveloping Algebras

### Algebras and Representation Theory (2011-08-01) 14: 601-608 , August 01, 2011

We study the problem of the existence of filtered multiplicative bases of a restricted enveloping algebra *u*(*L*), where *L* is a finite-dimensional and *p*-nilpotent restricted Lie algebra over a field of positive characteristic *p*.

## Strong Lefschetz Elements of the Coinvariant Rings of Finite Coxeter Groups

### Algebras and Representation Theory (2011-08-01) 14: 625-638 , August 01, 2011

For the coinvariant rings of finite Coxeter groups of types other than H_{4}, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.

## Fusion Formulas and Fusion Procedure for the Yang-Baxter Equation

### Algebras and Representation Theory (2017-04-17): 1-36 , April 17, 2017

We use the fusion formulas of the symmetric group and of the Hecke algebra to construct solutions of the Yang–Baxter equation on irreducible representations of $\mathfrak {gl}_{N}$ , $\mathfrak {gl}_{N|M}$ , $U_{q}(\mathfrak {gl}_{N})$ and $U_{q}(\mathfrak {gl}_{N|M})$ . The solutions are obtained via the fusion procedure for the Yang–Baxter equation, which is reviewed in a general setting. Distinguished invariant subspaces on which the fused solutions act are also studied in the general setting, and expressed, in general, with the help of a fusion function. Only then, the general construction is specialised to the four situations mentioned above. In each of these four cases, we show how the distinguished invariant subspaces are identified as irreducible representations, using the relevant fusion formula combined with the relevant Schur–Weyl duality.

## The Nodal Cubic is a Quantum Homogeneous Space

### Algebras and Representation Theory (2017-06-01) 20: 655-658 , June 01, 2017

The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring *B* can be embedded as a right coideal subalgebra into a Hopf algebra *A* such that *A* is faithfully flat as a *B*-module. In the present article such a Hopf algebra *A* is constructed for the coordinate ring *B* of the nodal cubic, thus further motivating the question which affine varieties are quantum homogeneous spaces.

## Torsion Pairs and Rigid Objects in Tubes

### Algebras and Representation Theory (2014-04-01) 17: 565-591 , April 01, 2014

We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Prüfer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs, maximal rigid objects and the bijection between them geometrically. We also give a similar geometric description in the case of the linear orientation of a Dynkin quiver of type A.

## Realizability of Two-dimensional Linear Groups over Rings of Integers of Algebraic Number Fields

### Algebras and Representation Theory (2011-04-01) 14: 201-211 , April 01, 2011

Given the ring of integers *O*_{K} of an algebraic number field *K*, for which natural numbers *n* there exists a finite group *G* ⊂ *GL*(*n*, *O*_{K}) such that *O*_{K}*G*, the *O*_{K}-span of *G*, coincides with *M*(*n*, *O*_{K}), the ring of (*n* × *n*)-matrices over *O*_{K}? The answer is known if *n* is an odd prime. In this paper we study the case *n* = 2; in the cases when the answer is positive for *n* = 2, for *n* = 2*m* there is also a finite group *G* ⊂ *GL*(2*m*, *O*_{K}) such that *O*_{K}*G* = *M*(2*m*, *O*_{K}).