The Two-Step Nilpotent Representations of the Extended Affine Hecke Algebra of Type A

We define a set of cell modules for the extended affine Hecke algebra of type A which are parametrised by SL_n()-conjugacy classes of pairs (s, N), where s SL_n() is semisimple and N is a nilpotent element of the Lie algebra which has at most two Jordan blocks and satisfies Ad(s)·N=q ² N. When q ²–1, each of these has irreducible head, and the irreducible representations of the affine Hecke algebra so obtained are precisely those which factor through its Temperley–Lieb quotient. When q ²=–1, the above remarks apply to a subset of the cell modules. Using our work on the cellular nature of those quotients, we are able to obtain complete information on the decomposition of the cell modules in all cases, even when q is a root of unity. They turn out to be multiplicity free, and the composition factors may be precisely described in terms of a partial order on the pairs (s, N). These results give explicit formulae for the dimensions of the irreducibles. Assuming our modules are identified with the standard modules earlier defined by Bernstein–Zelevinski, Kazhdan–Lusztig and others, our results may be interpreted as the determination of certain Kazhdan–Lusztig polynomials. [This has now been proved and will appear in a subsequent work of the authors.]The second author thanks the Australian Research Council and the Alexander von Humboldt Stiftung for support and the Universität Bielefeld for hospitality during the preparation of this work.     

Tiling bijections between paths and Brauer diagrams

There is a natural bijection between Dyck paths and basis diagrams of the Temperley–Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the two-dimensional integer lattice. We show that there is a natural bijection, extending the above tiling construction, between overhang paths and basis diagrams of the Brauer algebra.     

Representations of the odd affine Temperley-Lieb algebra

We define the odd affine Temperley–Lieb algebra (OATLA)to be the category defined by planar diagrams in an annuluswith an odd number of marked points on each boundary connectedin pairs by disjoint strings and a modulus . This algebra isthe odd part of the annularization of the Temperley–Liebplanar algebra. A positivity result is proved, which allowsus to completely characterize the Hilbert space representationsof OATLA when the parameter is of the form . The results finish the project of describing theirreducible Hilbert representations of the affine Temperley–Liebalgebra, which naturally arises in the study of subfactors.     

On algebras of the Temperley-Lieb type associated with algebras generated by generators with given spectrum

We introduce and study algebras of the Temperley—Lieb type associated with algebras generated by linearly connected generators with given spectrum. We study their representations and the sets of parameters for which representations of these algebras exist.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 634–641, May, 2004.     

Cyclotomic Extensions of Diagram Algebras

A uniform approach to cyclotomic extensions of diagram algebras is given, focussing on cellular structures. Cyclotomic Temperley–Lieb algebras, cyclotomic Brauer algebras and cyclotomic walled Brauer algebras are discussed as examples.     

On the radical of Brauer algebras

The radical of the Brauer algebra is known to be non-trivial when the parameter x is an integer subject to certain conditions (with respect to f). In these cases, we display a wide family of elements in the radical, which are explicitly described by means of the diagrams of the usual basis of . The proof is by direct approach for x  =  0, and via classical Invariant Theory in the other cases, exploiting then the well-known representation of Brauer algebras as centralizer algebras of orthogonal or symplectic groups acting on tensor powers of their standard representation. This also gives a great part of the radical of the generic indecomposable -modules. We conjecture that this part is indeed the whole radical in the case of modules, and it is the whole part in a suitable step of the standard filtration in the case of the algebra. As an application, we find some more precise results for the module of pointed chord diagrams, and for the Temperley–Lieb algebra—realised inside —acting on it.

“Ahi quanto a dir che sia è cosa dura lo radical dell’algebra di Brauer pur se’l pensier già muove a congettura” N. Barbecue, “Scholia”

Partially supported by the European RTN “LieGrits”, contract no. MRTN-CT-2003-505078, and by the Italian PRIN 2005 “Moduli e teorie di Lie”.     

On the Various Realizations of the Basic Representation of An−1(1) and the Combinatorics of Partitions

The infinite dimensional Lie algebra l _n = A _n–1 ⁽¹⁾ can be realized in several ways as an algebra of differential operators. The aim of this note is to show that the intertwining operators between the realizations of l _n corresponding to all partitions of n can be described very simply by using combinatorial constructions.     

Support Varieties for Selfinjective Algebras

Support varieties for any finite dimensional algebra over a field were introduced in (Proc. London Math. Soc. 88 (3) (2004) 705–732) using graded subalgebras of the Hochschild cohomology ring. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, the variety of periodic modules are lines and for symmetric algebras a generalization of Webbs theorem is true. As a corollary of a more general result we show that Webbs theorem generalizes to finite dimensional cocommutative Hopf algebras.Received November 2003Mathematics Subject Classifications (2000) Primary: 16E40, 16G10, 16P10, 16P20; Secondary: 16G70.     

Twisting Products in Algebras II

Let k be an arbitrary field, H a kbialgebra and A a kalgebra. In Comm.Algebra 23 (7) (1995), pp. 2719–2744, the first author defined, in the case that A is an (H,H*)bicomodule algebra, a new product in A called the twist of the original multiplication. We generalize these considerations and define the twist with respect to more general twisting data that need not come from a bicomodule algebra structure. This general setting enlarges the range of applications of the concept of twisting, in particular it is shown that most known examples of twisting, e.g. the twist by cocycles and the concept of biproduct as introduced by Radford and Majid appear as particular cases of our construction.     

Cellular Bases for Algebras with a Jones Basic Construction

We construct analogues for the Brauer, BMW, partition, and Jones–Temperley–Lieb algebras of the Murphy basis of the Hecke algebra of the symmetric group. The bases are cellular bases indexed by paths on branching diagrams, and compatible with restriction of cell modules. The Jucys–Murphy elements for each class of algebras act by triangular matrices on the Murphy basis.