Georgia Benkart

Tableau Switching

We define and characterizeswitching, an operation that takes two tableaux sharing a common border and “moves them through each other” giving another such pair. Several authors, including James and Kerber, Remmel, Haiman, and Shimozono, have defined switching operations; however, each of their operations is somewhat different from the rest and each imposes a particular order on the switches that can occur. Our goal is to study switching in a general context, thereby showing that the previously defined operations are actually special instances of a single algorithm. The key observation is that switches can be performed in virtually any order without affecting the final outcome. Many known proofs concerning the jeu de taquin, Schur functions, tableaux, characters of representations, branching rules, and the Littlewood?Richardson rule use essentially the same mechanism. Switching provides a common framework for interpreting these proofs. We relate Schutzenbergers evacuation procedure to switching and in the process obtain further results concerning evacuation. We definereversal, an operation which extends evacuation to tableaux of arbitrary skew shape, and apply reversal and related mappings to give combinatorial proofs of various symmetries of Littlewood?Richardson coefficients.

Tensor product representations for orthosymplectic Lie superalgebras

We derive a general result about commuting actions on certain objects in braided rigid monoidal categories. This enables us to define an action of the Brauer algebra on the tensor space V⊗k which commutes with the action of the orthosymplectic Lie superalgebra spo(V) and the orthosymplectic Lie color algebra spo(V, β). We use the Brauer algebra action to compute maximal vectors in V⊗k and to decompose V⊗k into a direct sum of submodules Tλ. We compute the characters of the modules Tλ, give a combinatorial description of these characters in terms of tableaux, and model the decomposition of V⊗k into the submodules Tλ with a Robinson-Schensted-Knuth-type insertion scheme.

Whittaker modules for generalized Weyl algebras

We investigate Whittaker modules for generalized Weyl algebras, a class of associative algebras which includes the quantum plane, Weyl algebras, the universal enveloping algebra of sl_2 and of Heisenberg Lie algebras, Smiths generalizations of U(sl_2), various quantum analogues of these algebras, and many others. We show that the Whittaker modules V = Aw of the generalized Weyl algebra A = R(phi,t) are in bijection with the phi-stable left ideals of R. We determine the annihilator Ann_A(w) of the cyclic generator w of V. We also describe the annihilator ideal Ann_A(V) under certain assumptions that hold for most of the examples mentioned above. As one special case, we recover Kostants well-known results on Whittaker modules and their associated annihilators for U(sl_2).

Quantized enveloping algebras for Borcherds superalgebras

The rst author was supported in part by NSF Grant #DMS-9300523. The second author was supported in part by the Nondirected Research Fund, Korea Research Foundation, 1996. The third author was supported in part by a Faculty Research Grant from St. Lawrence University.

Flexible Lie-admissible algebras☆

Abstract This paper investigates finite-dimensional flexible Lie-admissible algebras A over fields of characteristic 0. Under these hypotheses the vector space A with the Lie product[ x , y ] = xy − yx is a Lie algebra, denoted by A − . The main result of this work gives a characterization of those flexible Lie-admissible algebras for which the solvable radical of A − is a direct summand of A − . Included in this class of algebras are all flexible Lie-admissible A for which A − is a reductive Lie algebra. Our technique is to view A as a module for a certain semisimple Lie algebra of derivations of A and to see what restrictions the module structure imposes on the multiplication of A . A subsequent investigation will show that this module approach can also be used to determine the flexible Lie-admissible algebras A for which the radical of A − is abelian.

The universal central extension of the three-point sl2 loop algebra

We consider the three-point loop algebra, L = sl 2 ⊗ K[t,t -1 , (t-1) - 1], where K denotes a field of characteristic 0 and t is an indeterminate. The universal central extension L of L was determined by Bremner. In this note, we give a presentation for L via generators and relations, which highlights a certain symmetry over the alternating group A 4 . To obtain our presentation of L, we use the realization of L as the tetrahedron Lie algebra.

Level 1 perfect crystals and path realizations of basic representations at q = 0

We present a uniform construction of level 1 perfect crystals

The equitable basis for \({\mathfrak{sl}_2}\)

\mathcal B

The Lie inner ideal structure of associative rings

for all affine Lie algebras. We also introduce the notion of a crystal algebra and give an explicit description of its multiplication. This allows us to determine the energy function on

Lie algebras with self-centralizing ad-nilpotent elements☆

\mathcal B \otimes \mathcal B