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The algebra generated by the down and up operators on a differential partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinite-dimensional associative algebras called down-up algebras. We show that… (More)

We prove that the multiplicity of an arbitrary dominant weight for an integrable highest weight representation of the affine Kac-Moody algebra A (1) r is a polynomial in the rank r. In the process we show that the degree of this polynomial is less than or equal to the depth of the weight with respect to the highest weight. These results allow weight… (More)

- Georgia Benkart, Sarah Witherspoon, GEORGIA BENKART, SARAH WITHERSPOON
- 2004

We determine the finite-dimensional simple modules for two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl n and sl n , and give a complete reducibility result. These quantum groups have a natural n-dimensional module V. We prove an analogue of Schur-Weyl duality in this setting: the centralizer algebra of the… (More)

- Georgia Benkart, Sarah Witherspoon, GEORGIA BENKART, SARAH WITHERSPOON
- 2008

We investigate two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl n and sl n. We show that these quantum groups can be realized as Drinfel'd doubles of certain Hopf subalgebras with respect to Hopf pairings. Using the Hopf pairing, we construct a corresponding R-matrix. The quantum groups have a natural… (More)

- GEORGIA BENKART
- 2008

We consider the three-point loop algebra, L = sl2 ⊗ F[t, t −1 , (t − 1) −1 ], where F 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 A4. To… (More)

We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.

- Georgia Benkart, Thomas Gregory, Alexander Premet
- 2005

- Georgia Benkart
- 2008

This article contains an investigation of the equitable basis for the Lie algebra sl2. Denoting this basis by {x, y, z}, we have [x, y] = 2x + 2y, [y, z] = 2y + 2z, [z, x] = 2z + 2x. One focus of our study is the group of automorphisms G generated by exp(ad x *), exp(ad y *), exp(ad z *), where {x * , y * , z * } is the basis for sl2 dual to {x, y, z} with… (More)