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The algebra generated by the down and up operators on a diierential 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 innnite-dimensional associative algebras called down-up algebras. We show that… (More)
We construct quantum deformations of enveloping algebras of Borcherds superalgebras, their Verma modules, and their irreducible highest weight modules.
We deene and characterize switching, 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 deened switching operations; however, each of their operations is somewhat diierent from the rest and each imposes a… (More)
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)
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)
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.
To subscribe to email notification of new AMS publications, please go to Contents: Graded Lie algebras; Simple Lie algebras and algebraic groups; The contragredient case; The noncontragredient case; Bibliography. This volume contains fourteen cutting-edge research articles on algebraic and analytic aspects of singularities of spaces and maps. By reading… (More)