Quantized Enveloping Algebra Research Articles

Right Coideal Subalgebras of the Borel Part of a Quantized Enveloping Algebra

For the Borel part of a quantized enveloping algebra, we classify all right coideal subalgebras for which the intersection with the coradical is a Hopf algebra. The result is expressed in terms of characters of the subalgebras U+[w] of the quantized enveloping algebra, introduced by de Concini, Kac, and Procesi for any Weyl group element w. We explicitly determine all characters of U+[w] building on recent work by Yakimov on prime ideals of U+[w] which are invariant under a torus action.

Representations of the Twisted Quantized Enveloping Algebra of Type Cn

Representations of the Twisted Quantized Enveloping Algebra of Type Cn

An Algorithm to Compute the Canonical Basis of an Irreducible Module Over a Quantized Enveloping Algebra

AbstractThe paper describes an algorithm to compute the canonical basis of an irreducible module over a quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. The algorithm works for any module that is constructed as a submodule of a tensor product of modules with known canonical bases.

The Lusztig Cones of a Quantized Enveloping Algebra of Type A

We show that for each reduced expression for the longest word in the Weyl group of type An, the corresponding cone arising in Lusztig's description of the canonical basis in terms of tight monomials is simplicial, and construct explicit spanning vectors.

Regions of Linearity, Lusztig Cones, and Canonical Basis Elements for the Quantized Enveloping Algebra of Type A4

Let Uq be the quantum group associated to a Lie algebra g of rank n. The negative part U− of U has a canonical basis B with favourable properties (see M. Kashiwara (1991, Duke Math. J.63, 465–516) and G. Lusztig (1993. “Introduction to Quantum Groups,” Sect. 14.4.6, Birkhäuser, Boston)). The approaches of Lusztig and Kashiwara lead to a set of alternative parametrizations of the canonical basis, one for each reduced expression for the longest word in the Weyl group of g. We show that if g is of type A4 there are close relationships between the Lusztig cones, canonical basis elements, and the regions of linearity of reparametrization functions arising from the above parametrizations. A graph can be defined on the set of simplicial regions of linearity with respect to adjacency, and we further show that this graph is isomorphic to the graph with vertices given by the reduced expressions of the longest word of the Weyl group modulo commutation and edges given by long braid relations.

The Coradical Filtration for Quantized Enveloping Algebras

We determine the coradical filtration on a quantized enveloping algebra t/9(g) = U defined over Q(q). As applications, we determine the biideals of U and describe the group of Hopf algebra automorphisms of U.

A Completion of the Quantized Enveloping Algebra of a Kac–Moody Algebra

A Completion of the Quantized Enveloping Algebra of a Kac–Moody Algebra

Star Products and Quantum Algebras

I show explicitly that the star product on atriangular Poisson Lie group leads to a quantum algebrastructure (triangular Hopf algebra) on the quantizedenveloping algebra of the Lie algebra of the Lie group, and that equivalent star-productsgenerate isomorphic quantum algebras.

On the Center of Quantized Enveloping Algebras

LetUbe a quasitriangular Hopf algebra. One may use theR-matrix ofUin order to construct scalar invariants of knots. Analogously, Reshetikhin wrote down tangle invariants which take their values in the center ofU. Reshetikhin's expressions thus define central elements inU. We prove here an identity characterizing some of these elements, whenUis a quantized enveloping algebra. As an application, we give a proof for a statement of Faddeev, Reshetikhin, and Takhtadzhyan concerning the center of a quantized enveloping algebra.

The Coradical Filtration for Quantized Enveloping Algebras

We determine the coradical filtration on a quantized enveloping algebra Uq(g) = U defined over Q(q). As applications, we determine the biideals of U and describe the group of Hopf algebra automorphisms of U.

Root Vectors for Hecke Algebras and Quantized Enveloping Algebras

Root Vectors for Hecke Algebras and Quantized Enveloping Algebras