A Parameterization of the Canonical Bases of Affine Modified Quantized Enveloping Algebras

For a symmetrizable Kac-Moody Lie algebra g, Lusztig introduced the corresponding modified quantized enveloping algebra˙U and its canonical basis˙B given by Lusztig in 1992. In this paper, in the case that g is a symmetric Kac-Moody Lie algebra of finite or affine type, the authors define a set M which depends only on the root category R and prove that there is a bijection between M and ˙B, where R is the T^2-orbit category of the bounded derived category of the corresponding Dynkin or tame quiver. The method in this paper is based on a result of Lin, Xiao and Zhang in 2011, which gives a PBW-type basis of U^+.

The Extension of a Quantized Borcherds Superalgebra by a Hopf Algebra

A two-parameter quantum group is obtained from the usual enveloping algebra by adding two commutative grouplike elements. In this paper, we generalize this procession further by adding commutative grouplike elements b（ik）, c（ik）, g（ik）, h（ik）（i ∈I, k = 1,..., mi） of a Hopf algebra H to the quantized enveloping algebra Uq（G） of a Borcherds superalgebra G defined by a symmetrizable integral Borcherds–Cartan matrix A =（aij）i,j∈I. Therefore, we define an extended Hopf superalgebra HUq（G）. We also discuss the basis and the grouplike elements of HUqG.

Realization of Poisson enveloping algebra

For a Poisson algebra, the category of Poisson modules is equivalent to the module category of its Poisson enveloping algebra, where the Poisson enveloping algebra is an associative one. In this article, for a Poisson structure on a polynomial algebra S, we first construct a Poisson algebra R, then prove that the Poisson enveloping algebra of S is isomorphic to the specialization of the quantized universal enveloping algebra of R, and therefore, is a deformation quantization of R.

Quantum Weyl symmetric polynomials and the center of quantum group U_q(sl_4)

Suppose that q is not a root of unity, it is proved in this paper that the center of the quantum group Uq(sl4) is isomorphic to a quotient algebra of polynomial algebra with four variables and one relation.

Tight monomials for type B3

The global crystal basis or canonical basis plays an important role in the theory of the quantized enveloping algebras and their representations. The tight monomials are the simplest elements in the canonical basis. We discuss the tight monomials in quantized enveloping algebra of type B3.