Graphical Calculus for the Double Affine Q-Dependent Braid Group

Braid Group Action and Quantum Affine Algebras

We lift the lattice of translations in the extended affine Weyl group to a braid group action on the quantum affine algebra. This action fixes the Heisenberg subalgebra pointwise. Loop like generators are found for the algebra which satisfy the relations of Drinfel ′ d's new realization. Coproduct formulas are given and a PBW type basis is constructed. §0. Introduction. The purpose of this pape...

Categorical Affine Braid Group Action on Springer Resolutions

Double Affine Hecke Algebras for the Spin Symmetric Group

We introduce a new class (in two versions, Au and Bu) of rational double affine Hecke algebras (DaHa) associated to the spin symmetric group. We establish the basic properties of the algebras, such as PBW and Dunkl representation, and connections to Nazarov’s degenerate affine Hecke algebra and to a new degenerate affine Hecke algebra introduced here. We formulate a precise connection between t...

Quantum Affine Transformation Group and Covariant Differential Calculus

We discuss quantum deformation of the affine transformation group and its Lie algebra. It is shown that the quantum algebra has a noncocommutative Hopf algebra structure, simple realizations and quantum tensor operators. The deformation of the group is achieved by using the adjoint representation. The elements of quantum matrix form a Hopf algebra. Furthermore, we construct a differential calcu...

Affine Braid Groups and Hecke Algebras

These are notes for a class on double affine Hecke algebras and Macdonald polynomials given in the Fall of 2007 at the University of Minnesota. Here we introduce the (single and double) affine braid groups and Hecke algebras. We are following chapters 3,4, and 5 of Macdonald’s book [Mac]. There is also an (optional) aside giving applications to the T -equivariant K-theory of flag varieties G/B....