28 research outputs found
Dynamical Gelfand-Zetlin algebra and equivariant cohomology of Grassmannians
We consider the rational dynamical quantum group and introduce an
-module structure on \oplus_{k=0}^n
H^*_{GL_n\times\C^\times}(T^*Gr(k,n))', where
H^*_{GL_n\times\C^\times}(T^*Gr(k,n))' is the equivariant cohomology algebra
H^*_{GL_n\times\C^\times}(T^*Gr(k,n)) of the cotangent bundle of the
Grassmannian \Gr(k,n) with coefficients extended by a suitable ring of
rational functions in an additional variable . We consider the
dynamical Gelfand-Zetlin algebra which is a commutative algebra of difference
operators in . We show that the action of the Gelfand-Zetlin algebra
on H^*_{GL_n\times\C^\times}(T^*Gr(k,n))' is the natural action of the
algebra H^*_{GL_n\times\C^\times}(T^*Gr(k,n))\otimes \C[\delta^{\pm1}] on
H^*_{GL_n\times\C^\times}(T^*Gr(k,n))', where is the shift operator.
The -module structure on \oplus_{k=0}^n
H^*_{GL_n\times\C^\times}(T^*Gr(k,n))' is introduced with the help of
dynamical stable envelope maps which are dynamical analogs of the stable
envelope maps introduced by Maulik and Okounkov. The dynamical stable envelope
maps are defined in terms of the rational dynamical weight functions introduced
in [FTV] to construct q-hypergeometric solutions of rational qKZB equations.
The cohomology classes in H^*_{GL_n\times\C^\times}(T^*Gr(k,n))' induced by
the weight functions are dynamical variants of Chern-Schwartz-MacPherson
classes of Schubert cells.Comment: Latex, 27 pages, v2, v3, and v4: misprints are correcte
Combinatorics of rational functions and Poincare-Birkhoff-Witt expansions of the canonical U(n-)-valued differential form
We study the canonical U(n-)-valued differential form, whose projections to
different Kac-Moody algebras are key ingredients of the hypergeometric integral
solutions of KZ-type differential equations and Bethe ansatz constructions. We
explicitly determine the coefficients of the projections in the simple Lie
albegras A_r, B_r, C_r, D_r in a conviniently chosen Poincare-Birkhoff-Witt
basis. As a byproduct we obtain results on the combinatorics of rational
functions, namely non-trivial identities are proved between certain rational
functions with partial symmetries.Comment: More typos correcte