We found an explicit construction of a representation of the positive quantum group and its modular double by positive essentially self-adjoint operators. Generalizing Lusztig's parametrization, we found a Gauss type decomposition for the totally positive quantum group parametrized by the standard decomposition of the longest element . Under this parametrization, we found explicitly the relations between the standard quantum variables, the relations between the quantum cluster variables, and realizing them using non-compact generators of the q-tori by positive essentially self-adjoint operators. The modular double arises naturally from the transcendental relations, and an space in the von Neumann setting can also be defined.
We construct the positive principal series representations for Uq(gR) where g is of type Bn, Cn, F4 or G2, parametrized by Rn where n is the rank of g. We show that under the representations, the generators of the Langlands dual group Uq(LgR) are related to the generators of Uq(gR) by the transcendental relations. This gives a new and very simple analytic relation between the Langlands dual pair. We define the modified quantum group Uqq(gR)=Uq(gR) ⊗ Uq(LgR) of the modular double and show that the representations of both parts of the modular double commute with each other, and there is an embedding into the q-tori polynomials.
We study the root of unity degeneration of cluster algebras and quantum dilogarithm identities. We prove identities for the cyclic dilogarithm associated with a mutation sequence of a quiver, and as a consequence new identities for the noncompact quantum dilogarithm at |$b=1$|.Communicated by Michio Jimbo
We prove that certain parabolic Kazhdan-Lusztig polynomials calculate the graded decomposition matrices of v-Schur algebras given by the Jantzen ﬁltration of Weyl modules, conﬁrming a conjecture of Leclerc and Thibon.
We categorify a coideal subalgebra of the quantum group of S[2r+ 1 by introducing
a 2-category analogous to the one defined by Khovanov一Lauda-Rouquier， and show
that self-dual indecomposable l-morphisms categorify the canonical basis of this algebra.
This allows us to define a categorical action of this coideal algebra on the categories of
modules over cohomology rings of partial flag varieties and on the category ð of type