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Some filtrations of the tensor product of a highest weight module and a lowest weight module over quantum group Uq(g)are constructed in[1]and one can use them to define a two-sided ideal of the modified quantized enveloping algebra. It is shown that the quotient algebra inherits a canonical basis from the modified quantized enveloping algebra and is dual to the quantum coordinate algebra defined by Kashiwara for a symmetrizable Kac–Moody algebra g.

We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.

Let $\mathbf{U}$ be the quantum group and $\mathbf{f}$ be the Lusztig's algebra associated with a symmetrizable generalized Cartan matrix. The algebra $\mathbf{f}$ can be viewed as the positive part of $\mathbf{U}$. Lusztig introduced some symmetries $T_i$ on $\mathbf{U}$ for all $i\in I$. Since $T_i(\mathbf{f})$ is not contained in $\mathbf{f}$, Lusztig considered two subalgebras ${_i\mathbf{f}}$ and ${^i\mathbf{f}}$ of $\mathbf{f}$ for any $i\in I$, where ${_i\mathbf{f}}=\{x\in\mathbf{f}\,\,|\,\,T_i(x)\in\mathbf{f}\}$ and ${^i\mathbf{f}}=\{x\in\mathbf{f}\,\,|\,\,T^{-1}_i(x)\in\mathbf{f}\}$. The restriction of $T_i$ on ${_i\mathbf{f}}$ is also denoted by $T_i:{_i\mathbf{f}}\rightarrow{^i\mathbf{f}}$. The geometric realization of $\mathbf{f}$ and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig's symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of ${_i\mathbf{f}}$, ${^i\mathbf{f}}$ and $T_i:{_i\mathbf{f}}\rightarrow{^i\mathbf{f}}$ by using the language 'quiver with automorphism' introduced by Lusztig.

For each simple Lie algebra g, we construct an algebra embedding of the quantum group Uq(g) into certain quantum torus algebra Dg via the positive representations of split real quantum group. The quivers corresponding to Dg is obtained from an amalgamation of two basic quivers, each of which is mutation equivalent to one describing the cluster structure of the moduli space of framed G-local system on a disk with 3 marked points on its boundary when G is of classical type. We derive a factorization of the universal R-matrix into quantum dilogarithms of cluster monomials, and show that conjugation by the R-matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in a twice punctured disk.