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.

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.

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.

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Let $\mathbf{U}$ be the quantized enveloping algebra and $\dot{\mathbf{U}}$ its modified form. Lusztig gives some symmetries on $\mathbf{U}$ and $\dot{\mathbf{U}}$. Since the realization of $\mathbf{U}$ by the reduced Drinfeld double of the Ringel-Hall algebra, one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig's symmetries on $\mathbf{U}$ and their important properties, for instance, the braid relations. In this paper, we define a modified form $\dot{\mathcal{H}}$ of the Ringel-Hall algebra and realize the Lusztig's symmetries on $\dot{\mathbf{U}}$ by applying the BGP-reflection functors to $\dot{\mathcal{H}}$.

Nous montrons que le morphisme de Frobenius quantique construit par Lusztig dans le cadre des algèbres enveloppantes quantiques $U_{\mathcal{B}}$ spécialisées en une racine de l’unité admet un scindage multiplicatif (non unitaire). Nous utilisons pour ce faire une base de la partie torique de la petite algèbre quantique constituée d’idempotents orthogonaux deux à deux et de somme 1 et faisons de même dans le cas «modulaire» pour l’algèbre des distributions d’un groupe algébrique semi-simple.

In this article, we classify all standard invariants that can arise from a composed inclusion of an A3with an A4subfactor. More precisely, if N⊂P is an A3subfactor and P⊂M is an A4subfactor, then only four standard invariants can arise from the composed inclusion N⊂M. We answer a question posed by Bisch and Haagerup in 1994. The techniques of this paper also show that there are exactly four standard invariants for the composed inclusion of two A4subfactors.

We describe the generalized Casimir operators and their actions on the positive representations Pλ of the modular double of split real quantum groups Uqq (gR). We introduce the notion of virtual highest and lowest weights, and show that the central characters admit positive values for all parameters λ. We show that their image defines a semi-algebraic region bounded by real points of the discriminant variety independent of q, and we discuss explicit examples in the lower rank cases.

Let$p$be a prime integer, 1≤$s$≤$r$be integers and$F$be a field of characteristic different from$p$. We find upper and lower bounds for the essential$p$-dimension ed_{$p$}( $$ Al{{g}_{{{{p}^r},{{p}^s}}}} $$ ) of the class $$ Al{{g}_{{{{p}^r},{{p}^s}}}} $$ of central simple algebras of degree$p$^{$r$}and exponent dividing$p$^{$s$}. In particular, we show that ed($Alg$_{8,2})=ed_{2}($Alg$_{8,2})=8 and ed_{$p$}( $$ Al{{g}_{{{{p}^2},p}}} $$ )=$p$^{2}+$p$for$p$odd.

For a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$, Lusztig introduced the corresponding modified quantized enveloping algebra $\dot{\textbf{U}}$ and its canonical basis $\dot{\textbf{B}}$ in [13]. In this paper, in case $\mathfrak{g}$ is a symmetric Kac-Moody Lie algebra of finite or affine type, we define a set $\tilde{\mathcal{M}}$ which depends only on the root category $\mathcal{R}$ and prove that there is a bijection between $\tilde{\mathcal{M}}$ and $\dot{\textbf{B}}$, where $\mathcal{R}$ is the $T^2$-orbit category of the bounded derived category of corresponding Dynkin or tame quiver. Our method is based on a result of Lin, Xiao and Zhang in [10], which gives a PBW-type basis of $\textbf{U}^+$.