Cluster realization of $$\mathcal {U}_q(\mathfrak {g})$$ U q ( g ) and factorizations of the univers

Abstract

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 amalgamation of two basic quivers, where each of them is mutation equivalent to the cluster structure of the moduli space of framed G-local system on a disk with 3 marked points 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.

On the polynomiality conjecture of cluster realization of quantum groups

“…We mainly follow the notation in [FG09] and [GHK13] for the A and X cluster varieties. However, we slightly modify the definitions following [Nak21, Section 6], so that the mutation formulas for the A and X variables match that used by the first author in [Ip18], [Ip20] and also the original definitions in [FZ02]. (See Remark 2.8.)…”

“…In [Ip18], the first author constructed a quiver D(i 0 ) for any reduced word i 0 of w 0 , and showed that there is an embedding D q (g) into the quantum cluster algebra O q for this quiver. The image of F i are telescoping sums along certain F i -paths, and the image of the K ′ i generators are monomials along the F i -paths.…”

“…When S = ⊙ is the once punctured disk with two boundary special points s, t, the quiver for P G,⊙ is mutation equivalent to the quiver constructed in [Ip18]. There is an outer monodromy map µ out : P G,⊙ → H, such that the condition µ out = 1 corresponds to K s,i K t,i * = 1 for all i ∈ I.…”

“…Split real quantum groups were developed by Faddeev in [Fad95] and extended to its modular double in [Fad99], motivated from string theory and conformal field theory. Based on the idea by Schrader-Shapiro in the case of type A n [SS19], by forgetting the real structure the positive representations lead to an embedding of the quantum group U q (g) of general Lie type into a quotient of a certain quantum torus algebra O q (X D(i0) ) associated to a quiver D(i 0 ) [Ip18]. The quiver D(i 0 ) is mutation equivalent to the quiver for the moduli space of framed G-local systems P G,⊙ , where ⊙ denotes the once punctured disk with two special points on the boundary [GS22].…”

“…In Section 3, we give the definition of quantum groups. Then we provide a summary of the works by the first author and Man [Ip18], [Ip20], [IM22], as well as the works by Goncharov and Shen [GS22], [She22] on embeddings of U q (g) into cluster algebras. In Section 4, we use the sign coherence of c-vectors to give a sufficient condition for a regular element of a quantum cluster algebra O q (X ) to be universally polynomial.…”