Connected quantized Weyl algebras and quantum cluster algebras

Abstract

For an algebraically closed field K, we investigate a class of noncommutative K-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators {x 1 , . . . , x n } such that each pair satisfies a relation of the form x i x j = q ij x j x i + r ij , where q ij ∈ K * and r ij ∈ K, with, in some sense, sufficiently many pairs for which r ij = 0. For such an algebra it turns out that there is a single parameter q such that each q ij = q ±1 . Assuming that q = ±1, we classify connected quantized Weyl algebras, showing that there are two types linear and cyclic. When q is not a root of unity we determine the prime spectra for each type. The linear case is the easier, although the result depends on the parity of n, and all prime ideals are completely prime. In the cyclic case, which can only occur if n is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank.We apply connected quantized Weyl algebras to obtain presentations of two classes of quantum cluster algebras. Let n ≥ 3 be an odd integer. We present the quantum cluster algebra of a Dynkin quiver of type A n−1 as a factor of a linear connected quantized Weyl algebra by an ideal generated by a central element. We also consider the quiver P(1)

Freidel-Maillet type presentations of $U_q(sl_2)$

“…However, in recent years a different presentation for U q (g) has been introduced, called equitable, with generators {X ±1 i , Y i , Z i , i = 1, ..., n} [ ITW05,T05]. Since [ITW05], the equitable presentation of U q (sl 2 ) has been studied from various perspectives: algebra [T11, T15b,GVZ15,FJ16,LXJ19], combinatorics and representation theory [A11, MR12, Fu13, HG13, IRT13, BCT14, T14, Y15, N15, T15a, SG16,T20] for instance. Also, note that irreducible finite dimensional representations have been studied in details in [ ITW05,T09].…”