Quantum groups, quantum tori, and the Grothendieck–Springer resolution

Abstract

Abstract. We construct an algebra embedding of the quantum group Uq(g) into a central extension of the quantum coordinate ring Oq[G w 0 ,w 0 /H] of the reduced big double Bruhat cell in G. This embedding factors through the Heisenberg double Hq of the quantum Borel subalgebra U ≥0 , which we relate to Oq [G] via twisting by the longest element of the quantum Weyl group. Our construction is inspired by the Poisson geometry of the Grothendieck-Springer resolution studied in [10], and the quantum Beilinson-Bernstein theorem investigated in [3] and [34]. IntroductionA basic and much-studied problem in the theory of quantum groups concerns finding embeddings of them into certain simpler algebras, which often lead to insights into their ring-theoretic and representation-theoretic properties. A well-known example of such an embedding is provided by the Feigin homomorphisms [2,31] from the positive part U q (n + ) of a Drinfeld-Jimbo quantized enveloping algebra to a quantum torus algebra. For each reduced decomposition w 0 = s i 1 · · · s i l of the longest element of the Weyl group, one has an algebra embedding of U q (n + ) into the algebra generated by variables X ±1 1 , . . . , X ±1 l subject to the q-commutativity relations X i X j = q b ij X j X i . Other examples of quantum groups which have been shown to admit similar embeddings into quantum torus algebras include the quantum coordinate ring O q [G] of a simple Lie group G, as well as the quantum coordinate rings O q [G u,v ] of its double Bruhat cells constructed by Berenstein and Zelevinsky in [6] and shown to bear an explicit structure of a quantum cluster algebra in [17]. These realizations of quantum groups are closely connected with the theory of quantum cluster ensembles [11], the Feigin homomorphism playing the role of quantum factorization parameters, and the Berenstein-Zelevinsky realizations playing the role of generalized minors.The problem of embedding the full quantized enveloping algebra U q (g) into a quantum torus appears to be more subtle than the previous examples. In the construction of principal series representations for quantized enveloping algebras in [13,18,19], homomorphisms from a certain modular double of U q (g) to a quantum torus were obtained by explicitly writing formulas for the images of the Chevalley generators, and verifying by direct computation that the defining relations were satisfied. In particular, this method depends intricately upon the Dynkin type of g. When g is of type A, quantum torus embeddings can also be obtained from representations of U q (g) of Gelfand-Zetlin type [29], which led to the proof of the GelfandKirillov conjecture in [12]. These embeddings, too, are constructed by explicitly verifying the relations in the Chevalley-Serre presentation of U q (g). Subsequently, analogs of such representations for any quantum affine Kac-Moody algebra U q (g) were proposed in [14].In this paper, we present a new approach to the construction of quantum torus realizations of U q (g). Our construction is geometricall...

A cluster realization of $$U_q(\mathfrak {sl}_{\mathfrak {n}})$$ U q ( sl n )

“…Remark 4.3. As shown in [SS17a], for any semisimple Lie algebra g the algebra U q (g) can be embedded into the quantized algebra of global functions on the Grothendieck-Springer resolution G × B B, where B ⊂ G is a fixed Borel subgroup in G. On the other hand, the variety G× B B is isomorphic to the moduli space of G-local systems on the punctured disc, equipped with reduction to a Borel subgroup at the puncture, as well as a trivialization at one marked point on the boundary. Classically, this moduli space is birational to X S,G , and it would be interesting to understand the precise relation between the corresponding quantizations.…”

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“…. , z n on G * whose Poisson brackets are log canonical [30,54,55], meaning that {z i , z j } G * = c i,j z i z j for some c i,j ∈ C.…”