AN EXPLICIT ISOMORPHISM BETWEEN QUANTUM AND CLASSICAL s l n $$ \mathfrak{s}{\mathfrak{l}}_n $$
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Springer Science+Business Media New York (2019)
AN EXPLICIT ISOMORPHISM BETWEEN QUANTUM AND CLASSICAL sln A. APPEL
S. GAUTAM
School of Mathematics University of Edinburgh Edinburgh, EH9 3FD, UK [email protected]
Department of Mathematics The Ohio State University Columbus, OH 43210, USA [email protected]
Abstract. Let g be a complex semisimple Lie algebra. Although the quantum group U~ g is known to be isomorphic, as an algebra, to the undeformed enveloping algebra U g[[~]], no such isomorphism is known when g 6= sl2 . In this paper, we construct an explicit isomorphism for g = sln , for every n > 2, which preserves the standard flag of type A. We conjecture that this isomorphism quantizes the Poisson diffeomorphism of Alekseev and Meinrenken [2].
1. Introduction 1.1. Let g be a complex semisimple Lie algebra. The Drinfeld–Jimbo quantum group U~ g is a topological Hopf algebra over the ring of formal power series C[[~]], which deforms the universal enveloping algebra U g [12], [21], [22]. In [15], Drinfeld pointed out that the algebra structure of U g[[~]] remains unchanged under quantization, i.e., there exists an isomorphism of C[[~]]-algebras ψ : U~ g → U g[[~]], which is congruent to the identity modulo ~. Due to its cohomological origin, such an isomorphism is highly non-canonical and indeed unknown with the sole exception of sl2 (e.g., [15, §5] and [9, Prop. 6.4.6]). In this paper, we contruct an explicit algebra isomorphism ϕ : U~ sln → U sln [[~]] for any n > 2. We refer the reader to Section 2, Theorem 2.5, for the formulae defining ϕ. Here, we will explain how the isomorphism is obtained and state some of its properties. 1.2. Our construction relies on the homomorphism between the quantum loop algebra and the Yangian of sln from [17]. Namely, ϕ is defined to be the following composition: DOI: 10.1007/S00031-019-09543-6 Received June 20, 2018. Accepted April 26, 2019. Corresponding Author: S. Gautam, e-mail: [email protected]
A. APPEL, S. GAUTAM
U~ Lsl O n
Φ
/Y \ ~ sln ev
?
U~ sln
ϕ
/ U sln [[~]]
where • Φ is the algebra homomorphism between the quantum loop algebra and (an appropriate completion of) the Yangian of sln , defined and studied by the second author and V. Toledano Laredo in [17]. While the results of [17] hold for any semisimple Lie algebra g, in this paper we only need them for sln , and even further, only the restriction of Φ to U~ sln . We refer the reader to Section 3 for a brief review of [17]. • ev is the evaluation homomorphism at 0. It is well known that the evaluation homomorphism exists only in type A (see [9, Prop. 12.1.15] for both these statements). The homomorphism Φ in the diagram above is given explicitly in the loop presentation of Yangian (also known as Drinfeld’s new presentation [14]), while the evaluation homomorphism ev is known only in either the J-presentation, or the RT T presentation of Yangian (see, for example, [9, Chap. 12] and [24, Chap. 1]). Thus, in order to work out an explicit formula for ϕ, defined via the diagram above, we need to
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