Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety

JOURNAL OF GEOMETRY AND PHYSICS

卷:94

Article

Rimanyi, R.¹  Tarasov, V.^2,3  Varchenko, A.¹ 

[1] Univ N Carolina, Dept Math, Chapel Hill, NC 27599 USA

[2] Indiana Univ Purdue Univ, Dept Math Sci, Indianapolis, IN 46202 USA

[3] Steklov Math Inst, St Petersburg Branch, St Petersburg 191023, Russia

关键词: Partial flag varieties;    Equivariant K-theory;    Quantum loop algebra;    Bethe algebra;    Discrete Wronskian;   

DOI  :  10.1016/j.geomphys.2015.04.002

来源: Elsevier

【 摘 要 】

We consider the cotangent bundle T*F-lambda of a an partial flag variety, lambda = (lambda(1), ..., lambda(N)), vertical bar lambda vertical bar = Sigma(i)lambda(i) = n, and the torus T = (C-x)(n+1) equivariant K-theory algebra K-T (T*F-lambda) We introduce K-theoretic stable envelope maps Stab(sigma) : circle plus vertical bar(lambda vertical bar=n) K-T((T*F-lambda)(T)) -> circle plus(vertical bar lambda vertical bar=n) K-T(T*F-lambda), where sigma is an element of S-n. Using these maps we define a quantum loop algebra action on circle plus(vertical bar lambda vertical bar=n) K-T (T*F-lambda). We describe the associated Bethe algebra B-q(K-T(T*F-lambda)) by generators and relations in terms of a discrete Wronski map. We prove that the limiting Bethe algebra B-infinity(K-T(T*F-lambda), called the Gelfand-Zetlin algebra, coincides with the algebra of multiplication operators of the algebra K-T (T*F-lambda). We conjecture that the Bethe algebra B-q (K-T (T*F-lambda)) coincides with the algebra of quantum multiplication on K-T (T*F-lambda),) introduced by Givental (2000), Givental and Lee (2003). The stable envelope maps are defined with the help of Newton polygons of Laurent polynomials representing elements of K-T (T*F-lambda) and with the help of the trigonometric weight functions introduced in Varchenko and Tarasov (1994), Tarasov and Varchenko (2013) to construct q-hypergeometric solutions of trigonometric qKZ equations. The paper has five appendices. In particular, in Appendix E we describe the Bethe algebra of the XXZ model by generators and relations. (C) 2015 Elsevier B.V. All rights reserved.

【 授权许可】

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