Duality of Gaudin Models

Filipp Uvarov, Purdue University

Abstract

We consider actions of the current Lie algebras gln [t] and glk [t] on the space Pkn of polynomials in kn anticommuting variables. The actions depend on parameters z¯ = (z1, . . . , zk) and ¯α = (α1, . . . , αn), respectively. We show that the images of the Bethe algebras B‹n›α¯ ⊂ U(gln [t]) and B‹k› z¯ ⊂ U(glk [t]) under these actions coincide. To prove the statement, we use the Bethe ansatz description of eigenvectors of the Bethe algebras via spaces of quasi-exponentials. We establish an explicit correspondence between the spaces of quasi-exponentials describing eigenvectors of B‹n›α¯ and the spaces of quasi-exponentials describing eigenvectors of B‹k› z¯. One particular aspect of the duality of the Bethe algebras is that the Gaudin Hamiltonians exchange with the Dynamical Hamiltonians. We study a similar relation between the trigonometric Gaudin and Dynamical Hamiltonians. In trigonometric Gaudin model, spaces of quasi-exponentials are replaced by spaces of quasi-polynomials. We establish an explicit correspondence between the spaces of quasi-polynomials describing eigenvectors of the trigonometric Gaudin Hamiltonians and the spaces of quasi-exponentials describing eigenvectors of the trigonometric Dynamical Hamiltonians. We also establish the (glk , gln)-duality for the rational, trigonometric and difference versions of Knizhnik-Zamolodchikov and Dynamical equations.

Degree

Ph.D.

Advisors

Tarasov, Purdue University.

Subject Area

Mathematics|Transportation

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