Duality for Bethe algebras acting on polynomials in anticommuting variables
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Duality for Bethe algebras acting on polynomials in anticommuting variables V. Tarasov1,2 · F. Uvarov1 Received: 25 February 2020 / Revised: 16 July 2020 / Accepted: 27 August 2020 © Springer Nature B.V. 2020
Abstract We consider actions of the current Lie algebras gln [t] and glk [t] on the space of polynomials in kn anticommuting variables. The actions depend on parameters z¯ = (z 1 , . . . , z k ) and α¯ = (α1 , . . . , αn ), respectively. We show that the images of the n k Bethe algebras Bα¯ ⊂ U (gln [t]) and Bz¯ ⊂ U (glk [t]) under these actions coincide. To prove the statement, we use the Bethe ansatz description of eigenvalues of the actions of the Bethe algebras via spaces of quasi-exponentials and establish an explicit n k correspondence between these spaces for the actions of Bα¯ and Bz¯ . Keywords (glk , gln )-Duality · Quantum Gaudin model · Bethe subalgebra Mathematics Subject Classification 17B35 · 17B80 · 82B23
1 Introduction The classical (glk , gln )-duality plays an important role in the representation theory and the classical invariant theory, for example, see [2,14]. It states the following. n k Let ei j , i, j = 1, . . . , n, and eab , a, b = 1, . . . , k, be the standard generators of the Lie algebras gln and glk , respectively. Define gln - and glk -actions on the space
V. Tarasov: Supported in part by Simons Foundation grant 430235 and RFBR grant 18-01-00271.
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F. Uvarov [email protected] V. Tarasov [email protected]; [email protected]
1
Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA
2
St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, Russia 191023
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V. Tarasov, F. Uvarov
Pkn = C[x11 , . . . , xkn ] of polynomials in kn variables: n
k
k
a=1 n
ei j → eab →
xai
∂ , ∂ xa j
(1.1)
xai
∂ . ∂ xbi
(1.2)
i=1
Then actions (1.1) and (1.2) commute, and there is an isomorphism of glk ⊕ gln modules n k Vλ ⊗ Vλ , (1.3) Pkn ∼ = λ
n
k
where Vλ and Vλ are the irreducible representations of gln and glk of highest weight λ, respectively. It is interesting to study a similar duality in the context of current algebras, where n the central role is played by the commutative subalgebras Bα¯ ⊂ U (gln [t]) and k n k Bz¯ ⊂ U (glk [t]) called the Bethe algebras, see Sect. 4. The algebras Bα¯ and Bz¯ depend on parameters α¯ = (α1 , . . . , αn ) and z¯ = (z 1 , . . . , z k ), respectively. One can extend the actions of gln and glk on Pkn to the respective gln [t]- and glk [t]-actions by the following formulas: n
ψz¯
k
n
k
k
a=1 n
: ei j ⊗ t s →
ψα¯ : eab ⊗ t s →
i=1
z as xai
∂ , ∂ xa j
(1.4)
αis xai
∂ . ∂ xbi
(1.5)
Actions (1.4) and (1.5) do not commute anymore. However, the images of the subaln k gebras (Bα¯ ) and (Bz¯ ) under the corresponding actions coincide, see [6]. According to [7], the Bethe ansatz method gives a bijection between eigenvectors of n the action of Bα¯ on Pkn and n-t
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