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Degeneration of Bethe subalgebras in the Yangian

Aleksei Ilin National Research University Higher School of Economics Faculty of Mathematics Moscow, Russia

Washington, 2018

Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Yangian for gln

Let V “ Cn, Rpuq “ 1 ´ Pu´1 P End pV b V qrru´1ss, where Ppu b vq “ v b u. Definition Yangian Y pglnq for gln is a complex unital associative algebra with countably many generators tp1q

ij , tp2q ij , . . . where 1 ď i, j ď n, and the

defining relations Rpu ´ vqT1puqT2pvq “ T1puqT2pvqRpu ´ vq. where Tpuq “ ptijpuqqn

i,j“1,

tijpuq “ δij ` tp1q

ij u´1 ` tp2q ij u´2 ` . . . P Y pglnqrru´1ss.

Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Bethe subalgebras

Let Ak “ ř

σPSkp´1qσσ P CrSks.

Definition Consider C P gln. For any 1 ď k ď n introduce the series with coefficients in Y pglnq by τkpu, Cq “ 1 k!tr AkC1 . . . CkT1puq . . . Tkpu ´ k ` 1q, where we take the trace over all copies of End Cn. We call the subalgebra generated by the coefficients of τkpu, Cq Bethe subalgebra and denote it by BpCq. Lemma τkpu, Cq “ ÿ

1ďa1ă...ăakďn

λa1 . . . λakta1,...,ak

a1,...,akpuq,

where ta1,...,ak

b1,...,bk “ ř σPSkp´1qσ ¨ taσp1qb1puq . . . taσpkqbkpu ´ k ` 1q is

quantum minor of Tpuq.

Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Bethe subalgebras

Theorem (Nazarov, Olshanski, 1996) Suppose that C P hreg. Then Bethe subalgebra BpCq is free and maximal commutative. The coefficients of the series τkpu, Cq are free generators for BpCq.

Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Limit subalgebras

deg tprq

ij “ r

BrpCq :“ Yrpglnq X BpCq θr : hreg Ñ

r

ą

i“1

Grpdpiq, dim Yipglnqq, C Ñ pB1pCq, . . . , BrpCqq. Denote the closure of θrphregq (with respect to Zariski topology) by Zr. We have natural projections ρk : Zr Ñ Zr´1. Let us define inverse limit Z “ lim Ð Ý ρk. Z naturally parameterizes some new commutative subalgebras with the same Poincare series, called limit Bethe subalgebras.

Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Main theorem

Theorem 1) Z is a smooth algebraic variety isomorphic to M0,n`2. 2) For any point X P M0,n`2, the corresponding subalgebra BpXq in Y pglnq is free and maximal commutative.

Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Description of limit algebras

How to think about M0,n`2? The points of M0,n`2 are isomorphism classes of curves of genus 0, with n ` 2 ordered marked points and possibly with nodes, such that each component has at least 3 distinguished points (either marked points or nodes). Elements of M0,n`2 can be represented by pictures like the following on the right. Conditions:

• 1. n ` 2 – marked points 0, z1, . . . , zn, 8;
• 2. At least 3 marked points or nodes at every component;
• 3. Nodes are marked too.

Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Description of limit algebras

The limit Bethe subalgebra corresponding to the curve X P M0,n`2 is the tensor product of the following 3 commuting subalgebras: ipBpCqq bC ψpBpX8qq bZUpÀ

λ‰0 glkλq ˆ

FpXλq Here i and ψ some embedding of corresponding Yangians to Y pglnq, C some diagonal matrix, ˆ FpXλq – shift of argument subalgebras of Upglnq corresponding to Xλ.

Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Conjecture

If C is real, then BpCq acts with simple spectrum on certain class of finite-dimensional representations of Y pglnq.

Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Yangian for g

Let g be an arbitrary complex simple Lie algebra. Due to Drinfeld, there exists so-called pseudo-uneversal R-matrix Rpuq. Suppose we have any finite-dimensional representation ρ : Y pgq Ñ End pV q (not a sum of trivial). Evaluate Rpuq “ pρ b ρqRp´uq. Definition Extended Yangian Xpgq for g is a complex unital associative algebra with countably many generators tp1q

ij , tp2q ij , . . . where 1 ď i, j ď dim V , and the

defining relations Rpu ´ vqT1puqT2pvq “ T1puqT2pvqRpu ´ vq. where Tpuq “ ptijpuqqdim V

i,j“1 ,

tijpuq “ δij ` tp1q

ij u´1 ` tp2q ij u´2 ` . . . P Xpgqrru´1ss.

Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Yangian for g

Definition Yangian Y pgq for g is defined as factor of Xpgq by some relation Zpuq “ 1, where Zpuq P Xpgq b EndpV qrru´1ss. Wendlandt proved that this definition is correct, i.e. does not depends on representation V . In fact in the same work it was proven that Xpgq » ZpXpgqq b Y pgq.

Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Bethe subalgebras

Let V “ ‘iV pωi, aiq – sum of fundamental representations of Y pgq. Definition Let C P G. For any 1 ď k ď n introduce the series with coefficients in Y pgq by τkpu, Cq “ tr Vωi ρipCqT ipuq. We call the subalgebra generated by the coefficients of τkpu, Cq Bethe subalgebra and denote it by BpCq.

Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian