Degeneration of Bethe subalgebras in the Yangian Aleksei Ilin - PowerPoint PPT Presentation

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 gl n Let V “ C n , R p u q “ 1 ´ Pu ´ 1 P End p V b V qrr u ´ 1 ss , where P p u b v q “ v b u . Definition Yangian Y p gl n q for gl n is a complex unital associative algebra with countably many generators t p 1 q ij , t p 2 q ij , . . . where 1 ď i, j ď n , and the defining relations R p u ´ v q T 1 p u q T 2 p v q “ T 1 p u q T 2 p v q R p u ´ v q . where T p u q “ p t ij p u qq n i,j “ 1 , ij u ´ 1 ` t p 2 q ij u ´ 2 ` . . . P Y p gl n qrr u ´ 1 ss . t ij p u q “ δ ij ` t p 1 q Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Bethe subalgebras σ P S k p´ 1 q σ σ P C r S k s . Let A k “ ř Definition Consider C P gl n . For any 1 ď k ď n introduce the series with coefficients in Y p gl n q by τ k p u, C q “ 1 k !tr A k C 1 . . . C k T 1 p u q . . . T k p u ´ k ` 1 q , where we take the trace over all copies of End C n . We call the subalgebra generated by the coefficients of τ k p u, C q Bethe subalgebra and denote it by B p C q . Lemma ÿ λ a 1 . . . λ a k t a 1 ,...,a k τ k p u, C q “ a 1 ,...,a k p u q , 1 ď a 1 ă ... ă a k ď n σ P S k p´ 1 q σ ¨ t a σ p 1 q b 1 p u q . . . t a σ p k q b k p u ´ k ` 1 q is where t a 1 ,...,a k b 1 ,...,b k “ ř quantum minor of T p u q . Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Bethe subalgebras Theorem (Nazarov, Olshanski, 1996) Suppose that C P h reg . Then Bethe subalgebra B p C q is free and maximal commutative. The coefficients of the series τ k p u, C q are free generators for B p C q . Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Limit subalgebras deg t p r q ij “ r B r p C q : “ Y r p gl n q X B p C q r θ r : h reg Ñ ą Gr p d p i q , dim Y i p gl n qq , C Ñ p B 1 p C q , . . . , B r p C qq . i “ 1 Denote the closure of θ r p h reg q (with respect to Zariski topology) by Z r . We have natural projections ρ k : Z r Ñ Z r ´ 1 . Let us define inverse limit Z “ lim Ý ρ k . Z naturally parameterizes some Ð 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 M 0 ,n ` 2 . 2) For any point X P M 0 ,n ` 2 , the corresponding subalgebra B p X q in Y p gl n q is free and maximal commutative. Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Description of limit algebras How to think about M 0 ,n ` 2 ? The points of M 0 ,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 M 0 ,n ` 2 can be represented by pictures like the following on the right. Conditions: 1. n ` 2 – marked points 0 , z 1 , . . . , z n , 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 M 0 ,n ` 2 is the tensor product of the following 3 commuting subalgebras: λ ‰ 0 gl kλ q ˆ i p B p C qq b C ψ p B p X 8 qq b ZU p À F p X λ q Here i and ψ some embedding of corresponding Yangians to Y p gl n q , C some diagonal matrix, ˆ F p X λ q – shift of argument subalgebras of U p gl n q corresponding to X λ . Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Conjecture If C is real, then B p C q acts with simple spectrum on certain class of finite-dimensional representations of Y p gl n q . 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 R p u q . Suppose we have any finite-dimensional representation ρ : Y p g q Ñ End p V q (not a sum of trivial). Evaluate R p u q “ p ρ b ρ q R p´ u q . Definition Extended Yangian X p g q for g is a complex unital associative algebra with countably many generators t p 1 q ij , t p 2 q ij , . . . where 1 ď i, j ď dim V , and the defining relations R p u ´ v q T 1 p u q T 2 p v q “ T 1 p u q T 2 p v q R p u ´ v q . where T p u q “ p t ij p u qq dim V i,j “ 1 , ij u ´ 1 ` t p 2 q ij u ´ 2 ` . . . P X p g qrr u ´ 1 ss . t ij p u q “ δ ij ` t p 1 q Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Yangian for g Definition Yangian Y p g q for g is defined as factor of X p g q by some relation Z p u q “ 1 , where Z p u q P X p g q b End p V qrr u ´ 1 ss . 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 X p g q » Z p X p g qq b Y p g q . Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian

Bethe subalgebras Let V “ ‘ i V p ω i , a i q – sum of fundamental representations of Y p g q . Definition Let C P G . For any 1 ď k ď n introduce the series with coefficients in Y p g q by τ k p u, C q “ tr V ωi ρ i p C q T i p u q . We call the subalgebra generated by the coefficients of τ k p u, C q Bethe subalgebra and denote it by B p C q . Aleksei Ilin Degeneration of Bethe subalgebras in the Yangian