Degeneration of Bethe subalgebras in the Yangian

Transcription

1 Degeneration of Bethe subalgebras in the Yangian National Research University Higher School of Economics Faculty of Mathematics Moscow, Russia Washington, 2018

2 Yangian for gl n Let V C n, Rpuq 1 P u 1 P End pv b V qrru 1 ss, where P pu b vq v b u. Definition Yangian Y pgl n q for gl n is a complex unital associative algebra with countably many generators t p1q ij, tp2q ij,... where 1 ď i, j ď n, and the defining relations where T puq pt ij puqq n i,j 1, Rpu vqt 1 puqt 2 pvq T 1 puqt 2 pvqrpu vq. t ij puq δ ij ` t p1q ij u 1 ` t p2q ij u 2 `... P Y pgl n qrru 1 ss.

3 Bethe subalgebras Let A k ř σps k p 1q σ σ P CrS k s. Definition Consider C P gl n. For any 1 ď k ď n introduce the series with coefficients in Y pgl n q by τ k pu, Cq 1 k! tr A kc 1... C k T 1 puq... T k pu k ` 1q, where we take the trace over all copies of End C n. We call the subalgebra generated by the coefficients of τ k pu, Cq Bethe subalgebra and denote it by BpCq. Lemma τ k pu, Cq ÿ 1ďa 1ă...ăa k ďn λ a1... λ ak t a1,...,a k a 1,...,a k puq, where t a1,...,a k b 1,...,b k ř σps k p 1q σ t aσp1q b 1 puq... t aσpkq b k pu k ` 1q is quantum minor of T puq.

4 Bethe subalgebras Theorem (Nazarov, Olshanski, 1996) Suppose that C P h reg. Then Bethe subalgebra BpCq is free and maximal commutative. The coefficients of the series τ k pu, Cq are free generators for BpCq.

5 Limit subalgebras θ r : h reg Ñ deg t prq ij r B r pcq : Y r pgl n q X BpCq rą Grpdpiq, dim Y i pgl n qq, C Ñ pb 1 pcq,..., B r pcqq. i 1 Denote the closure of θ r ph 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 ÐÝ ρ k. Z naturally parameterizes some new commutative subalgebras with the same Poincare series, called limit Bethe subalgebras.

6 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 BpXq in Y pgl n q is free and maximal commutative.

7 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.

8 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: ipbpcqq b C ψpbpx 8 qq b ZUp Àλ 0 gl k λ q ˆF px λ q Here i and ψ some embedding of corresponding Yangians to Y pgl n q, C some diagonal matrix, ˆF pxλ q shift of argument subalgebras of Upgl n q corresponding to X λ.

9 Conjecture If C is real, then BpCq acts with simple spectrum on certain class of finite-dimensional representations of Y pgl n q.

10 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 t p1q ij, tp2q ij,... where 1 ď i, j ď dim V, and the defining relations Rpu vqt 1 puqt 2 pvq T 1 puqt 2 pvqrpu vq. where T puq pt ij puqq dim V i,j 1, t ij puq δ ij ` t p1q ij u 1 ` t p2q ij u 2 `... P Xpgqrru 1 ss.

11 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 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 Xpgq» ZpXpgqq b Y pgq.

12 Bethe subalgebras Let V i V pω i, a i q 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 τ k pu, Cq tr Vωi ρ i pcqt i puq. We call the subalgebra generated by the coefficients of τ k pu, Cq Bethe subalgebra and denote it by BpCq.