On classical limits of Bethe subalgebras in Yangians
aa r X i v : . [ m a t h . QA ] N ov ON CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS
ALEKSEI ILIN AND LEONID RYBNIKOV
To the memory of Ernest Borisovich Vinberg
Abstract.
The Yangian Y p g q of a simple Lie algebra g can be regarded as a deformation oftwo different Hopf algebras: the universal enveloping algebra of the current algebra U p g r t sq and the coordinate ring of the first congruence subgroup O p G rr t ´ ssq . Both of these algebrasare obtained from the Yangian by taking the associated graded with respect to an appropriatefiltration on Y p g q .Bethe subalgebras B p C q in Y p g q form a natural family of commutative subalgebras de-pending on a group element C of the adjoint group G . The images of these algebras in tensorproducts of fundamental representations give all integrals of the quantum XXX Heisenbergmagnet chain.We describe the associated graded of Bethe subalgebras in the Yangian Y p g q of a simpleLie algebra g as subalgebras in U p g r t sq and in O p G rr t ´ ssq for all semisimple C P G . Inparticular, we show that associated graded in U p g r t sq of the Bethe subalgebra B p E q assignedto the unity element of G is the universal Gaudin subalgebra of U p g r t sq obtained from thecenter of the corresponding affine Kac-Moody algebra ˆ g at the critical level. This generalizesTalalaev’s formula for generators of the universal Gaudin subalgebra to g of any type. Inparticular, this shows that higher Hamiltonians of the Gaudin magnet chain can be quantizedwithout referring to the Feigin-Frenkel center at the critical level.Using our general result on associated graded of Bethe subalgebras, we compute some lim-its of Bethe subalgebras corresponding to regular semisimple C P G as C goes to an irregularsemisimple group element C . We show that this limit is the product of the smaller Bethesubalgebra B p C q and a quantum shift of argument subalgebra in the universal envelopingalgebra of the centralizer of C in g . This generalizes the Nazarov-Olshansky solution ofVinberg’s problem on quantization of (Mishchenko-Fomenko) shift of argument subalgebras. Introduction
Yangians and Bethe subalgebras.
Let g be a simple complex Lie algebra, and G bethe corresponding adjoint group. The Yangian Y p g q is the unique homogeneous Hopf algebradeformation of the universal enveloping algebra U p g r t sq , see [D]. It is also a Hopf algebradeformation of the algebra O p G rr t ´ ssq of functions on first congruence subgroup G rr t ´ ss Ă G rr t ´ ss deforming the natural Poisson structure on G rr t ´ ss , see [KWWY].Bethe subalgebras is the family of commutative subalgebras of B p C q Ă Y p g q depending ona group element C P G . The particular cases of Bethe subalgebras were defined in [C], [D3],[KR], [MO], [M] and [NO]. The most general definition of Bethe subalgebras goes back toDrinfeld: namely, one can define B p C q as the subalgebra generated by all Fourier coefficientsof T r V p ρ p C q b qp ρ b Id qp R p u qq for all finite dimensional representations ρ : Y p g q Ñ End p V q ,where R p u q is the universal R -matrix with spectral parameter. In [IR2] we gave a detaileddescription of these subalgebras using the RT T -realization of the Yangian from [D] and [W].1.2.
The (universal) Gaudin subalgebra.
The universal enveloping algebra of the currentalgebra g r t s contains a large commutative subalgebra A g Ă U p g r t sq . This subalgebra comes fromthe center of the universal enveloping of the affine Kac–Moody algebra ˆ g at the critical level and gives rise to the construction of higher hamiltonians of the Gaudin model (due to Feigin, Frenkeland Reshetikhin, [FFR]). Though there are no explicit formulas for the generators of A g knownin general, the classical analogue of this subalgebra, i.e. the associated graded subalgebra in thePoisson algebra A g Ă S p g r t sq , can be easily described. Namely, it is generated by all Fouriercomponents of the C rr t ´ ss -valued functions Φ l p x p t qq on t ´ g rr t ´ ss “ Spec S p g r t sq for Φ l beingfree homogeneous generators of the algebra of adjoint invariants S p g q g Ă S p g q .1.3. The associated graded of a Bethe algebra.
Let C be any element of a maximal torus T Ă G . Denote by z g p C q be the centralizer of C in the Lie algebra g . It is a reductive Lie algebracontaining the Cartan subalgebra h Ă g . The generators of Bethe subalgebra B p C q Ă Y p g q areinvariant with respect to the adjoint action of z g p C q . Theorem A. ‚ The associated graded of B p C q in U p g r t sq is the universal Gaudin subalgebra A z g p C q Ă U p z g p C qr t sq Ă U p g r t sq ; ‚ The associated graded of B p C q in O p G rr t ´ ssq is generated by all Fourier componentsof T r V C ¨ g p t q for all finite dimensional G -modules V ; ‚ The Bethe subalgebra B p C q is a maximal commutative subalgebra in Y p g q z g p C q . In particular, this gives a construction of the universal Gaudin subalgebra independent of therepresentation theory of ˆ g at the critical level and for arbitrary simple g . We expect this leadsto explicit type-free formulas for higher Gaudin Hamiltonians generalizing those of Talalaev,Chervov and Molev, see [T], [CM]. Remark.
We believe that our Theorem A is a part of a more general picture describing allpossible degenerations of the affine quantum group U q p ˆ g q at the critical level. In particular,according to Ding and Etingof [DE] the center of U q p ˆ g q at the critical level is generated bytraces of the R -matrix, so it is natural to expect that both Bethe subalgebras in the Yangianand Gaudin subalgebras are degenerate versions of this center. We hope to return to this inforthcoming papers.1.4. Limit Bethe subalgebras.
Let T reg Ă T be the set of regular elements of the torus T .From Theorem A we see that the family of Bethe subalgebras B p C q Ă Y p g q is not flat, i.e.the Poincaré series of B p C q is not constant in C P T , because for non-regular C P T z T reg , thesubalgebra B p C q becomes smaller. On the other hand a natural way to assign a commutativesubalgebra of the same size as for C P T reg to any C P T z T reg by taking some limit of B p C q as C Ñ C (this idea goes back to Vinberg [V] and Shuvalov [Sh]). In general, such limitsubalgebra lim C Ñ C B p C q is not unique since it depends on the path C p ε q such that C p q “ C .The second goal of this paper is to study the simplest limits of Bethe subalgebras correspondingto C p ε q “ C exp p εχ q , C P T z T reg , χ P h as ε Ñ . It turns out that the resulting commutativesubalgebra is the product of B p C q and the quantum shift of argument algebra in the universalenveloping algebra U p z g p C qq Ă Y p g q .1.5. Shift of argument subalgebras and Vinberg’s problem.
The shift of argument sub-algebras defined by Mishchenko and Fomenko in [MF] are (generically) maximal Poisson com-mutative subalgebras in S p g q . For any χ P g ˚ the corresponding shift of argument subalgebra A χ Ă S p g q can be described as the subalgebra generated by all the derivatives along χ of alladjoint invariant in S p g q . More precisely, it is generated by all elements of the form B kχ Φ l , for allgenerators Φ l P S p g q g , l “ , . . . rk g , k “ , , . . . , m l , where m l “ deg Φ l ´ are the exponents N CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS 3 of the Lie algebra g . Then the number of generators is rk g ř l “ p m l ` q “ p dim g ` rk g q , which isthe maximal possible transcendence degree for Poisson commutative subalgebras in S p g q . Vinberg’s problem stated in [V] is the problem of lifting the Poisson commutative subalgebras A χ Ă S p g q to commutative subalgebras in the universal enveloping algebra U p g q . In [NO]Olshansky and Nazarov construct the lifting of a shift of argument subalgebra A χ to U p g q as the image of Bethe subalgebra B p χ q in the (twisted) Yangian of g under the evaluationhomomorphism to U p gl n q . This works only for classical g since for others there is no evaluationhomomorphism from the Yangian to U p g q .In [R2] Vinberg’s problem was solved affirmatively for arbitrary simple g and semisimple χ with the help of the Feigin-Frenkel center of U p ˆ g q at the critical level. Namely, the lifting A χ Ă U p g q , called quantum shift of argument subalgebra, was determined as the image of(a version of) the universal Gaudin subalgebra under some homomorphism. Moreover, it wasproved that, for generic χ , the subalgebras A χ Ă S p g q can be lifted to the universal envelopingalgebra U p g q uniquely .Our second main result is the following Theorem B.
Let C p ε q “ C exp p εχ q , C P T z T reg with χ P h Ă z g p C q being a genericsemisimple element of the centralizer of C . Then lim ε Ñ B p C p ε qq “ B p C q b Z p U p z g p C qqq A χ , where A χ Ă U p z g p C qq is the quantum shift of argument subalgebra corresponding to χ . Remark.
Theorem B can be regarded as the closest approximation to the Olshansky-Nazarovsolution of Vinberg’s problem for arbitrary simple g : indeed, now one can define the lifting of A χ Ă S p g q to the universal enveloping algebra as A χ : “ U p g q X lim ε Ñ B p C p ε qq for C p ε q “ exp p εχ q .1.6. The paper is organized as follows. In section 2 we study two classical limits of the Yangianand relations between them. In section 3 we define Bethe subalgebras and give the lower boundfor the size of a Bethe subalgebra. In section 4 we define the universal Gaudin subalgebra andstudy some its properties. In section 5 we prove Theorem A. In section 6 we study some limitsof Bethe subalgebras and prove Theorem B.1.7. Acknowledgements.
We thank Boris Feigin for stimulating discussions. The study hasbeen funded within the framework of the HSE University Basic Research Program and theRussian Academic Excellence Project ’5-100’. Both authors were supported in part by theRFBR grant 20-01-00515. Theorem B was proved under support of the RSF grant 19-11-00056.The first author is a Young Russian Mathematics award winner and would like to thank itssponsors and jury. The work of the second author was supported by the Foundation for theAdvancement of Theoretical Physics and Mathematics “BASIS”.2.
Two classical limits of the Yangian
Notations and definitions.
Let g be a complex simple Lie algebra, G be the correspond-ing connected adjoint group, ˜ G be the corresponding connected simply-connected group. Let T Ă G be a maximal torus, T reg Ă T be the set of regular elements of T . Let h Ă g be tangentCartan subalgebra of T . Let h ¨ , ¨ i be the Killing form on g and t x a u , a “ , . . . , dim g , be anorthonormal basis of g with respect to h ¨ , ¨ i . We identify g ˚ with g using the Killing form. Let m i , i “ , . . . , rk g be the set of exponents of Lie algebra g . Let O p G q and O p ˜ G q be the algebrasof polynomial functions on G and ˜ G respectively. ALEKSEI ILIN AND LEONID RYBNIKOV
Let Y p g q be the Yangian of g . Let V “ À ni “ V p ω i , q be the direct sum of fundamentalrepresentations of Y p g q . Let R p u ´ v q be the image of the universal R -matrix in End p V q b .Using this data we define the RT T -realization Y V p g q as follows. It turns out that Y V p g q » Y p g q ,see [D] and [W] for details. Definition 2.2.
The Yangian Y V p g q is a unital associative algebra generated by the elements t p r q ij , ď i, j ď dim V ; r ě with the defining relations R p u ´ v q T p u q T p v q “ T p v q T p u q R p u ´ v q in End p V q b b Y V p g qrr u ´ , v ´ ss ,S p T p u qq “ T p u ` c g q , where S p T p u qq “ T p u q ´ is the antipode map and c g is the value of the Casimir element of g on the adjoint representation.Here T p u q “ r t ij p u qs i,j “ ,..., dim V P End V b Y V p g q ,t ij p u q “ δ ij ` ÿ r t p r q ij u ´ r and T p u q (resp. T p u q ) is the image of T p u q in the first (resp. second) copy of End V . Two filtrations on the Yangian.
The first filtration F on Y V p g q is determined byputting deg t p r q ij “ r . More precisely, the r -th filtered component F p r q Y p g q is the linear spanof all monomials t p r q i j ¨ . . . ¨ t p r m q i m j m with r ` . . . ` r m ď r . By gr we denote the operationof taking associated graded algebra with respect to F . From the defining relations we seethat gr Y p g q is a commutative algebra. Moreover, we have a Poisson algebra isomorphism gr Y V p g q » O p G rr t ´ ssq where the grading on O p G rr t ´ ssq is given by the C ˚ action dilating t (see Section 2.13 for details): Theorem 2.4. [IR2, Proposition 2.24]
There is an isomorphism of graded Poisson algebras gr Y V p g q » O p G rr t ´ ssq . Corollary 2.5.
Poincaré series of gr Y V p g q is ś r “ p ´ q r q ´ rk g . The second filtration F on Y V p g q is determined by putting deg t p r q ij “ r ´ . Similarly,the r -th filtered component F p r q Y p g q is the linear span of all monomials t p r q i j ¨ . . . ¨ t p r m q i m j m with r ` . . . ` r m ď r ` m . By gr we denote the operation of taking associated graded algebrawith respect to F . Theorem 2.6. [W] gr Y V p g q » U p g r t sq where the grading is given by the C ˚ action dilating t .Moreover, we have t r ´ g Ă span t t p r q ij u{ F p r ´ q Y p g q . Bigraded quotient.
The filtration F on Y p g q produces a filtration on U p g r t sq “ gr Y p g q which we denote by the same letter F . Similarly, the filtration F on Y p g q descends to a filtra-tion F on O p G rr t ´ ssq “ gr Y p g q . The corresponding associated graded algebras gr gr Y p g q and gr gr Y p g q get a bigrading from the filtrations F and F . We begin with some generalfacts about algebras with multiple filtrations.For any algebra A endowed with two filtrations, F and F , one can define a bigraded quotient of A as bigr A “ à i,j p F p i q A X F p j q A q ä p F p i ´ q A X F p j q A ` F p i q A X F p j ´ q A q N CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS 5
We also use the following notation: gr A “ gr gr A, gr A “ gr gr A Lemma 2.8.
The bigraded quotient bigr A is canonically isomorphic to gr A and to gr A .Proof. Consider the algebra gr A “ F p q A ‘ F p q A ä F p q A ‘ . . . . The filtration F producesa filtration W Ă W Ă . . . on gr A such that W i “ ‘ j F p i q A X F p j q A ä F p i q A X F p j ´ q A .Note that F p i q A X F p j q A ä F p i q A X F p j ´ q A » F p i q A X F p j q A ` F p j ´ q A ä F p j ´ q A therefore W Ă W Ă . . . is indeed a filtration.We have the following canonical isomorphisms W i ä W i ´ “ ‘ j F p i q A X F p j q A ä F p i q A X F p j ´ q AF p i ´ q A X F p j q A ä F p i ´ q A X F p j ´ q A ““ ‘ j F p i q A X F p j q AF p i ´ q A X F p j q A ` F p i q A X F p j ´ q A .
Then associated graded algebra gr A “ W ‘ W ä W ‘ . . . is canonically isomorphic to bigr A . It is also isomorphic to gr A by the same argument. (cid:3) In contrast with last Lemma, if U Ă A is a subspace, it is not true in general that gr U “ gr U as subspaces of bigr A , since the associated homomorphism of bigraded spaces bigr U Ñ bigr A is not necessarily injective (indeed, consider the algebra A “ C r x, y s with two flirtationssetting by deg x “ and deg y “ and take U “ C r x ` y s . Then the image of bigr U in bigr A is C ¨ , gr U “ C r x s , gr U “ C r y s ). On the other hand, the following is still true: Proposition 2.9.
Let U be a vector subspace of A such that gr U Ă gr U as subspaces of bigr A . Then gr U “ gr U .Proof. Suppose that we have an element x P gr W z gr W . Suppose that deg x “ p k, l q . Let « x be a lifting of x to U Ă A . Then « x P F p k q A X F p l q A ` ř k ă k F p k q A X F p l q A , where l are someintegers, so there is a presentation « x “ N ř i “ « x i such that « x i P F p k i q A X F p l i q A with k “ k, k i ` ă k i . Take such a presentation of « x with the string p k , k , k , . . . , k N q being lexicographicallyminimal among all such presentations. Then we have « x i R F p k i ´ q A X F p l i q A ` F p k i q A X F p l i ´ q A and l i ` ą l i for all i . Moreover, we can assume that « x is a lifting of with the lexicographicallyminimal possible p k , k , k , . . . , k N q among all liftings of x to U . It is sufficient to show that N “ : indeed, then « x P F p k q A X F p l q A and gr « x “ x P gr A .Suppose that N ą and consider y “ gr « x . It has degree p k N , l N q . Let « y P U be a liftingof y as an element of gr A , i.e. « y P F p k N q A X F p l N q A ` ř k ă k N F p k q A X F p l q A Then, in thesame way as before, we have « y “ M ř i “ N « y i such that « y i P F p k i q A X F p l i q A with k i ` ă k i and « y i R F p k i ´ q A X F p l i q A ` F p k i q A X F p l i ´ q A . ALEKSEI ILIN AND LEONID RYBNIKOV So « x ´ « y “ N ´ ř i “ « x i ´ M ř j “ N ` « y i is a lifting of x such that to U such that the correspondingsequence of degrees of the summands x , . . . , x N ´ , ´ y N ` ,..., ´ y M is p k , . . . , k N ´ , k N ` , . . . , k M q ,hence lexicographically smaller than p k , . . . , k N q . This is a contradiction. (cid:3) Suppose additionally that gr A is a commutative algebra. Then bigr A is also commutativeand has a structure of a Poisson algebra. Let u and v be homogeneous elements of degree p i , j q and p i , j q respectively. Then t u ` p F p i ´ q A X F p j q A ` F p i q A X F p j ´ q A q , v ` p F p k ´ q A X F p l q A ` F p k q A X F p l ´ q A qu ““ r u, v s ` p F p i ` k ´ q A X F p j ` l q A ` F p i ` k ´ q A X F p j ` l ´ q A q Also gr A is a Poisson algebra, and this give a Poisson algebra structure on bigr A . It followsfrom definitions that these brackets are the same Poisson brackets as on bigraded quotient.Also on gr A one can obtain a Poisson bracket from the commutator on gr A which is alsothe same bracket as on the bigraded quotient. So we have the following Lemma 2.10. If gr A is commutative then the bigraded quotient bigr A is canonically isomor-phic to gr A and gr A as Poisson algebra. Suppose that dim F p i q A { F p i ´ q A is always finite and let P p q q : “ ř i “ q i dim F p i q A { F p i ´ q A be the Poincaré series of gr A . Let P p q, z q : “ ÿ i,j “ q i z j dim p F p i q A X F p j q A q ä p F p i ´ q A X F p j q A ` F p i q A X F p j ´ q A q be the Poincaré series of bigr A . Then we have Lemma 2.11. P p q q “ P p q, q . Proposition 2.12.
We have a bigraded Poisson algebra isomorphism gr Y p g q » gr Y p g q » S p g r t sq where the bigrading on S p g r t sq is given by deg x r r ´ s “ r, deg x r r ´ s “ r ´ andthe Poisson bracket on S p g r t sq is given by t x r r s , y r s su “ r x, y sr r ` s s .Proof. The first isomorphism gr Y p g q » gr Y p g q is a particular case of Lemma 2.8. FromTheorem 2.6 we have x r r ´ s P F p r q U p g r t sq . Hence the Poincaré series P p q, z q is greater orequal to ś r “ p ´ q r z r ´ q (in the sense that every coefficient of the former is greater or equal to thecorresponding coefficient of the latter), and it is equal if and only if x r r ´ s R F p r ´ q U p g r t sq forall x ‰ . So according to Lemma 2.11 P p q q “ ś r “ p ´ q r q if and only if x r r ´ s R F p r ´ q U p g r t sq for all x ‰ . On the other hand, we have P p q q “ ś r “ p ´ q r q by Theorem 2.4. This completesthe proof. (cid:3) Congruence subgroup G rr t ´ ss and its coordinate ring. Let us give a bit moredetails on gr Y p g q “ O p G rr t ´ ssq . By definition, the proalgebraic group G rr t ´ ss consist of C rr t ´ ss points of G . For any g P G rr t ´ ss we denote by ev g the corresponding homomorphism O p G q Ñ C rr t ´ ss . The first congruence subgroup G rr t ´ ss Ă G rr t ´ ss is the kernel of theevaluation homomorphism at the infinity G rr t ´ ss Ñ G . N CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS 7
To any function f P O p G q one can assign the C rr t ´ ss -valued function r f : G rr t ´ ss Ñ C rr t ´ ss , r f “ ř r “ f p r q t ´ r as follows: for any g P G rr t ´ ss we have r f p g q “ ev g p f q . The Fourier coefficients f p r q for all f P O p G q generate the coordinate ring O p G rr t ´ ssq . Notethat the group G rr t ´ ss depends only on the formal group scheme assigned to G , so one canproduce f p r q from any f in the local completion of O p G q at the unity.There is a natural Poisson bracket on O p G rr t ´ ssq coming from the Lie bialgebra structurestructure on the loop algebra g pp t ´ qq (or, equivalently, from the rational r -matrix). To writethis bracket explicitly, we set, for any x P g , the corresponding momenta vector fields of theleft and right action on G , ξ Lx and ξ Rx , respectively. Then for f , f P O p G q , the bracket ofcorresponding C rr t ´ ss -valued functions reads(1) t r f p u q , r f p v qu “ u ´ v p Ć ξ Lx a f p u q Ć ξ Lx a f p v q ´ Ć ξ Rx a f p u q Ć ξ Rx a f p v qq . The C ˚ action on G rr t ´ ss by dilations of the variable t determines a grading on O p G rr t ´ ssq such that deg f p r q “ r for any f P O p G q . The Poisson bracket has degree ´ with respect tothis grading. A more precise statement of Theorem 2.4 is the following Proposition 2.14. [IR2, Proposition 2.24]
There is an isomorphism of graded Poisson algebras gr Y V p g q » O p G rr t ´ ssq such that gr t p r q ij “ ∆ p r q ij where ∆ ij P O p G q are the matrix elementsof the representation V . Any formal coordinate system in the formal neighborhood of the unity E P G (i.e. any formaldiffeomorphism ϕ : p g , q Ñ p G, E q ) determines an isomorphism Φ : t ´ g rr t ´ ss Ñ G rr t ´ ss which preserves the grading defined by the C ˚ action by dilations on both sides. The coordinatering of t ´ g rr t ´ ss is the symmetric algebra of its graded dual space i.e. g r t s with the pairinggiven by p x p t q , y p t qq : “ Res t “ x x p t q , y p t qy dt @ x p t q P g r t s , y p t q P t ´ g rr t ´ ss . This means that any formal diffeomorphism ϕ : p g , q Ñ p G, E q identifies O p G rr t ´ ssq with S p g r t sq .The filtration F on Y p g q induces a filtration on gr Y p g q “ O p G rr t ´ ssq . Slightly abusingnotations, we denote this filtration by F as well. Let O p G q ` be polynomial functions on G consist of f P O p G q such that f p E q “ . Proposition 2.15.
For any f P O p G q , we have f p r q P F p r ´ q O p G rr t ´ ssq . Moreover, if f “ f ¨ f ¨ . . . ¨ f k such that f , . . . , f k P O p G q ` then f p r q P F p r ´ k q O p G rr t ´ ssq .Proof. It suffices to check the first assertion on generators of O p G q . According to Peter-Weyltheorem, ∆ ij generate O p G q and we have ∆ p r q ij “ gr t p r q ij P F p r ´ q O p G rr t ´ ssq by definition.To prove the second assertion, notice that f p r q is a linear combination of f p r q f p r q ¨ . . . ¨ f p r k q k with r i ą and r ` r ` . . . ` r k “ r . (cid:3) Corollary 2.16.
Let f a P O p G q be a collection of functions such that t x a “ d E f a u is the basisof g “ g ˚ “ T ˚ E G . Then f p r q a P F p r ´ q O p G rr t ´ ssq , and gr O p G rr t ´ ssq is freely generated by gr f p r q a with i “ , . . . , n, r “ , , . . . . Moreover, gr f p r q a “ x a r r ´ s . Corollary 2.17.
Let ϕ : p g , q Ñ p G, E q and ϕ : p g , q Ñ p G, E q be formal diffeomorphismssuch that d ϕ “ d ϕ . Let Φ ˚ , Φ ˚ : O p G rr t ´ ssq Ñ S p g r t sq be the corresponding ring isomor-phisms. Then we have gr Φ ˚ “ gr Φ ˚ . ALEKSEI ILIN AND LEONID RYBNIKOV
This means that under any identification O p G rr t ´ ssq » S p g r t sq as above, the grading F on S p g r t sq is given by deg x r r ´ s “ r and the filtration F on S p g r t sq is given by deg x r r ´ s “ r ´ ,for any x P g . The Poisson bracket on O p G rr t ´ ssq » S p g r t sq descends to gr S p g r t sq “ S p g r t sq .We denote the latter bracket by t¨ , ¨u ℓ . Lemma 2.18.
We have t x r m s , y r l su ℓ “ r x, y sr n ` m s , for any x, y P g , m, l ě .Proof. For any x P g , denote by r x p u q the formal series ř r “ x r r ´ s u ´ r P S p g r t sqr u ´ s . Let f x P O p G q be a function such that d E f x “ x under the identification g “ g ˚ . According toCorollary 2.16, we have f p r q x P F p r ´ q O p G rr t ´ ssq and gr f p r q x “ x r r ´ s . Slightly abusingnotations we will write gr r f x p u q “ r x p u q .For x, y P g we take the functions f x , f y P O p G q as above and write the Poisson bracket t r f x p u q , r f y p v qu “ u ´ v p dim g ÿ a “ Č ξ Lx a f x p u q Č ξ Lx a f y p v q ´ Č ξ Rx a f x p u q Č ξ Rx a f y p v qq . Taking the leading term of each coefficient in the expansion in the variables u and v on bothsides of the equation, with respect to the filtration F , we get t r x p u q , r y p v qu ℓ “ u ´ v p dim g ÿ a “ pp x a , x qr x a , y p v qs`r x a , x p u qsp x a , y q´p x a , x qr y p v q , x a s´r x p u q , x a sp x a , y qqq . Since dim g ř a “ pp x a , x qr x a , y s “ ´ dim g ř a “ r x a , x sp x a , y q “ r x, y s we finally get t r x p u q , r y p v qu ℓ “ r x p u q , y p v qs . (cid:3) gr Y V p g q » gr Y V p g q as Poisson algebras andare isomorphic to S p g r t sq with the standard Kirillov-Kostant Poisson bracket.We identify gr Y V p g q with S p g r t sq and gr Y V p g q with S p g r t sq thus obtain an automorphismof Poisson algebra ψ : S p g r t sq Ñ S p g r t sq . Lemma 2.20. ψ p x r k sq “ c k x r k s , c P C ˚ .Proof. The filtrations F , F are g -invariant, therefore ψ is g -invariant. Let us identify gr and gr with bi-graded quotient. We know that F p q Y p g q X F p q Y p g q “ C ¨ ,F p q Y p g q X F p q Y p g q ä F p q Y p g q X F p q Y p g q “ g , p F p q Y p g q X F p q Y p g qq ä p F p q Y p g q X F p q Y p g q ` F p q Y p g q X F p q Y p g qq “ t ¨ g » g as g -modules. Note also that F p q Y p g q X F p q Y p g q ä F p q Y p g q X F p q Y p g q “ g is a Lie algebra isomorphism with respect to the Poisson bracket on the left hand side.Using the fact that g is simple we see that the only isomorphism of g with itself is identityand isomorphism of t ¨ g as g -module is the scalar of identity. N CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS 9
From Lemma 2.18 by induction on r we have ψ pr x, y sr r sq “ ψ pt x r r ´ s , y r suq “ t ψ p x r r ´ sq , cy r su “ t c r ´ x r r ´ s , cy r su “ c r r x, y sr r s . Since g is simple, we have r g , g s “ g therefore ψ has the desired form. (cid:3) It will be useful for what follows to have a way to write the leading term with respect to F of any function of the form f p r q P O p G rr t ´ ssq for any f P O p G q . For this, we fix a formalcoordinate system in the neighborhood of E P G , i.e. let ϕ : p g , q Ñ p G, E q be a formaldiffeomorphism such that d ϕ “ Id . Then to any function f P O p G q one can assign its Taylorexpansion at E P G , namely a collection of homogeneous polynomials f l P S l p g q , l “ , , . . . such that ϕ ˚ f “ ř l “ f l . We denote by D the derivation on S p g r t sq determined by D p x r r ´ sq “ rx r r s . Lemma 2.21.
Suppose f k P S k p g q is the first nonzero term in the Taylor series of f P O p ˜ G q ` at E P G , k ą .Then we have (1) f p r q “ for r ă k ; (2) f p r q P F p r ´ k q O p G rr t ´ ssq and f p r q R F p r ´ k ´ q O p G rr t ´ ssq for r ě k ; (3) gr f p r q “ p r ´ k q ! D r ´ k f k where f k P S p g q Ă S p g r t sq as a polynomial of the x r s ’s.Proof. The first assertion follows immediately from Proposition 2.15. The second one followsfrom Corollary 2.16. To show the last equality, note that Φ ˚ r f p u q “ ř l “ k r f l p u q . According tothe assertions (1-2) and by Corollary 2.17 the leading term of any Fourier coefficient withrespect to the filtration F is given by that of Φ ˚ r f k p u q . On the other hand for any x P g thecorresponding series r x p u q “ ř r “ x r r ´ s u ´ r rewrites as r x p u q “ exp p u ´ D q x r s . So we have Φ ˚ r f k p u q “ exp p u ´ D q f k as well, hence the assertion. (cid:3) Bethe subalgebras in Yangian
Definition.
Let ρ i : Y p g q Ñ End V p ω i , q be the i -th fundamental representation of Y p g q .Let π i : V Ñ V p ω i , q be the projection.Let T i p u q “ π i T p u q π i be the submatrix of T p u q -matrix, corresponding to the i -th fundamen-tal representation. Definition 3.2.
Let C P ˜ G . Bethe subalgebra B p C q Ă Y V p g q is the subalgebra generated by allcoefficients of the following series with the coefficients in Y V p g q τ i p u, C q “ tr V p ω i , q ρ i p C q T i p u q , ď i ď n. Remark.
In fact B p C q depends only on the class of C in ˜ G { Z p ˜ G q , i.e. on an element of adjointgroup G . Proposition 3.3. ( [IR2] , [I] ) (1) Bethe subalgebra B p C q is commutative for any C P G . (2) B p C q is a maximal commutative subalgebra of Y p g q for C P T reg . Bethe subalgebras in O p G rr t ´ ssq . Here we follow [IR2]. Let t V ω i u ni “ be the set ofall fundamental representations of g . We also consider t V ω i u ni “ as a representations of thecorresponding simply-connected group ˜ G . Let Λ i be some basis of V ω i . For any v P V wedenote the corresponding element of dual basis by v ˚ P V ˚ . By ∆ v,v ˚ P O p ˜ G q we denote thecorresponding matrix coefficient of V ω i . Definition 3.5.
Let C P ˜ G . Bethe subalgebra ˜ B p C q of O p G rr t ´ ssq is the subalgebra generatedby of the coefficients of the following series: σ i p u, C q “ tr V ωi ρ i p C q ρ i p g q “ ÿ v P Λ i ∆ v,v ˚ p Cg q “ ÿ r “ ÿ v P Λ i ∆ p s q v,v ˚ p Cg q u ´ r , where Λ i is some basis of V ω i , g P G rr t ´ ss . Remark.
This subalgebra depends only on the class of C in ˜ G { Z p ˜ G q as well. Remark.
One can define the same subalgebra using all finite-dimensional representations of ˜ G . Proposition 3.6. ( [IR2] ) We have gr B p C q “ ˜ B p C q for any C P T reg . We generalize this Proposition 3.6 to any C P T below.3.7. Size of a Bethe subalgebra.
Consider B p C q with C P T . In the next proposition weuse the filtration F . Proposition 3.8. (Lower bound for the size of Bethe subalgebra, see also [IR] ) Bethe subalgebra B p C q contains rk g infinite series of algebraically independent elements such that every seriesconsist of elements with the degrees m i ` , m i ` , . . . , where m i are the exponents of g , i “ , . . . , rk g .Proof. Analogous to [IR2, Proposition 4.8] we have gr B p C q Ą ˜ B p C q . We are going to find a setof algebraically independent elements in ˜ B p C q of the same degrees as in Proposition statementwith respect to the grading obtained from filtration F .Let σ i p u, C q be generators of Bethe subalgebra ˜ B p C q Ă O p G rr t ´ ssq . One can extend σ i p u, C q to the group G pp t ´ { qq by means of Definition 3.5. Denote by σ i p C q p r q the coefficientof u ´ r in σ i p u, C q .Let e be the principal nilpotent element of the reductive algebra z g p C q . Consider also theelement t ˜ ρ P G pp t ´ { qq , ˜ ρ “ ř i ˜ ω i , where ˜ ω i are fundamental co-weights of z g p C q . Note thatthis is well-defined element because ρ belong to co-weight lattice of g .The differential of σ i p C q p r q at the point exp p e q P G pp t ´ { qq is naturally a linear functional onthe tangent space T exp p e q G pp t ´ { qq » g pp t ´ { qq . Hereinafter we identify T g p t q G pp t ´ { qq with g pp t ´ { qq by the left G pp t ´ { qq -action for any g p t q P G pp t ´ { qq .Then we have d exp p t ´ e q σ i p C q p r q “ d t ˜ ρ ¨ exp p e q t ´ ˜ ρ σ i p C q p r q “ p Ad t ˜ ρ q d exp p e q σ i p C q p r q . The last equality follows from the invariance of σ p r q i p C q under conjugation by t ˜ ρ .We are now consider the restriction of differentials to T E G rr t ´ ss » t ´ g rr t ´ ss . Let χ ω i becharacters of ˜ G -modules V ω i , i “ , . . . , rk g . Lemma 3.9. d exp p e q σ i p C q p r q p xt ´ s q “ δ r,s d C ´ exp p e q χ ω i p x q for any x P g . N CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS 11
Proof.
We have d exp p e q σ i p C q p r q “ ÿ v P Λ i d exp p e q ∆ p r q v,v ˚ p C qp xt ´ s q “ δ r,s tr V ωi ρ i p C q ρ i p x q “ δ r,s d C ´ exp p e q χ ω i p x q for any x P g . (cid:3) Note that C ´ ¨ exp p e q is a regular element of ˜ G .As in [IR2] the key point here is the fact that differentials of characters of fundamentalrepresentations at regular point are linearly independent (see [St, Theorem 3, p.119]). Lemma 3.10. span (cid:10) d C ´ exp p e q χ ω i (cid:11) “ z z g p C q p e q under the identification g ˚ » g .Proof. Note that z g p C exp p e qq “ z z g p C q p e q . It is sufficient to show now that span (cid:10) d C ´ exp p e q χ ω i (cid:11) “ z g p C exp p e qq . It is obvious that span (cid:10) d C ´ exp p e q χ ω i (cid:11) Ă z g p C exp p e qq and dimensions coincideaccording to linear independence of differentials at regular point. (cid:3) Under the correspondence from previous Lemma one can express eigenvector v j of ˜ ρ witheigenvalue m j as a linear combination of d C ´ exp p e q χ ω i . Let σ p r q v j p C q be the corresponding linearcombination of σ i p C q p r q , i “ , . . . , rk g . Lemma 3.11. p Ad t ˜ ρ q d exp p e q σ v i p C q p r q p xt ´ s q “ δ r,s ´ m i h v i , x i for any x P g .Proof. It follows from Lemma 3.9 and the fact that σ p r q v i p C q is an eigenvector of t ˜ ρ with eigenvalue t m i . (cid:3) From the last Lemma the statement of Proposition follows. (cid:3)
Remark.
We will also give another proof of Proposition 3.8 in Section 5, see Proposition 5.10. Universal Gaudin subalgebra
Commutative subalgebra from the center on critical level.
We regard the Liealgebra g r t s as a “half” of the corresponding affine Kac–Moody algebra ˆ g which is a centralextension of the loop Lie algebra g pp t ´ qq . According to Feigin and Frenkel, the local com-pletion of the universal enveloping algebra U p ˆ g q on the critical level k “ ´ h _ has a hugecenter Z . The image of natural homomorphism from Z to the quantum Hamiltonian reduction p U p ˆ g q{ U p ˆ g q t ´ g r t ´ sq t ´ g r t ´ s is a commutative subalgebra there. The latter naturally embedsinto U p g r t sq , so the image of Z can be regarded as a commutative subalgebra A g Ă U p g r t sq ,which we call the universal Gaudin subalgebra of U p g r t sq .Though there are no explicit formulas for for the generators of A g in general, one can describeexplicitly the associated graded subalgebra A g Ă S p g r t sq “ O p t ´ g rr t ´ ssq . Namely, A g is freelygenerated by all Fourier components of C rr t ´ ss -valued functions Φ l p x p t qq for all generators Φ l of the algebra of adjoint invariants S p g q g . The subalgebra A g Ă S p g r t sq can be obtained viaMagri-Lenard scheme ([Ma]) from a pair of compatible Poisson brackets on S p g rr t ssq (see nextsubsection).4.2. Two Poisson brackets on S p g rr t ssq . Let g rr t ss be a Lie algebra of formal power serieswith coefficients in g . Consider two Poisson brackets on S p g rr t ssq : t x r n s , x r m su “ r x, y sr n ` m s ; t x r n s , y r m su “ r x, y sr n ` m ` s , for any x, y P g . Note that bracket t , u is bracket we obtain on gr Y V p g q if we restrict it to S p g r t sq . Note also that g r t s with t¨ , ¨u is isomorphic to t ¨ g r t s as a Lie algebra. We call a pair of Poisson brackets on S p g rr t ssq compatible if every linear combination of themis a also a Poisson bracket. The following Lemma is well-known. Lemma 4.3. (1)
Poisson brackets t¨ , ¨u and t¨ , ¨u are compatible. (2) Every linear combination of these brackets restricts to S p g rr t ssq g . By S p g rr t ssq u,v we denote a Poisson algebra S p g rr t ssq with Poisson bracket u t , u ` v t , u .From the pair of compatible Poisson brackets one can obtain a Poisson commutative subalgebraof S p g rr t ssq g with respect to t , u and t , u at the same time, see e.g. [R1]. The constructionis as follows: subalgebra is generated by all centers of S p g rr t ssq g u,v for u, v P C except the case u “ , v “ .4.4. Universal Gaudin subalgebra A g . Consider the derivation D of S p g r t sq g : D p x r n sq “ p n ` q x r n ` s . Let Φ i , i “ , . . . , rk g be free generators of S p g r sq g . Definition 4.5.
Universal Gaudin subalgebra A g is the subalgebra generated by all D k Φ i , k ě , i “ , . . . , rk g . Proposition 4.6.
Subalgebra A g is commutative and elements D k Φ i , k ě , i “ , . . . , rk g arefree generators of A g .Proof. It is easy to check that the map ϕ ,v : S p g rr t ssq , Ñ S p g rr t ssq ,v ; x r m s ÞÑ x r m s ` ÿ k “ p´ q k v k x r m ` k s , for all x P g , m ě , is an isomorphism of Poisson algebras. Indeed, the inverse map is x r m s ÞÑ x r m s ` vx r m ` s , for all x P g , m ě . One can restrict ϕ ,v to S p g rr t ssq g , to obtain the isomorphism S p g rr t ssq g , » S p g rr t ssq g ,v .If Φ P S p g r sq g then it is central in S p g rr t ssq g , therefore ϕ ,v p Φ q is central in S p g rr t ssq g ,v . Itimplies that the elements of the form ϕ ,v p Φ q , Φ P S p g r sq g , v P C commutes with respect to anybracket u t¨ , ¨u ` v t¨ , ¨u . This implies that the coefficients of degrees of v are commutes withrespect to any bracket u t¨ , ¨u ` v t¨ , ¨u , in particularly t¨ , ¨u and t¨ , ¨u . Note that coefficientsbelongs to S p g r t sq g , hence these coefficients generate some commutative subalgebra of S p g r t sq g .It is easy to see that for any f P S p g r sq ϕ ,v p f q “ exp p´ vD q f. Therefore the coefficient of v k of ϕ ,v p Φ q is proportional to D k Φ . This means that oursubalgebra coincides with A g .The statement that elements of the form D k Φ i are free generators of A g is well-known, seee.g. [BD], [R1]. (cid:3) N CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS 13
Properties of subalgebra A g . By definition put ω g “ ÿ a x a r s P S p g r sq g , Ω g “ ÿ a x a r s x a r s P S p g r t sq g , where t x a u , a “ , . . . , dim g is an orthonormal basis of g with respect to x¨ , ¨y .Note that ω g P A g by construction. Also Dω g “ g therefore Ω g P A g too. Proposition 4.8. ( [R1] ) Subalgebra A g is the centralizer of ω g in S p g r t sq with respect to t , u . Proposition 4.9.
Subalgebra A g is the centralizer of Ω g in S p g r t sq g with respect to t , u .Proof. Let us again consider the isomorphism ϕ ,v : S p g rr t ssq , Ñ S p g rr t ssq ,v . Note that ϕ ,v p ω g q “ ω g ` Ω g v ` . . . and ϕ ,v p ω g q belong to the center of S p g rr t ssq g ,v . Then for any z P S p g r t sq g Ă S p g rr t ssq g wehave t ϕ ,v p ω g q , z u ` v t ϕ ,v p ω g q , z u “ . Considering the coefficient of v we get t Ω g , z u ` t ω g , z u “ . Therefore the centralizer of ω g in S p g r t sq g with respect to t¨ , ¨u coincides with the centralizerof Ω g in S p g r t sq g with respect to t¨ , ¨u . (cid:3) Corollary 4.10. A g is a maximal commutative subalgebra of S p g r t sq g , and S p g r t sq g , . Proposition 4.11.
There exists no more than one lifting A g of A g to U p g r t sq g .Proof. Up to scaling and additive constant there exists a unique lifting of Ω g to U p g r t sq g .Moreover, any lifting of subalgebra A g is the centralizer of the lifting of the element Ω g . Butthe centralizer does not depend on a constant therefore the lifting is unique. (cid:3) Remark.
We will assign to any C P T the subalgebra A z g p C q Ă S p z g p C qq Ă S p g q and considerthe elements ω z g p C q , Ω z g p C q in it, i.e. consider the above objects for a reductive Lie algebra,not necessarily semisimple. All the statements and definitions of the present section remain thesame for z g p C q with the following conventions: we take the restriction of x¨ , ¨y to z g p C q as theinvariant scalar product on z g p C q , rk z g p C q “ rk g , the exponents of z g p C q aren the exponentsof the semisimple algebra r z g p C q , z g p C qs plus additional rk g ´ rk r z g p C q , z g p C qs of zeros.5. Bethe subalgebras and universal Gaudin subalgebras
Let E P G be the identity element. We are going to prove Theorem A for C “ E . Theorem 5.1. gr B p E q is the universal Gaudin subalgebra, i.e. gr B p E q “ A g .Proof. Lemma 5.2.
The element Ω g belongs to gr B p E q and to gr B p E q . Proof.
Firstly we consider gr B p E q Ă S p g r t sq g . From Proposition 3.8 we know that thereare 2 algebraically independent elements of degree 3 for type A and 1 element of degree 3 forother types in gr B p E q . All degree 3 elements in S p g r t sq g are from S p g q g ` p g ¨ t g q g . In type A the spaces S p g q g and p g ¨ t g q g are -dimensional. In other types we have S p g q g “ and dim p g ¨ t g q g “ . So gr B p E q contains the spaces S p g q g and p g ¨ t g q g .Any element from p g ¨ t g q g has the form c ¨ Ω g , c P C ˚ . Finally, Ω g is homogeneous withrespect to the second grading hence Ω g P gr B p E q .We identify gr Y V p g q and gr Y V p g q with S p g r t sq then obtain the automorphism ψ : S p g r t s Ñ S p g r t sq of Poisson algebra. From Lemma 2.20 it follows that ψ maps any graded(with respect to the first grading) vector subspace of S p g r t sq to itself.We have F p k q Y V p g q X F p l q Y V p g q “ F p k q Y V p g q X F p k ´ q Y V p g q for l ě k . Then any lifting of Ω g to Y V p g q belongs to F p q X F p q ` ÿ k ă ,l ă k F p k q X F p l q “ F p q X F p q Then for any lifting « Ω g we have gr « Ω g “ Ω g . (cid:3) Proposition 5.3. gr B p E q “ A g .Proof. We know that Ω g P gr B p E q and that A g is the centralizer of Ω g . Moreover gr B p E q Ă S p g r t sq g thus gr B p E q Ă A g .From Proposition 3.8 and the definition of A g we see that Poincaré series of gr B p E q andof A g coincide. Hence we have gr B p E q “ A g . Moreover, gr B p E q Ă gr B p E q , because Ω g P gr B p E q and gr B p E q is Poisson commutative. Then by Proposition 2.9 we have gr B p E q “ gr B p E q “ A g . (cid:3) From the last proposition and Proposition 4.11 it follows that gr B p E q “ A g and we aredone. (cid:3) Remark.
From Theorem 5.1 we get the construction of A g independent of center on criticallevel of ˆ g . Corollary 5.4. B p E q is a maximal commutative subalgebra of Y p g q g . Application to the Gaudin model.
Let z P C . Let ev z : U p g r t sq Ñ U p g q be anevaluation map. Let z i , i “ , . . . , n be different complex numbers. Let d be the diagonalembedding d : U p g r t sq Ñ U p g r t sq b n . We define a map ev z ,...,z n : “ ev z b . . . b ev z n ˝ d : U pr g r t sq Ñ U p g q b . . . b U p g q . Definition 5.6.
Gaudin subalgebra A p z , . . . , z n q Ă U p g q b n is ev z ,...,z n p A g q . It is known (see [FFR]) that this subalgebra is commutative and gives the complete set ofintegrals for the quantum Gaudin magnet chain.From Theorem 5.1 we have the following
Corollary 5.7. ev z ,...,z n p gr B p E qq “ A p z , . . . , z n q . This generalizes Talalaev’s formulas [T] for higher Gaudin Hamiltonians to g of arbitrarytype. N CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS 15
Proof of Theorem A in the general case.
Let C P T and consider gr B p C q Ă U p g r t sqq .As before z g p C q is the infinitesimal centralizer of C . In this subsection we are going to proofTheorem A in the full generality. Theorem 5.9. gr B p C q is the universal Gaudin subalgebra in U p z g p C qr t sq z g p C q , i.e. gr B p C q “ A z g p C q . Proposition 5.10.
All generators of A z g p C q belong to gr B p C q .Proof. The idea of proof is as follows. Consider any linear combination of functions σ i p C qp g q : “ Tr V ωi ρ i p C q ρ i p g q , g P ˜ G and obtain from it the series of functions on O p G rr t ´ ssq . Then these functions by definitionbelongs to ˜ B p C q Ă gr B p C q . We will find all generators of A z g p C q using this construction.According to Lemma 2.21 it is sufficient to show that there are functions in O p ˜ G q ˜ G ` such thatfirst non-zero term in Taylor expansion at the point C ´ are Φ i , i “ , . . . , rk g , where Φ i arefree generators of S p z g p C qr sq z g p C q .Indeed, it is equivalent to find f P O p ˜ G q with Φ i being the first non-zero term in Taylorexpansion of f at the unity, as a linear combination of the functions σ i p C q .Consider the decomposition g “ z g p C q ‘ n , where n is sum of eigenvalues of Ad p C q whichis not equal to 1 as follows. We choose a formal coordinate system in the neighborhood of C ´ P G with the help of the map: Ψ : z g p C q ‘ n Ñ G, p h, x q ÞÑ exp p´ x q C ´ exp p h q exp p x q The differential of Ψ at C ´ is p Ad C ´ ´ Id q ‘ Id hence is non-degenerate.Central functions on G depend only on the conjugacy class, so the Taylor expansion coeffi-cients the central functions in our coordinates depend only on z g p C q , not on n .Let f P O p ˜ G q ˜ G ` be a central function. Let r S p g q z g p C q be the completion of S p g q z g p C q . Considerthe Taylor series expansion of f at point C ´ as an element of the completion r S p g q z g p C q . Let N be the ideal, generated by the coordinate functions on n and let J “ S p z g p C qq z g p C q` . Let e be the principal nilpotent of z g p C q . Consider the following composite map Θ : O p ˜ G q ˜ G ` Ñ r S p g q z g p C q` Ñ r S p g q z g p C q {p N ` p r S p g q z g p C q qq » J { J » z z g p C q p e q . Here the first arrow is taking the Taylor expansion and the last isomorphism is due to Kostant[Ko]. We claim that Θ is surjective. Indeed, one can obtain z z g p C q p e q by considering d C ´ exp p e q f and applying Lemma 3.10.Let t e, h, f u be the corresponding sl -triple in z g p C q . One can split the centralizer z z g p C q p e q into the eigenspaces of the operator ad h : z z g p C q p e q “ rk g à i “ V i . Note that dim V i is the number of algebraically independent generators of S p z g p C qq z g p C q ofdegree m i ` , see [Ko]. For any f P O p ˜ G q ˜ G ` whose Taylor series expansion starts from the k -thterm we have Θ p f q P rk g à m i ě k ´ V i . By the surjectivity of Θ we have functions with Taylor series on C ´ starting from Φ i , i “ , . . . , rk g where Φ i , i “ , . . . , rk g are free generators of S p z g p C qq z g p C q and we are done. (cid:3) Proposition 5.10 implies that A z g p C q Ă gr B p C q . To prove that in fact we have an equalitywe are going to prove that A z g p C q is a maximal commutative subalgebra of S p g r t sq z g p C q .We mostly follow the argument of [R1] below. Let t e, h, f u be a principal sl -triple of g . Letus recall two classical facts. Proposition 5.11. ( [Ko] ) Let π be the restriction homomorphism π : C r g s Ñ C r f ` z g p e qs . If we restrict π to C r g s g we obtain an isomorphism C r g s g » C r f ` z g p e qs . The next proposition is well-known.
Proposition 5.12.
Let ψ be the restriction homomorphism ψ : C r g s Ñ C r h s . If we restrict ψ to C r g s g we obtain an isomorphism C r g s g » C r h s W , where W is the Weyl groupof g . Particularly, C r h s is an algebraic extension of ψ p C r g s g q . Proposition 5.13. A z g p C q is a maximal commutative subalgebra of S p g r t sq z g p C q with respect to t¨ , ¨u . Moreover, A z g p C q is centralizer of the element Ω z g p C q .Proof. Let ψ : S p g r t sq Ñ S p h r t sq be a h -invariant projection. Lemma 5.14. (1) π p A z g p C q q » S p z z g p C q p e qr t sq ; (2) A z g p C q is algebraically closed in S p g r t sq ; (3) ψ p A q Ă S p h r t sq is an algebraic extension.Proof.
1) Let g “ z z g p C q p e q ‘ n , where n is any complement subspace, and let Φ i , i “ , . . . , rk g be algebraically independent generators of S p z g p C qr sq z g p C q . Consider π : S p g r t sq Ñ S p z z g p C q p e qr t sq such that π p x r n sq “ x r n s , x P z z g p C q p e q , π p x r n sq “ δ n h x, f i , x P n We have π p C r Φ , . . . , Φ k sq » S p z z g p C q p e qr sq by Proposition 5.11. Moreover, D commuteswith π hence we have π p C r D s Φ , . . . , D s Φ k sq » S p z z g p C q p e qr s sq and hence π p A z g p C q q » S p z z g p C q p e qr t sq .
2) Suppose that A z g p C q is not algebraically closed. Let a P S p g r t sq be an element which isalgebraic over A z g p C q . Then by the first statement of this Lemma we can assume that π p a q “ .Suppose that p n a n ` . . . ` p a ` p “ , where p i P A z g p C q and n is minimal. Then π p p q “ .But S p g r t sq does not have zero divisors and we have contradiction with minimality of n .3) Note that ψ p C r Φ , . . . , Φ rk g sq Ă S p h r sq is the algebraic extension from Proposition 5.12.Using the fact that D commutes with ψ we see that ψ p C r Φ , . . . , Φ rk g , D Φ , . . . , D Φ rk g , . . . , D s Φ , . . . , D s Φ rk g sq Ă S p s à i “ h r i sq is the algebraic extension as well for any s . (cid:3) N CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS 17
From Lemma 4.8 it follows that the centralizer of ω z g p C q in S p g r t sq z g p C q , contains the subal-gebra A z g p C q .Now let us define the family of automorphisms of S p g r t sq with respect to the bracket t¨ , ¨u .Let ϕ s p x r m sq “ x r m s ` sδ m h h, x i . It is a straightforward computation that ϕ s is an automor-phism. Lemma 5.15.
We have h r s P lim s Ñ8 ϕ s p A z g p C q q .Proof. Recall that an element h is an element from principal sl -triple of g . It is straightforwardcomputation that lim s Ñ8 ϕ s p ω z g p C q q ´ s x h, h y s “ h r s . (cid:3) Lemma 5.16. ( [R1] ) The algebra S p h r t sq is the centralizer of h r s is S p g r t sq with respect to t¨ , ¨u . Now return to the proof of the Proposition. The centralizer of Ω z g p C q with respect to t¨ , ¨u coincide with the centralizer of ω z g p C q with respect to t¨ , ¨u . Suppose that we have some element a R A z g p C q in centralizer of ω z g p C q with respect to t¨ , ¨u . Then a should be transcendent over A z g p C q so we can assume that ψ p a q “ . For some k we have a non-zero limit ˜ a “ lim s Ñ8 ϕ s p a q s k P ϕ s p A z g p C q q . This limit should lie in the centralizer of the element h r s , and then lie in S p h r t sq .It means that ψ p a q ‰ which is a contradiction and completes the proof. (cid:3) Corollary 5.17.
We have gr B p C q “ A z g p C q and gr B p C q “ A z g p C q .Proof. Analogous to Lemma 5.2 we can find Ω z g p C q in gr B p C q . Then from Proposition 5.13and the fact that gr B p C q is Poisson commutative it follows that gr B p C q Ă gr B p C q andthen by Proposition 2.9 we have gr B p C q Ă gr B p C q “ A z g p C q . From the uniqueness of liftingof Ω z g p C q to U p g r t sq (up to scalar and additive constant) we see that gr B p C q “ A z g p C q . (cid:3) This finishes the proof of Theorem 5.9.
Corollary 5.18. B p C q is a maximal commutative subalgebra of Y p g q z g p C q and ˜ B p C q is a max-imal Poisson commutative subalgebra in O p G rr t ´ ssq z g p C q . Corollary 5.19. gr B p C q “ ˜ B p C q for any C P T . Some limits of Bethe subalgebras.
Definition of limit subalgebras.
Let C be an element of G reg . Recall that the formula deg t p r q ij “ r defines the filtration F on Y V p g q . Recall that F p r q Y V be an r -th filtered component.Consider B p r q p C q : “ F p r q Y V X B p C q . In the paper [IR2] (in course of proof of Theorem 2.6)it is proved that the images of the coefficients of τ p u, C q , . . . , τ n p u, C q freely generate thesubalgebra gr B p C q Ă gr Y V p g q . Hence the dimension d p r q of B p r q p C q does not depend on C .Therefore for any r ě we have a map θ r from G reg to ś ri “ Gr p d p i q , dim F p i q Y V q such that C ÞÑ p B p q p C q , . . . , B p r q p C qq . Here Gr p d p i q , dim F Y p i q V q is the Grassmannian of subspaces ofdimension d p i q of a vector space of dimension dim F Y p i q V . Denote the closure of θ r p G reg q (withrespect to Zariski topology) by Z r . There are well-defined projections ζ r : Z r Ñ Z r ´ for all r ě . The inverse limit Z “ lim ÐÝ Z r is well-defined as a pro-algebraic scheme and is naturally a parameter space for some family of commutative subalgebras which extends the family of Bethesubalgebras.Indeed, any point z P Z is a sequence t z r u r P N where z r P Z r such that ζ r p z r q “ z r ´ . Every z r is a point in ś ri “ Gr p d p i q , dim Y p i q V q i.e. a collection of subspaces B p i q r p z q Ă Y p i q V such that B p i q r p z q Ă B p i ` q r p z q for all i ă r . Since ζ r p z r q “ z r ´ we have B p i q r p z q “ B p i ´ q r ´ p z q for all i ă r .Let us define the subalgebra corresponding to z P Z as B p z q : “ Ť r “ B p r q r p z q . Proposition 6.2. ( [IR2, Proposition 4.11] ) For any z P Z subalgebra B p z q is a commutativesubalgebra of Y V p g q . The Poincaré series of B p z q is (lexicographically) not smaller that thePoincaré series series of B p C q for C P G reg . We call a subalgebra of the form B p z q , z P Z limitsubalgebra. Remark. In [Sh] the limits of subalgebras are defined in the analytic topology, just as limits of1-parametric families of subalgebras. But it is well known (see e.g. [Se] ) that the closure of analgebraic variety under a regular map with respect to Zariski topology coincides with its closurewith respect to the analytic topology. Shift of argument subalgebras.
Let g be a reductive Lie algebra. To any χ P g ˚ one canassign a Poisson-commutative subalgebra in S p g q with respect to the standard Poisson bracket(coming from the universal enveloping algebra U p g q by the PBW theorem). Let ZS p g q “ S p g q g be the center of S p g q with respect to the Poisson bracket. The algebra A χ Ă S p g q generatedby the elements B nχ Φ , where Φ P ZS p g q , (or, equivalently, generated by central elements of S p g q “ C r g ˚ s shifted by tχ for all t P C ) is Poisson-commutative and has maximal possibletranscendence degree. More precisely, we have the following Theorem 6.4. [MF]
For regular semisimple χ P g the algebra A χ is a free commutativesubalgebra in S p g q with p dim g ` rk g q generators (this means that F p C q is a commutativesubalgebra of maximal possible transcendence degree). One can take the elements B nχ Φ k , k “ , . . . , rk g , n “ , , . . . , deg Φ k ´ , where Φ k are basic g -invariants in S p g q , as free generatorsof A χ . Theorem 6.5. ( [R2] ) For any regular semisimple χ P g there exist a lifting A χ Ă U p g q , i.e. acommutative subalgebra A χ Ă U p g q such that gr A χ “ A χ . Moreover, this lifting is unique forgeneric regular χ . Theorem 6.6. ( [R2] ) For generic regular semisimple χ P g , the subalgebra A χ is the centralizerof its quadratic part which is the linear span of the elements ř α P Φ ` p α,h qp α,χ q e α e ´ α for all h P h . Certain limits of Bethe subalgebras.Theorem 6.8.
Let C p ε q “ exp p εχ q , χ P h and C p ε q P T reg if ε ‰ . Then lim ε Ñ B p C p ε qq “ B p E q b Z p U p g qq A χ for generic χ P h .Proof. According to [I] the quadratic part of Bethe subalgebra contains the following elements: σ i p C q “ J p t ω i q ´ ÿ α P Φ ` e α p C q ` e α p C q ´ p α, α i q x α x ´ α P Y p g q , i “ , . . . , rk g . Here J p t ω i q is an element of Y p g q which does not depend on C . In the limit ε Ñ , the leadingterm has the form ÿ α P Φ ` p α, α i qp α, χ q x α x ´ α , N CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS 19 i.e. the quadratic part of shift of argument subalgebra A χ (see [V]). As we state before forgeneral χ shift of argument subalgebra A χ is a centralizer of its quadratic part. Lemma 6.9. (1)
Suppose that g is a reductive Lie algebra, g – a reductive subalgebraof g . Then the subalgebras Y p g q g and U p g q in Y p g q are both free U p g q g -modules.Moreover, the product of these subalgebras in Y p g q is: Y p g q g ¨ U p g q » Y p g q g b U p g q g U p g q ; (2) z Y p g q p ZU p g qq “ Y p g q g b U p g q g U p g q . Proof.
1) Let us consider associated bigraded factor with respect to the filtrations F , F . Then gr Y p g q g “ S p g r t sq g , gr U p g q g “ S p g q g , gr U p g q “ S p g q .Let f n p t q be a polynomial of degree n with n pairwise different roots. Consider the quotient S n p g r t sq : “ S p g r t sq{ f n p t q . From [IR, Lemma 4.8] it follows that S n p g r t sq g ¨ S p g q » S n p g r t sq g b S p g q g S p g q . We see that the statement of Lemma holds for any filtered component and hence for thewhole algebra.2) It follows from [K, Main Theorem (d)] if we consider the associated graded with respectto filtration F and follow the proof of first statement. (cid:3) In our limit we have B p E q Ă Y p g q g and it is a maximal subalgebra of Y p g q g (Lemma 5.4).Moreover, Z p U p g qq lie in B p E q . From Lemma 6.9 we see that the limit subalgebra should liesin B p E q ¨ U p g q » B p E q b ZU p g q U p g q . But then it should lie in the centralizer of the quadraticpart of the Bethe subalgebra in the latter tensor product i.e. in B p E q ¨ A χ » B p E q b ZU p g q A χ .Subalgebra B p E q b ZU p g q A χ has the same Poincaré series as B p C q , C P T reg . Then the limitcoincides with B p E q b ZU p g q A χ . (cid:3) Theorem 6.10.
Let C p ε q “ C exp p εχ q , C P T z T reg , χ P h and C p ε q P T reg if ε ‰ . Then lim ε Ñ B p C p ε qq “ B p C q b Z p U p z g p C qqq A χ for generic χ .Proof. The proof is the same as the proof of previous Theorem, with the only difference thatwe use Corollary 5.18 instead of Corollary 5.4. (cid:3)
Remark.
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Leonid Rybnikov
National Research University Higher School of Economics,Russian Federation,Department of Mathematics, 6 Usacheva st, Moscow 119048;Institute for Information Transmission Problems of RAS; [email protected]
Aleksei Ilin