RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES RIMS Bethe subalgebras in Hecke algebra and Gaudin models. A.P. ISAEV and A.N.

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1 RIMS-1775 Bethe subalgebras in Hecke algebra and Gaudin models By A.P. ISAEV and A.N. KIRILLOV February 2013 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan

2 Bethe subalgebras in Hecke algebra and Gaudin models 1 A.P. Isaev and A.N. Kirillov Bogoliubov Laboratory of Theoretical Physics, JINR, , Dubna, Moscow region, and ITPM, M.V.Lomonosov Moscow State University, Russia isaevap@theor.jinr.ru Research Institute of Mathematical Sciences, RIMS, Kyoto University, Sakyo-ku, , Japan Abstract. The generating function for elements of the Bethe subalgebra of Hecke algebra is constructed as Sklyanin s transfer-matrix operator for Hecke chain. We show that in a special classical limit q 1 the Hamiltonians of the Gaudin model can be derived from the transfer-matrix operator of Hecke chain. We consruct a non-local analogue of the Gaudin Hamiltonians for the case of Hecke algebras The work of A.P.Isaev was supported by the grant RFBR a and grant Higher School of Economics No

3 1 Introduction The Gaudin models were firstly introduced by M.Gaudin in [1]. These models were also investigated as limiting cases of integrable quantum inhomogeneous su2-chains in [2]. Here we use an algebraic approach and obtain Gaudin s Hamiltonians from the transfer-matrix operator for open inhomogeneous chain models which formulated in terms of generators of affine Hecke algebra ĤM+1q. In our chain model an inhomogeneity appears as well as in [2] as different shifts in spectral parameters related to different sites of the Hecke chain. The Gaudin Hamiltonians are obtained from the generating function which defines a Bethe subalgebra in the Hecke algebra ĤM+1q, by taking a special classical limit q 1. The Bethe subalgebras in the group ring of symmetric groups have been studied, for example, in [13], [11]. In the present paper we construct a lift of the Bethe subalgebras studied in the papers mentioned above, to the cases of the Hecke and affine Hecke algebras. Our construction of the Bethe subalgebras is based on some special properties of the trace maps [7],[8],[9] in the tower of the affine Hecke algebras, see Section 3, formulae 15. Non formally speaking, the main idea behind our construction, is to define a set of baxterized Jucys Murphy elements in the affine Hecke algebra. To realize this idea we treat the Jucys Murphy elements in the affine Hecke algebra as a classical limit of the canonical free abelian subgroup in the affine braid group, see Section 2. The plan of the paper is as follows. In Section 2 we review some basic facts about braid and affine braid groups we need, namely, definitions and the construction of the maximal free abelian subgroups in these groups. Sections 3,4 contain our main results, namely, the construction of Bethe s subalgebras in the Hecke and affine Hecke algebras. In particular, Theorem 1 in Section 4 describes the Hecke version of the Gaudin Hamiltonians. We treat the Bethe subalgebras obtained as a baxterization of the canonical free abelian subgroup in the corresponding braid group. In other words, we introduce a spectral parameter dependences in definition of the Jucys Murphy elements keeping the commutativity property of the deformed elements. A similar construction can be done for the Birman Murakami-Wenzl algebras, cyclotomic Hecke algebras and some other quotients of braid groups. In Section 5 we study the classical and Yangian limits of the Bethe subalgebras in the Hecke and affine Hecke algebras correspondingly. We thank S.Krivonos for valuable discussions. 2 Braid group Denote by S n the symmetric group on n letters, and by s i the simple transposition i, i + 1 for 1 i n 1. The well-known Moore Coxeter presentation of the symmetric group has the form s 1,..., s n 1 s 2 i = 1, s i s i+1 s i = s i+1 s i s i+1, s i s j = s j s i, if i j 2. Transpositions s ij := s i s i+1 s j 2 s j 1 s j 2 s i+1 s i, 1 i < j < j n, satisfy the following set of defining relations: s 2 ij = 1, s ij s kl = s kl s ij, if {i, j} {k, l} =, s ij s ik = s jk s ij = s ik s jk, s ik s ij = s ij s jk = s jk s ik, i < j < k. The Artin braid group on n strands B n is defined by generators σ 1,..., σ n 1 and relations σ i σ i+1 σ i = σ i+1 σ i σ i+1, 1 i n 2, σ i σ j = σ j σ i if i j 2. 1 Proposition 2.1 Let us introduce elements D i,j := σ j 1 σ j 2 σ i+1 σ 2 i σ i+1 σ j 2 σ j 1, where 1 i < j n. F i,j := σ n j σ n j+1... σ n i 1 σ 2 n i σ n i 1... σ n j+1 σ n j, 2

4 For example, D i,i+1 = σi 2, D i,i+2 = σ i+1 σi 2σ i+1, F i,i+1 = σn i 2, F i,i+2 = σ n i 1 σn i 2 σ n i 1, and so on. Then For each j = 3,..., n, the element D 1,j commutes with σ 1,..., σ j 2. The elements D i,i+1, D i,i+2,..., D i,n resp. F i,i+1, F i,i+2,..., F i,n 1 i n 1, generate a free abelian subgroup in B n. The elements D 1,2, D 2,3,..., D 1,n resp. F 1,2, F 2,3,..., F 1,n generate a maximal free abelian subgroup in B n. If n 3, the element D 1,j = F 1,j = σ 1 σ n 1 n 2 j n 2 j n generates the center of the braid group B n. D i,j D i,j+1 D j,j+1 = D j,j+1 D i,j+1 D i,j, if i < j. Consider the elements s := σ 1 σ 2 σ 1, t := σ 1 σ 2 in the braid group B 3. Then s 2 = t 3 and the element c := s 2 generates the center of the group B 3. Moreover, B 3 / c = P SL 2 Z, B 3 / c 2 = SL 2 Z. The affine Artin braid group Bn aff, is an extension of the Artin Braid group on n strands B n by the element τ subject to the set of crossing relations Proposition 2.2 The elements generate a free abelian subgroup in B aff n. σ 1 τ σ 1 τ = τ σ 1 τ σ 1, σ i τ = τ σ i for 2 i n 1. ˆD 1 := τ, ˆDj = σ j 1 ˆDj 1 σ j 1, 2 j n, Therefore, for a unital commutative algebra F, any quotient F [B n ]/J of the group algebra F [B n ] of the braid group B n resp. a quotient F [Bn aff ]/I of the affine braid group Bn aff group algebra F [Bn aff ] by a two-sided ideal J F [B n ] resp I F [Bn aff ] contains distinguish commutative subalgebra generated by the images of elements D 1,2,..., D 1,n 1 resp. ˆD1,..., ˆD n. It is well-known that the Hecke and affine Hecke algebras are certain quotients of the braid and affine braid groups correspondingly, see the next Section for details. In these cases the images of elements D 1,2,..., D 1,n 1 and those ˆD 1,..., ˆD n coincide with the Jucys Murphy elements in the Hecke and affine Hecke algebras correspondingly. Our main objective of the next Section is to construct an analogue of the Bethe subalgebras in the affine Hecke algebras. 3 Bethe subalgebras for affine Hecke algebra The Hecke algebra H M+1 q see, e.g., [5] and [7] is generated by invertible elements T i i = 1,..., M subject to the set of relations: T i T i+1 T i = T i+1 T i T i+1, T i T j = T j T i for i j ± 1. 2 Let x be a spectral parameter. We define Baxterized elements T 2 i 1 = q q 1 T i, 3 T i x := T i xt 1 i = 1 xt i + λx H M+1 q, 4 which in view of 2 and 3 solve the Yang-Baxter equation T i x T i 1 xz T i z = T i 1 z T i xz T i 1 x, 5 3

5 and satisfy relations T i xt i z = λt i xz + 1 x1 z, 6 T i 1 = λ, T i x = 1 x 1 z T iz + λ x z 1 z, 0 7 where λ := q q 1. Note that from 6 for Baxterized elements we have the condition T i x T i x 1 = q 1 x qq 1 x 1 q, which can be written as unitarity condition T i x T i x 1 = 1 for modified baxterized elements T i x = 1 q q 1 x T ix. 8 The affine Hecke algebra ĤM+1q see, e.g., Chapter 12.3 in [5] and [10] is an extension of the Hecke algebrah M+1 q by additional affine elements ŷ k k = 1,..., M + 1 subject to relations: ŷ k+1 = T k ŷ k T k, ŷ k ŷ j = ŷ j ŷ k, ŷ j T i = T i ŷ j j i, i The elements {ŷ k } form a commutative subalgebra in ĤM+1, while the symmetric functions in ŷ k form the center in ĤM+1. The Jucys Murphy elements {ŷ k } coincide with the images of elements ˆD 1,..., ˆD n considered in previous Section. Here and below we omit the dependence on q in the notations H M+1 q and Ĥ M+1 q of the Hecke algebras. The Ariki-Koike algebra [3],[4] H M+1 q, Q 1,..., Q m is the quotient of the affine Hecke algebra Ĥ M+1 by the characteristic identity where Q 1,..., Q m are parameters. Definition 3.1 ŷ 1 Q 1 ŷ 1 Q m = 0, 10 Let ξ n = ξ 1,..., ξ n be n parameters and y 1 x ĤM+1, define the elements x y n x; ξ n 1 = T n 1 ξ n 1 T 2 x ξ 2 T 1 x ξ 1 y 1 x T 1 xξ 1 T 2 xξ 2 T n 1 xξ n 1 = = T n 1 x ξ n 1 y n 1 x; ξ n 2 T n 1 xξ n 1, 11 which we call as baxterized Jucys Murphy elements. Proposition 3.1 Assume that the element y 1 x ĤM+1 in 11 is any local i.e., [y 1 x, T k ] = 0, k > 1 solution of the reflection equation T 1 x/z y 1 x T 1 x z y 1 z = y 1 z T 1 x z y 1 x T 1 x/z, 12 Then the elements 11 satisfy the reflection equation T n x/z y n x; ξ n 1 T n x z y n z; ξ n 1 = y n z; ξ n 1 T n x z y n x; ξ n 1 T n x/z, 13 Proof The case n = 1 of the equation 13 corresponds to our assumption that y 1 x satisfies the equation 12. The general case follows by induction using the definition 11 of elements y n x; ξ n 1. For example, in the case of the affine Hecke algebra, one can use the local solution see [8]: y 1 x = ŷ1 ξx, 14 ŷ 1 ξx 1 where ξ is a parameter. In the case of the Ariki-Koike algebra this rational solution is represented in the polynomial form by writing the characteristic identity 10 as 1 ŷ 1 ξx 1 = v 1 ŷ m v 2 ŷ m v m 1 ŷ 1 + v m, 4

6 where v 1,..., v m are functions of ξ, x, Q 1,..., Q m. Consider the following inclusions of the subalgebras Ĥ1 Ĥ2 ĤM+1: {ŷ 1 ; T 1,..., T n 1 } Ĥn Ĥn+1 {ŷ 1 ; T 1,..., T n 1, T n }. Define for the algebra ĤM+1 linear mappings Tr n+1 : Ĥ n+1 Ĥn, n = 1, 2,..., M, such that for all X, X Ĥn and Y Ĥn+1 we have Tr n+1 T ±1 n X T 1 n = Tr n X, Tr n+1 X Y X = X Tr n+1 Y X, Tr n Tr n+1 T n Y = Tr n Tr n+1 Y T n, Tr n+1 T n = 1, Tr 1 y k 1 = D k, Tr n+1 X = D 0 X, 15 where k Z and D k C\{0} are constants. Note that D 0 is independent of n and all D k can be considered as central elements for certain central extension ExtĤM+1 of Ĥ M+1. The elements D k generate an abelian subalgebra we denote this subalgebra Ĥ0 in ExtĤM+1. Using the properties 15 of the map Tr n+1 and relations 6, one can show Lemma 3.1 For all X Ĥn and x, z, the following identity is true: Tr n+1 T n x X T n z = 1 x 1 z Tr n X + λ 1 p x z X, 16 where T n x are Baxterized elements 4 and p = 1 λd 0 = 1 q q 1 Tr n+1 1. From eq. 16, for p x z = 1, we obtain the crossing-symmetry relation where F p x = p x 1 xp x 1. Tr n+1 T n x X T n 1/px = 1 F p x Tr X, 17 n Proposition 3.2 see also [8], [9]. Let y n x Ĥn be any solution of the RE 13. The operators τ n 1 x = Tr n y n x Ĥn 1, 18 form a commutative family of operators [ ] τ n 1 x, τ n 1 z = 0 x, z, 19 in the subalgebra Ĥn 1 ĤM+1. Proof. Using 15, 17 and 13 we find where F p x is defined in 17. τ n 1 x τ n 1 z = T r n y n x τ n 1 z = = F p x z Tr n yn x Tr n+1 Tn xz y n zt n pxz 1 = = F p x z Tr n Tr n+1 T 1 n x/z y n x T n xz y n z T n x/z T n pxz 1 = = F p x z Tr n Tr n+1 yn z T n xz y n x T n pxz 1 = = Tr n y n z τ n 1 x = τ n 1 z τ n 1 x, 5

7 Now we consider the operators 18, where solution y n x of the reflection equation is taken in the form 11: τ n x; ξ n = Tr n+1 y n+1 x; ξ n Ĥn 20 We stress that the elements 20 are nothing but the analogs of Sklyanin s transfer-matrices [12] and the coefficients in the expansion of τ n x; ξ n over the variable x for the homogeneous case ξ k = 1 are the Hamiltonians for the open Hecke chain models with nontrivial boundary conditions which was considered e.g. in [9]. Consider this expansion of τ n x; ξ n for inhomogeneous case: τ n x; ξ n = k= Φ k ξ n x k Ĥn. 21 According to the Proposition 3.2, for fixed parameters ξ n = ξ 1,..., ξ n, the elements Φ k ξ n generate a commutative subalgebra ˆB n ξ n Ĥn. These elements are interpreted as Hamiltonians for the inhomogeneous open Hecke chain models. Following [13] we call the subalgebras ˆB n ξ n as Bethe subalgebras of the affine Hecke algebra Ĥn. First we obtain more explicit form for the generating function of the elements Φ k ξ n ˆB n ξ n. For this we substitute the solution y n+1 x; ξ n of the reflection equation in the form 11 to the transfer-matrix operators 20. Using relation 16 we obtain τ n x; ξ n = Tr n+1 T n x ξ n T 2 x ξ 2 T 1 x ξ 1 y 1 x T 1 xξ 1 T 2 xξ 2 T n xξ n = ξ n xξn 1 x τ n 1 x; ξ n 1 + λ1 p x 2 y n x; ξ n 1 = = n ξ k xξ 1 k x τ n 2 x; ξ n 2 + k=n 1 +λ1 p x 2 ξ n xξn 1 xy n 1 x; ξ n 2 + y n x; ξ n 1 = = n = ξ k xξ 1 k x τ 0 x + λ1 p x 2 J n x; ξ n, k=1 where the element τ 0 x = Tr 1 y 1 x Ĥ0 by definition is the central element in Ĥn. In equation 22 we have introduced the notation J n x; ξ n for new explicit generating function of the commutative elements Φ k ξ n Ĥn: J n x; ξ n = n d n r x; ξ y r x; ξ r 1 = = n r=1 r=1 d n r x; ξ T r 1 x ξ r 1 T 1 x ξ 1 y 1 x T 1 xξ 1 T r 1 xξ r 1. where we have used the concise notation d n r x; ξ for coefficient functions d n r x; ξ n = n k=r+1 x ξ k x ξ 1 k = n k=r+1 = ρk x + x 2, 24 ρ k = ξ k + ξ 1 k. 25 Remark. From Yang-Baxter equation 5 and reflection equation 13 we deduce τ n x; ξ n T k ξ k+1 /ξ k = T k ξ k+1 /ξ k τ n x; s k ξ n, k = 1,..., n 1, τ n x; ξ n ȳ k ξ k ; ξ k 1 = ȳ k ξ k ; ξ k 1 τ n x; I k ξ n, k = 1,..., n, 26 where s k ξ n ξ 1,..., ξ k 1, ξ k+1, ξ k, ξ k+2,... ξ n, i.e. s k is the transposition of two parameters ξ k and ξ k+1, and I k ξ n ξ 1,..., ξ k 1, ξ 1 k, ξ k+1,... ξ n. It means that the Bethe subalgebras generated by transfer-matrix type elements τ n x; ξ n, τ n x; s k ξ n and τ n x; I k ξ n are equivalent. It is clear that the symmetry 26 is also valid for the generating functions 23. 6

8 4 Bethe subalgebra for the Hecke algebra The Hecke algebra H n is the quotient of the affine Hecke algebra Ĥn by the relation ŷ 1 = 1. Thus, one can obtain the generating function for the elements of Bethe subalgebra of usual Hecke algebra H n if we substitute into 23 the trivial solution y 1 x = 1 of the reflection equation. Then to simplify the function 23 for the case of H n we first transform the elements y r x; ξ r 1 given in 11. For this we use relations 4 and identities T k x ξ k T k xξ k = λt k x 2 + x ξ k x ξ 1 k = = λ1 x 2 T k + [λ 2 x 2 + x ξ k x ξ 1 k ], 27 which can be deduced from 6. For y 1 x = 1, applying 27 many times, we obtain new representation for the elements 11: y n+1 x; ξ n = T n x ξ n T 2 x ξ 2 T 1 x ξ 1 T 1 xξ 1 T 2 xξ 2 T n xξ n = = T n x ξ n T 2 x ξ 2 λ1 x 2 T 1 + [λ 2 x 2 + x ξ 1 x ξ 1 1 ] T 2 xξ 2 T n xξ n = = λ1 x 2 ỹ n+1 x; ξ n + c n+1 x; ρ n, 28 where ỹ n x; ξ n 1 = n 1 k=1 c k x; ρ k 1 T n 1 x ξ n 1 T k+1 x ξ k+1 T k T k+1 xξ k+1 T n 1 xξ n 1, 29 parameters ρ k were defined in 25 and for the coefficient functions c k we have c 1 = 1, c k x; ρ k 1 k 1 j=1 1 xρj λ 2 x 2 k 2. Using representation 28 it is convenient to redefine the generating function 23 once again J n x; ξ n = n d n r x; ξ r=1 λ1 x 2 ỹ r x; ξ r 1 + c r x; ρ r 1 = = λ1 x 2 J n x; ξ n + n d n r x; ξ c r x; ρ r 1. r=1 30 For new function J n x; ξ n which generate elements of the Bethe subalgebra B n ξ n H n we obtain the recurrent relations J 2 = ỹ 2 = T 1, Jn = 1 xρ n + x 2 J n n 1 + ỹ n = d n kx; ρ ỹ k, 31 where coefficients d n k x; ρ were defined in 24. Hamiltonian of our integrable system, namely, x J n x; ξ n x=0 = 1 i<j n ρ i + ρ j T ij + λ k=2 Using the recurrence relation 30 we can compute the 1 i<j<k<n ξ 1 j T ij T jk + ξ j T jk T ij. At the end of this Section we present the explicit expressions for first few elements ỹ n and J n for n 2: ỹ 2 = T 1, ỹ 3 = T 2 x ξ 2 T 1 T 2 xξ xρ λ 2 x 2 T 2, ỹ 4 = T 3 x ξ 3 T 2 x ξ 2 T 1 T 2 xξ 2 T 3 xξ xρ λ 2 x 2 T 3 x ξ 3 T 2 T 3 xξ 3 +[1 xρ λ 2 x 2 ][1 xρ λ 2 x 2 ]T 3, 32 ỹ 5 = T 4 x ξ 4...T 2 x ξ 2 T 1 T 2 xξ 2...T 4 xξ 4 + c 2 x; ρ 1 T 4 x ξ 4 T 3 x ξ 3 T 2 T 3 xξ 3 T 4 xξ 4 + +c 3 x; ρ 1, ρ 2 T 4 x ξ 4 T 3 T 4 xξ 4 + c 4 x; ρ 1, ρ 2, ρ 3 T 4. 7

9 J 2 = T 1, J3 = 1 xρ λ 2 x 2 T xρ λ 2 x 2 T xρ 2 + x 2 T 2 T 1 T 2 + λxξ 2 x T 2 T 1 + λx ξ 1 2 x T 1 T 2 = T 1 + T 2 + T 1 T 2 T 1 ρ 3 T 1 + ρ 2 T 1 T 2 T 1 + ρ 1 T 2 λξ 2 T 2 T 1 λξ2 1 T 1T 2 x+ 1 + λ 2 T 1 + T 2 + T 1 T 2 T 1 λt 1 T 2 + T 2 T 1 + λ T 1 T 2 T 1 x Therefore the Bethe subalgebra B 3 ξ 3 is generated by the central elements C 1 = T 1 + T 2 + T 13 and C 2 = T 1 T 2 + T 2 T 1 + λt 13, and the element D = ρ 3 T 1 + ρ 2 T 13 + ρ 1 T 2 λξ 1 2 T 2T 1 λξ 2 T 1 T 2. Here we used notation T 13 := T 1 T 2 T 1. Now let us introduce the following elements One can check that ξ θ 1 := θ 3 D ρ 1 C 1 1 = ρ 2 ρ 1 ρ 3 ρ 1 = T 1 + T 13 λξ 2 T 2 T 1 + λξ2 1 T 1T 2, ρ 2 ρ 1 ρ 3 ρ 1 ρ 2 ρ 1 ρ 3 ρ 1 ξ θ 2 := θ 3 D ρ 2 C 1 1 = ρ 1 ρ 2 ρ 3 ρ 2 = T 1 + T 2 λξ 2 T 2 T 1 + λξ2 1 T 1T 2, ρ 1 ρ 2 ρ 3 ρ 2 ρ 1 ρ 2 ρ 3 ρ 2 ξ θ 3 := θ 3 D ρ 3 C 1 1 = ρ 1 ρ 3 ρ 2 ρ 3 = T 2 + T 13 λξ 2 T 2 T 1 + λξ2 1 T 1T 2. ρ 2 ρ 3 ρ 1 ρ 3 ρ 2 ρ 3 ρ 1 ρ 3 θ 1 + θ 2 + θ 3 = 0, ρ 1 θ 1 + ρ 2 θ 2 + ρ 3 θ 3 = C 1, λ 2 + 3C 2 = C 2 1 2λC 1 3, and the elements θ ξ 3 1, θ ξ 3 2, θ ξ 3 3 pairwise commute and generate the Bethe subalgebra B 3 ξ 3. Our goal is to show that a similar set of generators exist for the Bethe algebra B n ξ n for arbitrary n. To state our main result of this Section we need to introduce a bit of notation. First of all, for a pair of integers 1 i < j n let us introduce the elements T ij := T j 1 T i+1 T i T i+1 T j 1, 1 i < j n. ξ Now let B [1, 2,..., n] be a subset, define inductively the elements T B := T nb as follows T {b} = 0, T {a < b} = T ab, T {a < b < c <... < d} = ξ a T {b < c <... < d} T ab + ξa 1 T ab T {b < c <... < d}. For example, T {a < b < c < d} = ξ a ξ b T cd T bc T ab +ξ a ξ 1 b T bc T cd T ab +ξa 1 ξ b T ab T cd T bc +ξa 1 ξ 1 b T ab T bc T cd. Using the notation introduced above, let us define the following elements Theorem 1 The elements satisfy the following properties θ ξ n θ ξ n n = 0, θ ξ n a = B [1,...,n] a B λ B 2 b B b a T B, a = 1,..., n. ρ a ρ b θ ξ n a, a = 1,..., n mutually commute, generate the Bethe subalgebra B n ξ n and n j=1 ξ jθ ξ n j = 1 i<j n T ij, the elementary symmetric polynomials e j θ ξ n 1,..., θ ξ n n, j = 2,..., n, generate the center of the Hecke algebra H n, Clearly, θ ξ n a = b a T ab ρ a ρ b + λ.... In other words the elements {θ ξ n a, a = 1,..., n} are a lift of the Gaudin elements {g a ξ n := j a ρ a ρ j 1 s aj, a = 1,..., n} from the group algebra C[S n ] of the symmetric group S n to the Hecke algebra H n C. 8

10 ξ Using the recurrence relation 30 we can compute the Hamiltonian H n of our integrable model, namely, ξ H n = x J n x; ξ n x=0 = ρ i + ρ j T ij + λ B 2 T B. 1 i<j n B [1,...,n] B 3 Finally we remark that the example above shows that the set of all Bethe s subalgebras in the Hecke algebra H n does not coincide with the set of all maximal commutative subalgebras in H n, if n 3. 5 Symmetric group limit and Gaudin model. Let us consider the special classical limit when q 1 while parameters x and ξ k are fixed. For q = 1 or λ = 0 in view of 3, 27 the Hecke algebra H M+1 is degenerated to the symmetric group algebra S M+1, i.e. T k = T 1 k = s k,k+1 = s k are elementary transpositions of k and k +1. In this limit we have T k x = 1 xs k and formulas 29 and 32 are simplified ỹ 2 = s 1, ỹ 3 = [1 xρ 2 + x 2 ]s 2 s 1 s 2 + [1 xρ 1 + x 2 ]s 2, ỹ 4 = [1 xρ 2 + x 2 ][1 xρ 3 + x 2 ]s 3 s 2 s 1 s 2 s 3 + [1 xρ 1 + x 2 ][1 xρ 3 + x 2 ]s 3 s 2 s [1 xρ 1 + x 2 ][1 xρ 2 + x 2 ]s 3,,..., k 1 k 1 ỹ k = [1 xρ k + x 2 s ] j,k [1 xρ j +x 2 ], 35 m=1 where s j,k = s k 1...s j+1 s j s j+1...s k 1 are transpositions in S M+1, i.e. s j,k = s k,j. Substitution of 35 into 31 gives = n k 1 n k=2 j=1 m=1 m j,k After the renormalization J n x; ρ n = n n k=2 m=k+1 1 xρm + x 2 s j,k = n m=1 J n x; ρ n J nx; ρ n = j=1 1 xρm + x 2 ỹ k = 1 xρm + x 2 n k>j x 2 Jn x; ρ n n 1 xρ m + x 2 m=1 s j,k [1 xρ j +x 2 ][1 xρ k +x 2 ], and change of variables u = x + 1/x we obtain the generating function for Bethe subalgebra of symmetric group algebra S n in the form J nx; ρ n = x2 2 n k,j=1 = 1 2 s j,k [1 xρ j +x 2 ][1 xρ k +x 2 ] = 1 2 n k,j=1 n k,j=1 s j,k 1 ρ j ρ k [u ρ j ] 1 [u ρ k ] s j,k [u ρ j ][u ρ k ] = 1 2 = n k,j=1 n k,j=1 s j,k ρ j ρ k ρ j ρ k [u ρ j ][u ρ k ] = s j,k ρ j ρ k 1 [u ρ j ]. 36 The commuting Hamiltonians H [n] j for Gaudin model are obtained from 36 as residues for u ρ j : j = res J nx; ρ n = u=ρj H [n] n k=1 s j,k ρ j ρ k. The right hand side of 36, after the change of variables ρ k z k, can be represented in the form 1 n s j,k [u z m ], [u z m ] z j z k m k,j=1 m m j 9

11 and the expression in the brackets is related up to the shift by a scalar function to the generating function u of the Bethe subalgebra which was presented in [13] see Remark after the Theorem 4.3 in [13]. Φ [n] 2 Remark 1. There is another semi-classical limit q 1 for the constructions considered above which is called Yangian limit. In this case we have to consider substitution and take the limit h 0. Then we have x = q 2u, ξ j = q 2z j, q = e h, x = 1 2hu +..., λ = 2h +..., T k x = 2hus k , T k = s k + h s k +..., T k 2 = h s k +..., s k s k + s k s k = 2s k, 1 ρ j x + x 2 = 1 ξ j x1 x ξ j = 4h 2 u 2 zj , 1 ρ j x λ 2 x 2 = 4h 2 u 2 zj , ŷ 1 = 1 + 2h l , ŷ k = 1 + 2h l k +..., T k x ξ k = 2hu z k s k + 1, T k xξ k = 2hu + z k s k + 1, 37 where s k S M+1 are elementary transpositions, dots... means the higher orders in h and we have used notation l 1 = l 1, l k+1 = s k + s k l ks k, 38 for commutative set of elements {l k }. Moreover it is easy to see that s kl k = l k+1 s k 1, k = 1,..., M, and the elements {s k, l 1, l k+1, k = 1,..., M} generate the degenerate affine Hecke algebra [6]. We call the limit 37 as Yangian limit since for the modified Baxterized element T k x given in 8 we obtain see the first line in 37: T k x = us k + 1 u h In the matrix representation ρ of the symmetric group S M+1 we have ρs k = P kk+1, where P kk+1 are permutation matrices which act in the products of n-dimensional vector spaces V M+1 and permute factors with numbers k and k + 1. In this representation the Baxterized element 39 is nothing but Yangian R-matrix which is used for the definition of the gln-type Yangian. Remark 2. Now we show that the elements l 1 commute as following: [l 1, l 2 ] = s 1 l 1 l 1 s 1, 40 where we denoted l 2 := s 1 l 1 s 1. It follows from the relation [l 1, l 2] = 0, but we check this identity directly. First, we start with the relation see 9 in the affine Hecke algebra and change here the affine generators ŷ 1 K 1 : The substitution of 42 into 41 and using of 3 gives for q q 1 0: T 1 ŷ 1 T 1 ŷ 1 = ŷ 1 T 1 ŷ 1 T 1, 41 ŷ 1 = 1 + q q 1 K T 1 K 1 T 1 K 1 K 1 T 1 K 1 T 1 = K 1 T 1 T 1 K Now we take here the Yangian limit 37, i.e. we put K 1 = l 1 + h... and T 1 = s 1 + h.... Then, in the zero order approximation in h, eq. 43 is converted to 40. In the matrix representation see the end of the Remark 1. when ρs k = P kk+1, ρl 1 = L 1 1 EndV }{{} M+1 M 10

12 and L = L ij i,j=1,...,n is a n n matrix with noncommutative entries, relation 40 gives the defining relations for generators L ij of gln Lie algebra. Remark 3. The using of the Yangian limit 37 in 23 will give the generating function for generators of Bethe subalgebras for degenerate affine Hecke algebra. Here we present first few cases for n = 1, 2, 3: J 1 x, ξ 1 = y 1 x = ŷ1 ξ x ŷ 1 ξ x 1 h 0 l 1 + z + u l 1 + z u + h..., J 2 x, ξ 2 = x 2 ρ 2 x + 1 y 1 x + T 1 x/ξ 1 y 1 x T 1 xξ 1 4h 2 u 2 z2 2 l 1 + z + u l 1 + z u + u z 1s l 1 + z + u l 1 + z u u + z 1s h 3..., J 3 x, ξ 3 = x 2 ρ 3 x + 1 x 2 ρ 2 x + 1 y 1 x + T 1 x/ξ 1 y 1 x T 1 xξ 1 + h 0 +T 2 x/ξ 2 T 1 x/ξ 1 y 1 x T 1 xξ 1 T 2 xξ 2 2h 4 u 2 z3u 2 2 z2 2 l 1 + z + u l 1 + z u + u2 z3u 2 z 1 s l 1 + z + u h 0 l 1 + z u u + z 1s h 5..., +u z 2 s 2 + 1u z 1 s l 1 + z + u l 1 + z u u + z 1s 1 + 1u + z 2 s where we have used 14 and taken ξ = q 2z. The expansion of elements J n x, ξ n over the spectral h 0 parameter u gives the set of commuting elements Φ k z 1,..., z n cf. eq. 21: J n x, ξ n = 2h 2n 1 h 0 k=0 Φ k z 1,..., z n u k + h 2n The elements Φ k z 1,..., z n are generators of the Bethe subalgebras in the degenerate affine Hecke algebra {s m, l 1, m = 1,..., n 1} and Φ k z 1,..., z n can be interpreted as Hamiltonians for inhomogeneous open spin chains. In particular for homogeneous case z m = 0, z 0 of such open spin chain we obtain the local Hamiltonian Hz: u J n q 2u, 1,..., 1 h 0,u=0 References h 2n 1 Hz + h 2n 1..., Hz = n 1 m=1 s m + 1. l 1 + z [1] M.Gaudin, Diagonalisation dùne classe d Hamiltoniens de spin, J. Physique 37, No ; M.Gaudin, La fonction d onte de Bethe, Masson, Paris, 1983 in French; Mir, Moscow, 1987 in Russian. [2] E.K.Sklyanin, Separation of variables in the Gaudin model, Zap. Nauch. Sem. LOMI, Vol [3] S. Ariki and K. Koike, A Hecke algebra of Z/rZ S n and construction of its irreducible representations, Adv. in Math [4] M. Brouè and G. Malle, Zyklotomische Hecke algebren, Asterisque [5] V. Chari and A. Pressley, A guide to quantum groups, Cambrige Univ. Press [6] V.G.Drinfeld, Degenerate affine Hecke algebra and Yangians, Func. Anal. Appl. 20 No [7] V.F.R. Jones, Hecke algebra representations of braid groups and link polynomials, Annals Math ,

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