Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model

In this paper we construct the quantum spectral curve for the quantum dynamical elliptic gl(n) Gaudin model. We realize it considering a commutative family corresponding to the Felder's elliptic quantum group and taking the appropriate limit. The approach of Manin matrices here suits well to the problem of constructing the generation function of commuting elements which plays an important role in SoV and Langlands concept.


Introduction
The Gaudin model plays intriguingly important role in the modern mathematical physics as like as in purely mathematical subjects as the Geometric Langlands correspondence. In fact, it is shown that the separation of variables of the quantum rational Gaudin model is equivalent to the categorical part of the geometric Langlands correspondence over the rational curve with punctures over C. Its physical importance lies in the condense matter field, this system provides an example of the spin interacting magnetic chain. This paper deals with a new useful formalism applicable to the dynamical elliptic case of the Gaudin model. This case provides an interpretation for the Langlands correspondence over an elliptic curve. From the physical point of view this case is responsible for the periodic spin chains.
The main strategy of the paper goes throw the classical ideas of [1,2]. We start with the formalism of L-operators corresponding to the Felder "elliptic quantum groups" E τ, (gl n ) [3,4]. Using the RLL-relations we construct the Bethe elliptic commutative subalgebra from these L-operators. To make the proof of commutativity more apparent we use technique of Manin matrices developed in [5,6]. For some particular cases this commutative subalgebra can be found in [7] in a slightly different form. It is worth to mention the detailed work [8] described the centre of E τ, (sl n ). We refer also to the rational version in [9,10], to the trigonometric dynamical case in [11,12]. Then we degenerate these families to obtain a commutative family for the elliptic gl n Gaudin model 1 . To do it we consider a quantum characteristic polynomial, which has a suitable form for the degeneration. This approach extends the method applied in the rational case in [13].
The article is organized as follows: in Subsection 2.1 we consider the dynamical elliptic Loperators defined by the RLL relation with the Felder elliptic R-matrix. Then in Subsection 2.2 we define Manin matrices following [5,6] and demonstrate that the L-operator multiplied by some shift operator is a Manin matrix. In Subsection 2.3 we consider a fusion procedure of the L-operators and construct a commutative family as traces of the "fused" L-operators. In Subsection 2.4 we show that a characteristic polynomial of the considered Manin matrix generates the obtained commutative family. Subsection 2.5 is devoted to commutative families corresponding to the traces of powers of Manin matrices related to the families constructed in 2.3 via Newton identities. In Subsection 2.6 we consider briefly a trigonometric degeneration of the characteristic polynomial and of the commutative families. We discuss their connection with the Hopf algebra U q ( gl n ).
In Subsection 3.1 we obtain a characteristic polynomial which generates a commutative family by passing to the limit → 0. In Subsection 3.2 we consider L-operators that gives commutative families for the elliptic gl n Gaudin model. Subsection 3.3 is devoted to a twist relating these L-operators with standard L-operators for the elliptic gl n Gaudin model. We consider sl 2 case as an example in Subsection 3.4. In Subsection 3.5 we present the result of application of the Newton identities to the elliptic gl n Gaudin model.

Elliptic quantum groups and commutative families
Here we use the gl n dynamical RLL relation to construct the commutative families for dynamical L-operators. We consider an arbitrary L-operator that defines representation of the "elliptic quantum group" E τ, (gl n ) introduced in [3,4]. The commutativity in the dynamical case is understood modulo Cartan elements. Restricting these "commutative" families to the space annihilating by Cartan elements one obtains a corresponding integrable system.
We use the matrix (or "Leningrad") notations. Let T = j t j · a 1,j ⊗ · · · ⊗ a N,j be a tensor over a ring R (a complex algebra or the field C), where t j ∈ R and a i,j belong to a space End C n . Then the tensor T (k 1 ,...,k N ) is the following element of R ⊗ (End C n ) ⊗M for some M N : where each element a i,j is placed to the k i -th tensor factor, the numbers k i are pairwise different and 1 k i M . (The condition k 1 < · · · < k N is not implied.) We also use the following notation. Let F (λ) = F (λ 1 , . . . , λ n ) be a function of n parameters λ k taking values in an algebra A: that is F : C n → A. Then we define F (λ + P ) = F (λ 1 + P 1 , . . . , λ n + P n ) for some P = (P 1 , . . . , P n ), P k ∈ A. We omit here the convergence question considering only such situations where this is the case.

Elliptic dynamic RLL-relations
First of all we introduce a notion of a dynamical elliptic L-operator corresponding to the Felder R-matrix. Using a lemma about the products of these L-operators and the particular choice of the L-operator we prove the commutativity of the family of operators under consideration. Let {e i } be a standard basis of C n and {E ij } be a standard basis of End C n , that is E ij e k = δ j k e i . In [3,4] Felder introduce the following element of End C n ⊗ End C n depending meromorphically on the spectral parameter u and n dynamic parameters λ 1 , . . . , λ n : and the relations ii , (2.5) We also should comment that we always mean by λ the vector λ 1 , . . . , λ n and the expression of the type λ + E (s) as the argument of (2.1) with ii . The relation (2.5) is obvious. The relation (2.6) follows from (2.5). Let R be a C[[ ]]-algebra, L(u; λ) be an invertible n × n matrix over R depending on the spectral parameter u and n dynamical parameters λ 1 , . . . , λ n . Let h 1 , . . . , h n be a set of some pairwise commuting elements of R . If the matrix L(u; λ) satisfies dynamical RLL-relation then it is called a dynamical elliptic L-operator with Cartan elements h k . The argument λ + h is always meant in the sense of (2.1) with P i = h i . Let us introduce an equivalent but more symmetric form of RLL relations. For each Loperator we introduce the following operator (similar to an operator introduced in [11]): The equation (2.7) can be rewritten in terms of this operator: (2.10) Lemma 1. If L 1 (u; λ) ∈ End(C n ) ⊗ R 1 and L 2 (u; λ) ∈ End(C n ) ⊗ R 2 are two dynamical elliptic L-operators subject to the two sets of Cartan elements: h 1 = (h 1 1 , . . . , h 1 n ) and h 2 = (h 2 1 , . . . , h 2 n ) then their matrix product L 2 (u; λ)L 1 (u; λ+ h 2 ) ∈ End(C n )⊗R 1 ⊗R 2 is also a dynamical elliptic L-operator with Cartan elements h = h 1 + h 2 = (h 1 1 + h 2 1 , . . . , h 1 n + h 2 n ). Thus, if L 1 (u; λ), . . . , L m (u; λ) are dynamical elliptic L-operators with Cartan elements h 1 , . . . , h m then the matrix Remark 1. The arrow in the product means the order of the factors with respect to the growing index value: the expression ← − The basic example of the dynamical elliptic L-operator is the dynamical Felder R-matrix: L(u) = R(u − v; λ). In this case the second space End(C n ) plays the role of the algebra R.
Here v is a fixed complex number and the Cartan elements are h k = E (2) kk . Lemma 1 allows to generalize this example: let v 1 , . . . , v m be fixed numbers, then the matrix The relation (2.4) gives another important example of the dynamical elliptic L-operator -the matrix L(u) = R (21) (v − u) −1 with the second space End C n considering as R (the number v is fixed), or equivalently, the matrix L(v) = R (12) (u − v) −1 with the first space End C n considering as R. Let us introduce the notation A more general class of the dynamical elliptic L-operators associated with small elliptic quantum group e τ, (gl n ) was constructed in the work [14]. This is an C[[ ]]((λ 1 , . . . , λ n ))-algebra generated by the elementst ij and h k satisfying the commutation relations and the elements h 1 , . . . , h k , λ 1 , . . ., λ k commute with each other. The authors of [14] consider the matrix T (−u) with the entries Representing this matrix in the form we obtain a dynamical elliptic L-operator L 0 (u; λ) over the algebra commute with h i and do not commute witht ij .

Manin matrices and L-operators
The RLL relations allow to construct the Manin matrices and q-Manin matrices investigated in [5,6,15] starting from the L-operators. In particular, the dynamical RLL relation with Felder R matrix leads to the Manin matrices (here q = 1). We use the properties of these matrices to prove the commutativity of some function familyt m (u) which will be constructed in the next subsection. We shall suppose in what follows that all matrix entries belong to some non-commutative ring which satisfies basically the conditions described precisely in [5,6,15]. Let S m be the symmetric group and π : S n → End(C n ) ⊗m be its standard representation:
Proof . Substituting u = − to (2.2) we obtain the formula (12) . Consider the relation (2.10) at u − v = − . The multiplication of this relation by A (12) from the right does not change its right hand side, hence this does not change its left hand side: Multiplying (2.16) by e 2 ∂ ∂u from the right we obtain Let us note that the matrix (2.15) is related to A (12) by the formula B(λ)R(− ; λ) = A (12) , where So, multiplying (2.17) by B(λ) from the left one obtains (2.13).
Lemma 2 ( [5,6]). If M is a Manin matrix invertible from the left and from the right then its inverse M −1 is also a Manin matrix.
In particular, the matrix inverse to (2.14) having the form is a Manin matrix.
Proof . The idea of the proof of (2.19) is the following. It is sufficient to prove that the left hand side does not change if we multiply it by (−1) σ i,i+1 π(σ i,i+1 ) from the right, where σ i,i+1 is an elementary permutation. But the last fact follows from (2.13). (See details in [15].) Multiplying (2.19) by (−1) σ lst π(σ lst ) from the both sides, where σ lst is a longest permutation, one yields the relation (2.20).
Proof . The relations (2.21) follows from the formula (2.19) for Manin matrix (2.14): Let us consider the following matrix Taking into account (2.5) we obtain the relations R (ij) R (ab) = R (ab) R (ij) , where 1 i < a m, m + 1 j < b N . Using it we derive the formula to the formulae (2.21) and (2.22) (with m = 0, N = m) respectively, we obtain

Commutative families
The integrals of motion for an integrable system related with an L-operator are obtained often as coefficients of a decomposition for some operator functions. These functions are constructed as traces of tensor products of L-operators multiplied by the "alternator" A [0,m] . In the dynamical case one should consider the tensor product of the operators (2.9). Now we use the facts proved below to establish the commutativity of these operator functions. Let us first fix the dynamical elliptic L-operator with Cartan elements h k and assign the notation L(u; λ) to this L-operator. Its values at points u belong to the algebra End C n ⊗ R.
Introduce the following operators where m < N . Using the dynamical RLL relations (2.7) and taking into account (2.6) we derive the commutation relations for them: in this relation we obtain the main relation which we need to construct a commutative family: . . , λ n )) be a completed space of functions. The operators D λ acts on the space A n ⊗ C n , so that the operators L [a,b] (u; λ) act from the space Consider the subalgebra h ⊂ R ⊂ A n generated by elements h k and its normalizer in A n : Let us note that A n h is a two-side ideal in N n .
where the trace is implied over all m spaces C n . They commute with Cartan elements h k : (2.30) Hence they take values in the subalgebra N n . And they pair-wise commute modulo the ideal A n h ⊂ N n : Proof . The relation (2.30) follows from the formulae and from the trace periodicity. Let us stress that the commutativity (2.31) modulo A n h is a corollary of the formula where N = m + s and the trace is considered over all N spaces C n . This formula can be deduced as follows. Consider the left hand side of (2.33). Applying the relation (2.29) we can rewrite it in the form Note that by virtue of the formula (2.32) the relations (2.27), (2.28) with the shifted dynamical where the matrix Y (λ) does not contain the shift operators. Using the formula where the matrix X(λ) does not contain the shift operators and elements of the ring R, one can rewrite (2.35) as the right hand side of (2.33).
Thus we obtain the family of functions t m (u) with values in the algebra N n . Their images by the canonical homomorphism N n → N n /A n h commute and we obtain the commutative family of functions in the algebra N n /A n h. Decomposing these functions in some basis functions φ m,j (u) one yields a commutative family of elementsÎ m,j of this algebra: t m (u) = jÎ m,j φ k,j (u) mod A n h. In this way one can construct an integrable system corresponding to a given Loperator.

Characteristic polynomial
The functions t m (u) can be gathered to a generating function called the quantum characteristic polynomial. This polynomial is defined in terms of determinants of Manin matrices. Considering the Gaudin degeneration we shall calculate the degeneration of the characteristic polynomial, although the degeneration of the functions t m (u) can not be obtained explicitly.
Introduce the following notion of a determinant for a matrix with non-commutative entries. Let M be an arbitrary n × n matrix, define its determinant by the formula

Newton identities and quantum powers
In the theory of classical integrable systems commutative families are usually provided by traces of powers of a classical L-operator. Their quantization can be presented as traces of deformed powers of the corresponding quantum L-operators. These deformed powers are called quantum powers of L-operator. For rational L-operators these quantum powers were described in details in [5]. The main tool to obtain the quantum powers are Newton identities for Manin matrices. They allow to express the quantum powers of L-operators through the coefficients of the characteristic polynomial.

Trigonometric limit
Up to now we have considered L-operators satisfying dynamical RLL-relations with a certain dynamical elliptic R-matrix. But the presented technique is sufficiently universal to be applied to many other cases. This approach is directly generalized to the case of arbitrary dynamical R-matrix such that R(− ; λ) = B(λ)A (12) for some invertible matrix B(λ). The corresponding L-operators give us a family t m (u) commuting modulo A n h. In particular, it works for the trigonometric and rational degenerations of the Felder R-matrix. Using the same scheme for their non-dynamical limits we obtain commuting functions t m (u) (modulo 0).
Here we briefly discuss the dynamical and non-dynamical trigonometric limits. The Felder R-matrix in the limit τ → i∞ takes the form where z = e 2πiu , w = e 2πiv , q = e πi , µ kj = e 2πiλ kj . An L-operator satisfying the relations (2.7) and (2.8) with this R-matrix -a dynamical trigonometric L-operator -defines the commutative family t m (u) (modulo corresponding A n h) by means of the formula (2.37), for instance. One can also consider the commuting traces of quantum powers in the same way as in the Subsection 2.5. Consider the limit λ k − λ k+1 → −i∞, that is µ ij → ∞ for i < j and µ ij → 0 for i > j. The limit of the matrix (2.42) is a non-dynamical trigonometric R-matrix Let L(z) be an L-operator satisfying the usual RLL-relations with this matrix: The matrix M = L(z)q 2z ∂ ∂z is a Manin matrix. The corresponding characteristic polynomial det(1 − L(u)q 2z ∂ ∂z ) = This is a standard R-matrix for the quasi-triangular Hopf algebra U q ( gl n ). This algebra can be described by two L-operators L ± (z) ∈ End C n ⊗ U q ( gl n ) satisfying RLL-relation with the R-matrix (2.44). Each L-operator L(z) ∈ End C n ⊗ R satisfying RLL-relation with this Rmatrix defines a homomorphism from the certain Hopf subalgebra B ⊂ U q ( gl n ) to R. 2 The R-matrix (2.44) considered as an L-operator defines a representation π w : B → End C n . The 2 The algebra B is the algebra described by one of those L-operators, for example, by e L + (z). In this case the homomorphism is defined by the formula e L + ij (z) → e Lij (z).
subalgebra B contains elementsĥ 1 , . . . ,ĥ n such that π v (ĥ k ) = E kk , ε(ĥ k ) = 0, ∆(ĥ k ) =ĥ k ⊗ 1 + 1 ⊗ĥ k , where ε and ∆ are co-unity and co-multiplication of U q ( gl n ). The element satisfies the cocycle condition (the Drinfeld equation) F (12) (∆ ⊗ id)(F) = F (23) (id ⊗∆)(F), the condition (ε⊗id)F = (id ⊗ε)F = 1 and, therefore, defines a twist between two co-multiplications of the algebra U q ( gl n ), the standard one ∆ and twisted one ∆ F defined as ∆ F (x) = F∆(x)F −1 . Let φ : B → R is the homomorphism defined by a given L-operator L(z). Define the matrix where h k = φ(ĥ k ). From the theory of quasi-triangular Hopf algebras it follows that the twisted L-operator L(z) = G L(z)G satisfies RLL-relations with the matrix (2.43). This means that each L-operator L(z) defines the commutative family t m (u) by means of the characteristic polynomial for the Manin matrix M = G L(z)Gq 2z ∂ ∂z . Let us also remark that the commutative families for L(z) can be constructed without twisting. In this case one should use some generalization of the notion of Manin matrices. The matrix M = L(z)q 2z ∂ ∂z belongs to such kind of generalization called q-Manin matrices (see [15] for details). Nevertheless the approach of q-Manin matrices is a little more complicated. Let us also remark that the same situation occurs for the U q,p ( gl n ) L-operators considered in [16], there exist a dynamical elliptic analogue of the twist F relating the U q,p ( gl n ) R-matrix with the Felder R-matrix. So we expect that there exists a generalization of the q-Manin matrices corresponding to the dynamical elliptic quantum group U q,p ( gl n ). 3

Characteristic polynomial for elliptic Gaudin model
We consider a degeneration of the dynamical elliptic RLL relations: → 0. In particular, this degeneration describes the dynamical elliptic gl n Gaudin model. To relate this degenerated Loperator with the elliptic Gaudin L-operator in the standard formulation we introduce a twist of this L-operator. Degenerating the commutative family obtained in the previous section we yield a commutative family for the elliptic Gaudin model. We see that the obtained results generalize the elliptic sl 2 Gaudin model case investigated in [18,19].

→ 0 degeneration
Here we consider the degeneration of the previous section formulae obtained in the limit → 0: we compare the coefficients at the minimal degree of that does not lead to a trivial identity.
In the degeneration of the dynamical RLL-relation, for example, we consider 2 . And the determinant (2.37) is proportional to the n-th power of . We avoid using the term the classical degeneration because we don't want to mismatch the considering degeneration with the classical mechanics degeneration. Suppose that L(u; λ) is a dynamical elliptic L-operator of the form L(u; λ) = 1 + L(u; λ) + o( ). (3.1) where the entries of L(u; λ) belongs to the algebra R 0 = R/ R. The matrix L(u; λ) is called dynamical classical elliptic L-operator. It satisfies the dynamical classical rLL-relations with the dynamical classical elliptic r-matrix The matrix (3.3) is related with the Felder R-matrix (2.2) via the formula The relation (3.2) follows directly from the relation (2.7). In turn, the relation (2.8) implies Let us remark that two different Lie bialgebras described by the relations (3.2) and (3.4) are considered in great details for the sl 2 -case in [20]. Theorem 2. Let A n = R 0 ⊗ A n [∂ λ ] and N n = N An (h) = {x ∈ A n | hx ⊂ A n h}, where A n = C((λ 1 , . . . , λ n )). Define the N n -valued functions s m (u) by the formula where s 0 (u) = 1. They commute with Cartan elements h k : and they pair-wise commute modulo A n h: The decomposition coefficients of the functions s 1 (u), s 2 (u), . . . , s n (u) form a commutative family in the algebra N n at the level h k = 0. This means that their images by the canonical homomorphism N n → N n /A n h commute with each other. Below we consider some special class of L-operators L(u; λ).

Elliptic Gaudin model
Theorem 2 provides the functions s m (u) generating the quantum integrals of motion for the elliptic Gaudin model. This model is defined by a representation of the loop Lie algebra gl n [z, z −1 ]. Let us first consider a homomorphism ρ : U (gl n [z, z −1 ]) → R 0 from U (gl n [z, z −1 ]) to some algebra R 0 . To this homomorphism we associate the elliptic half-currents ρ(e ij z m ), (3.9) where i = j, {e ij } is a basis in gl n for which e ij → E ij is a faithful representation. The set {e ii z m } is a basis in gl n [z, z −1 ]. The matrix L(u; λ) with the elements where h k = ρ(e kk ), satisfies the relations (3.2) and (3.4).
In the case of homomorphism ρ = ρ {v} the corresponding commutative family can be obtained as follows. Let L 0 (u; λ) be the dynamical elliptic L-operator defined by (2.11). Shifting the argument we again get a dynamical elliptic L-operator L 0 (u − v; λ). This is also a ma- . Then applying Lemma 1 we obtain a dynamical elliptic L-operator L [N ]  Consider the homomorphism ϕ 0 : T 0 → U (gl n ) ⊗ A n defined by the formulae  7). Analogously, considering ϕ = (Π 1 ⊗ · · · ⊗ Π N ) • ϕ [N ] we obtain the Gaudin L-operator corresponding to the representation Π {v} = (Π 1 ⊗ · · · ⊗ Π N ) • ρ {v} . Thus we derive the following proposition. So we have shown that the L-operator (3.10) corresponding to the certain representation ρ : U (gl n [z, z −1 ]) → R 0 = End V defines a commutative family s m (u). We conjecture that this is true for any homomorphism ρ. For the case n = 2 this conjecture can be proved using a certain dynamical elliptic L-operator constructed in [21]. In notations of that paper this is the L-operatorL(u, λ) constructed from the L-operator L + λ (u). The entries ofL(u, λ) belong to the certain algebra R = U g (via some embedding) and the degeneration of this L-operator gives the L-operator (3.10) for ρ = id. 4 Thus the L-operator for ρ = id defines functions s id m (u) satisfying (3.6) and (3.7). The Loperator for the arbitrary homomorphism ρ defines function s ρ m (u) = ρ s m (u) , applying ρ to the relations (3.6) and (3.7) for s id m (u) we conclude that s ρ m (u) satisfy the same relations. For the general n this conjecture can be proved in the same way. For example, it follows from the Conjecture 5.1 of the paper [16]. There the authors present an analogue of the L-operator L + λ (u) for the quantum group U q,p ( sl n ). The existence of a twist relating this L-operator with an analogue ofL(u, λ) can be easily proved. Theorem 2 with Proposition 3 provide us a method to construct the quantum elliptic Gaudin model for general n in analogy to the rational case treated in [13]. Let us firstly note that the connection whose determinant Q(u, ∂ u ) provides a generating function for quantum Hamiltonians is the KZB connection. Moreover this construction allows to establish important relations of the elliptic Gaudin model with the Langlands correspondence program. The construction of the characteristic polynomial Q(u, ∂ u ) called the Universal G-oper in this case provides us • a universal construction of the commutative family; • a universal description of the eigenvalue problem; • a relation between the solutions of the KZB equation and the wave-functions of the Gaudin model; • a relation with the centre of the affine algebra at the critical level analogously to the results of [22,23] in the rational case.
Usually the elliptic Gaudin model defined by the L-operator with the currents (3.11), (3.12) or, more generally, with the currents (3.8), (3.9). In the next subsection we describe the relation of the L-operator (3.13) with the L-operator L(u; λ).

Twisting of a classical dynamical L-operator
First we consider the twists of dynamical r-matrices for arbitrary Lie algebras. Let g be a (in general, infinite-dimensional) Lie algebra, g ⊗ g be a tensor product completed in some way and h ⊂ g be an n-dimensional commutative subalgebra called Cartan subalgebra. Let us fix a basis {ĥ 1 , . . . ,ĥ n } of h. An element r(λ) ∈ g ⊗ g depending on n dynamical parameters λ 1 , . . . , λ n is called a dynamical classical r-matrix if r (12) (λ) + r (21) (λ) ∈ g ⊗ g g and it satisfies the classical dynamical Yang-Baxter equation The space g ⊗ g g consists of the elements x ∈ g ⊗ g such that [x, y ⊗ 1 + 1 ⊗ y] = 0 for all y ∈ g.
The brackets in the right hand side of (3.15) mean the commutators in U (g) ⊗ U (g) ⊗ U (g).
Lemma 6. Letr(λ) and f(λ) be elements of g ⊗ g. Suppose that f (12) (λ) + f (21) (λ) ∈ g ⊗ g Consider the loop algebra g = gl n [z, z −1 ] with Cartan subalgebra h spanned by the elementŝ h k = e kk . This is the Lie algebra of constant diagonal matrices. Let r(λ) be a dynamical classical r-matrix and let π u : gl n [z, z −1 ] → End C n be the standard evaluation representation defined as π u : e ij z k → E ij u k . Since π u is faithful the matrix r can be represented by the matrix r(u, v; λ) = (π u ⊗ π v )r(λ) ∈ End C n ⊗ End C n [[u, u −1 , v, v −1 ]], where the dependency of u and v is understood in the formal way. Each homomorphism ρ : U (gl n [z, z −1 ]) → R 0 defines the Loperator L(u; λ) = (π u ⊗ ρ)r(λ) satisfying the dynamical rLL-relations (3.2) with the r-matrix r(u, v; λ).
The twisted L-operator L(u; λ) = (π u ⊗ ρ)r(λ) satisfies the rLL-relations (3.2) with rmatrixr(u, v; λ) and the relation (3.4) with h k = ρ(ĥ k ) = ρ(e kk ). It coincides with the Loperator (3.13) related with the L-operator L(u; λ) via the formulae where j = i. The characteristic polynomial from Theorem 2 in these terms takes the form θ(λ ij ) h j in the determinant. However for n 3 one can not simplify it in this way, because even in the case n = 3 omitting this sum we lose the following term of Q(u,
Since the L-operator depends only on the difference λ = λ 1 − λ 2 we can restrict Q(u, ∂ u ) to the where h = h 1 − h 2 and S λ (u) is the following N -valued function which commutes with itself: [S λ (u), S λ (v)] = 0 mod Ah. Using the commutation relation ′ h one obtains the generating function for the sl 2 elliptic Gaudin Hamiltonians: This generating function was considered and used in [18,19] to find the eigenfunctions of this model.

Quantum powers for the Gaudin model
The quantum powers for the elliptic Gaudin models are defined in terms of the operator L D (u) = L(u; λ) − D λ . The first two quantum powers of this operator coincide with its ordinary powers: L The proof of this formula is the same as for the rational Gaudin model. See details in [5]. Instead of the quantum powers of L D (u) one can consider "simplified" quantum powers L D (u)L

Conclusion
In Section 2 we have constructed a commutative family starting with the Felder R-matrix. All results of that section are valid if we would start with another dynamical R-matrix satisfying R(− ; λ) = B(λ)A (12) for some invertible matrix B(λ). The universality of this method also implies that one can consider an arbitrary L-operator satisfying dynamical RLL-relations. In other words, it can be applied to a wide class of models which are described by using a dynamical R-matrix. Moreover, the integrals of motion for the models corresponding to a non-dynamical R-matrix can be obtained in the same way, or more precisely as it was done in [13] for the rational case. In particular, it should work for the XY Z-model or other models described by the Belavin R-matrix. The use of the Manin matrices is important to provide the universality of our method due to the universality of that notion. First, the characteristic polynomials of Manin matrices are key objects to obtain the commutative family of the (elliptic) gl n Gaudin model by the degeneration. Then the Manin matrices are used to relate the constructed commutative families with another important class of commutative families -the traces of the quantum powers. The Manin matrices and their characteristic polynomials find their applications to many important problems in the theory of integrable systems. It is also convenient to use the properties of Manin matrices to prove commutativity of the constructed families.
Analysing the trigonometric limit we have noticed that L-operators related with the Hopf algebra U q ( gl n ), such as the L-operator of the XXZ-model, gives also a Manin matrix after some simple twisting. This means that the corresponding quantum determinants, characteristic polynomials, quantum powers and Newton identities can be written in terms of the ordinary anti-symmetrizers A (1,...,m) by using the theory of Manin matrices, while consideration of the non-twisted L-operators leads to the using of q-deformations of these anti-symmetrizers and of a q-analogue of the Manin matrices.
In Section 3 we showed that the degeneration of each dynamical elliptic operator L(u; λ) gives a commutative family for the corresponding degenerated L-operator L(u; λ). To justify the obtained results for a given L(u; λ) one have to prove the existence of the corresponding L(u; λ). We considered the L-operators L(u; λ) defined by some homomorphism ρ : U (gl n [z, z −1 ]) → R 0 or a representation ρ : U (gl n [z, z −1 ]) → End V . We discuss the existence of the corresponding L-operators L(u; λ) presenting them explicitly or referring to other works.
For the representations ρ = Π {v} we present the corresponding L(u; λ) using the notion of the small elliptic quantum group suggested in [14]. To obtain the commutative family in the case of arbitrary homomorphism ρ : U (gl n [z, z −1 ]) → R 0 it is sufficient to construct L(u; λ) corresponding to ρ = id. In the gl 2 (technically sl 2 ) case the L-operator L(u; λ) corresponding to ρ = id in fact constructed in [21]. For the general gl n (technically sl n ) case the (nontwisted) L-operator L(u; λ) corresponding to ρ = id is written in [16] as a conjecture. One also need to relate the R-matrix using in [16] with the Felder R-matrix by some twist as in Subsection 2.3. The existence this twist can be quite easily established using the ansatz F = exp 2 i =j ϕ ij (λ)ĥ i ⊗ĥ j , where ϕ ij (λ) = −ϕ ji (λ) are C[[ ]]-valued functions. Let us finally remark that the degeneration of this twist gives the classical twist considered in Subsection 3.3: