“Manin” matrices and quantum spin models
Gregorio Falqui
Dipartimento di Matematica e Applicazioni Universit`adi Milano-Bicocca
March 23rd 2009, QIDS, Cambridge Outline
1 Introduction and (brief) overview The Gaudin model ... and their limits Lax matrices of ”Gaudin” and ”Yangian” type
2 Manin matrices Main properties On Quantum separation of variables Introduction and (brief) overview
We discuss some properties of Lax (and Transfer) matrices associated with quantum integrable systems. In the present talk we will consider the Lax matrix of the Gaudin system, and namely the problem of what happens when the arbitrary points z1,..., zN appearing in the Lax matrix, and in the quadratic Hamiltonians Hi glue together. (Joint work with A. Chervov and L. Rybnikov, SIGMA 2009) This will lead us to the discuss - in some more greater detail - the notion of Manin matrix. (Joint work with A. Chervov , J. Phys. A.: Math. Theor. 41,n.19. May 2008, paper no. 194006, and with A. Chervov and V. Rubtsov, Angers arXiv:0901.0235 to appear in Adv. Appl. Math. , and work in progress also with A. Sylantyev) Introduction and (brief) overview
Our point of view stems from the fact that Lax matrices satisfy special commutation properties, considered by Yu. I. Manin some twenty years ago at the beginning of Quantum Group Theory. They are the commutation properties of matrix elements of linear homomorphisms between polynomial rings; more explicitly they read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: [Mij , Mkl ] = [Mkj , Mil ] (e.g. [M11, M22] = [M21, M12]). Twofold Main aim : 1) Such matrices (which we call Manin matrices for short) behave almost as well as matrices with commutative elements. Namely theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, the Newton identities and so on and so forth) have a straightforward counterpart in the case of Manin matrices. Introduction and (brief) overview
2) Such matrices often enter theory of quantum integrable spin systems. For instance, Manin matrices include matrices satisfying the Yang-Baxter relation ”RTT=TTR” and the so–called Cartier-Foata matrices. Idea/Hope: Theorems of linear algebra, after being established for such matrices, have (or might have) various applications to quantum integrable systems and Lie algebras. Introduction and (brief) overview
The Gaudin model was introduced by M. Gaudin as a spin model related to the Lie algebra sl2, and later generalized to the case of arbitrary semisimple Lie algebras. The Hamiltonian is
dim g (i) a(j) HG = xa x , (1) a X=1 Xi=6 j
where {xa}, a = 1,..., dim g, is an orthonormal basis of g with respect to the Killing form (and xa its dual). These objects are regarded as elements of the polynomial algebra S(g∗)⊗ N in the classical case, and as elements of the universal envelopping algebra U(g)⊗N in the quantum case, as
(i) xa = 1 ⊗···⊗ xa ⊗1 ···⊗ 1. (2) i−th factor |{z} Introduction and (brief) overview
Gaudin himself found that the quadratic Hamiltonians
dim g (i) (k) xa xa H = . (3) i z − z a i k Xk=6 i X=1
provide a set of “constants of the motion” for HG . Later it was shown (Jurco) that - in the classical case - the spectral invariants of the Lax matrix (i) xa LG (z)= z − zi i,a X encode a (basically complete) set of invariant quantities for the corresponding model on an arbitrary simple Lie algebra g. Feigin Frenkel and Reshetikhin proved the existence of a large ⊗N commutative subalgebra A(z1,..., zN ) ⊂ U(g) containing Hi . For g = sl2, the algebra A(z1,..., zN ) is generated by Hi and the ⊗N central elements of U(sl2) . Introduction and (brief) overview
In other cases, the algebra A(z1,..., zN ) has also some new generators known as higher Gaudin Hamiltonians. Their explicit construction for g = gln was obtained in 2004 by D. Talalaev. Let us we consider the problem of discussing what happens when the arbitrary points z1,..., zN appearing in the Lax matrix, and in the (quadratic) Hamiltonians Hi glue together. Our limits of the Gaudin algebras when some of the points z1,..., zN glue together are as follows: We keep some points z1,..., zk ”fixed”, and let the remaining N − k points glue to a new point w, via
zk+i = w + sui , i = 1,..., N − k, zi 6= zj ; ui 6= uj , s → 0. (4) Introduction and (brief) overview ... and their limits Limits of the Lax matrix:
k N Xi i=k+1 Xi LG (z) → L2(z)= + , s → 0. (5) z − zi z − w i=1 P X Too na¨ıve: the number of Hamiltonians obtained from L2 is not sufficient to yield complete integrability. Rescaling: let us introduce a new variablez ˜ s.t. z = w + sz˜, and rewrite the Lax matrix k N Xi Xi LG = + . w + sz˜ − zi w + sz˜ − w − sui i X=1 i=Xk+1 Get the Lax matrix N Xi L1(z)= Ress=0LG (˜z)= z˜ − ui i=Xk+1 Introduction and (brief) overview ... and their limits
Summing up: to the Lax matrix with generic (distinct) points z1,..., zN , we can associated, to the gluing {zk+1,... zN}→ w the following pair of “Lax matrices”:
N k N Xi Xi i=k+1 Xi L1(z)= ; L2(z)= + . (6) z − ui z − zi z − w i=k+1 i=1 P X X We can choose the gluing procedure to be explicitly given by, e.g.,
zk+i = w + s(zk+i − w), s ∈ (0, 1) (7)
and, using invariance w.r.t. transformation of the spectral parameter z → z − w, trade the matrix L1 of (6) for N Xi L˜1(z)= . z − zi i k =X+1 Introduction and (brief) overview ... and their limits
In particular, in the example N = 5 and z3, z4, z5 → w, we would 5 X associate, to the Lax matrix L = i the two matrices z − zi i X=1 X3 X4 X5 L1(z)= + + , z − z3 z − z4 z − z5
X1 X2 X3 + X4 + X5 L2(z)= + + . z − z1 z − z2 z − w The number of independent Hamiltonians gotten in this way is enough to ensure complete integrability of the model. In some sense we recover integrability by adding one more pole. Introduction and (brief) overview ... and their limits
Proposition
For every choice of w ∈ C the family of spectral invariants (1) (2) H , H associated with the Lax matrices L1 and L2 satisfy the following properties: 1 The elements of H(1), H(2) commute w.r.t. the standard (diagonal) Poisson brackets on gN ;
2 The dimension of the Poisson commutative subalgebra H1,2,w generated by the spectral invariants H(1) and H(2) respectively associated with the Lax matrices L1 and L2, coincides with that of the spectral invariants associated with the generic Lax matrix LG .
3 The physical Hamiltonian HG lies in H1,2,w .
4 Suitable spectral invariants obtained from L1 and L2 commute among themselves also in the quantum case. Introduction and (brief) overview ... and their limits
Quantization of the spectral invariants: Problem: in the quantum Gaudin case (linear r matrix structure)
[TrL2(z), TrL4(u)] 6= 0!
”Good” (that is, commuting) quantum Hamiltonians (QH) are obtained via the prescription
n i ”Det”(∂z − L(z)) = QHn−i ∂z i X=0 (Talalaev(04), Chervov-Talalaev 2006). Question: is there a (possibly natural) simple and manageable algebraic framework for these and related models? Introduction and (brief) overview Lax matrices of ”Gaudin” and ”Yangian” type Some more examples .
Let K be an associative algebra over C. Let Π ∈ Matn ⊗ Matn be the permutation matrix: Π(a ⊗ b)= b ⊗ a, and let L(z) be a matrix 1 2 with elements in K((z)), and L(z)= L(z) ⊗ 1, L(u) = 1 ⊗ L(u). We say that L(z) is of Gaudin type if
Π [L(z) ⊗ 1, 1 ⊗ L(u)] = [ , L(z) ⊗ 1 + 1 ⊗ L(u)], z − u
Π (linear r-matrix structure, the r-matrix being r = z−u ). Introduction and (brief) overview Lax matrices of ”Gaudin” and ”Yangian” type
Let K be an arbitrary constant matrix, and n, k ∈ N, and z1,..., zk arbitrary points in the complex plane. Consider
qˆ 1 1,i L(z)= K + ... ⊗ pˆ1,i ... pˆn,i = z − zi i=1,...,k qˆ X n,i 1 1 K + Qˆ diag( , ..., ) Pˆ t (z − z1) (z − zk )
wherep ˆi,j , qˆi,j , i = 1, ..., n; j = 1, ..., k are standard generators of the standard Heisenberg algebra
[ˆpi,j , qˆk,l ]= δi,k δj,l , [ˆpi,j , pˆk,l ]=[ˆqi,j , qˆk,l ] = 0,
collected in n × k-rectangular matrices Qˆ , Pˆ with elements Qˆi,j =q ˆi,k , Pˆi,j =p ˆi,j . Introduction and (brief) overview Lax matrices of ”Gaudin” and ”Yangian” type Example 2 - standard
a Consider gln ⊕ ... ⊕ gln and denote by ekl the standard basis N−times element from the a-th copy of the direct sum gln ⊕ ... ⊕ gln. The standard Lax| matrix{z for} the Gaudin system is:
ei ... ei 1 1,1 1,n Lgln−Gaudin standard (z)= ...... (8) z − za i i a=1,...,N e ... e X n,1 n,n C Lgln (z) ∈ Matn ⊗ U(gln ⊕ ... ⊕ gln) ⊗ (z). za are a set of arbitrary but distinct complex parameters. Introduction and (brief) overview Lax matrices of ”Gaudin” and ”Yangian” type
Let K be an associative algebra over C. Let us call a matrix T (z) with elements in K((z)) a Lax matrix of Yangian type if
Π (1 ⊗ 1 − )(T (z) ⊗ 1) (1 ⊗ T (u)) = z − u Π (1 ⊗ T (u)) (T (z) ⊗ 1)(1 ⊗ 1 − ) z − u Or shortly:
1 2 2 1 R(z − u)T (z)T (u)= T (u)T (z)R(z − u). Introduction and (brief) overview Lax matrices of ”Gaudin” and ”Yangian” type
Examples: Consider the Heisenberg algebra generated byp ˆi , qˆi , i = 1, ..., n and relations [ˆpi , qˆj ]= δi,j , [ˆpi , pˆj ]=[ˆqi , qˆj ] = 0. Define
z − pˆ e−qˆi T (z) = i (9) Toda −eqˆi 0 i ,...,n =1Y
As it is known, this is a limit of the XXX Heisenberg sl2 Transfer matrix,
ei ... ei 1 1,1 1,n Tgln (z)= 1n×n + ...... (10) z − zi i i i=1,...,k e ... e Y n,1 n,n Manin matrices Manin Matrices
Formally: matrices associated with linear maps between commutative rings. Operative definition: Mij is (column) Manin if: Elements in the same column commute among themselves; Commutators of the cross terms in any 2 × 2 submatrix are equal:
[Mij , Mkl ] = [Mkj , Mil ] e.g. [M11, M22] = [M21, M12]. Manin matrices Matrix Notation
As usual, let 1 2 M = M ⊗ 1, M = 1 ⊗ M.
Proposition A matrix M is a Manin matrix if
1 2 1 2 [M , M ] = Π[M , M ]
Semiclassical case (straightforward) Proposition
A Matrix M is a Poisson-Manin matrix iff: 1 2 1 2 {M ⊗, M } = Π{M ⊗, M }. Manin matrices
A matrix with elements in a noncommutative ring K is called a Cartier-Foata matrix if elements from different rows commute with each other. Proposition A Cartier-Foata matrix is a Manin matrix. The characteristic conditions for Manin matrices are trivially satisfied in this case. Proposition
∂z − Lgln (z) is Manin, where Lgln−Gaudin(z) is the Lax matrix for the L ie algebra gln (as well as its generalization to the affine algebra gln[t]). −∂z e Tgln (z) is Manin, where Tgln (z) is the Lax (or ”transfer”) matrix for the Yangian algebra Y (gln). Manin matrices Main properties The determinant of a Manin matrix.
Let M be a Manin matrix. Define the determinant of M by column expansion: y column σ detM = det M = (−1) Mσ(i),i , (11) i ,...,n σX∈Sn =1Y where Sn is the group of permutations of n letters, and the symbol y means that in the product i=1,...,n Mσ(i),i one writes at first the elements from the first column, then from the second column Q and so on and so forth. Prop The determinant of a Manin matrix does not depend on the order of the columns in the column expansion, i.e., y column σ ∀p ∈ Sn det M = (−1) Mσ(p(i)),p(i) (12) i ,...,n σX∈Sn =1Y Manin matrices Main properties Cramer’s formula
Let M be a Manin matrix and denote by M∨ the adjoint matrix ∨ k+l column defined in the standard way, (i.e. Mkl = (−1) det (Mlk ) where Mlk is the (n − 1) × (n − 1) submatrix of M obtained removing the l-th row and the k-th column. Then the sameb formulab as in the commutative case holds true, that is,
M∨M = detc (M) Id (13)
Remark: We can consistently define determinant of the minors since any submatrix of a Manin matrixis a Manin matrix. Manin matrices Main properties An application to the Knizhnik-Zamolodchikov equation
We can give a very simple proof of the formula relating the solutions of KZ to the solutions of the equation defined by the formula: det(∂z − L(z))Q(z) = 0.
In ”general”, the standard KZ-equation for gln, is given by:
ab (i) E ⊗ πi (eab ) ∂z − Ψ(z)= π(∂z − κLG (z))Ψ(z) = 0(14) z − zi i ...k ! =1X ⊗k where π = (V1 ⊗ ... ⊗ Vk ) is a representation of U(gln) and n Ψ(z) is a C ⊗ V1 ⊗ ... ⊗ Vk -valued function. LG (z) is the Lax matrix of the quantum Gaudin system. Manin matrices Main properties
Proposition Let Ψ(z) be a solution of the KZ-equation; Then:
∀i = 1, ..., n π(det(∂z − LG (z)))Ψi (z) = 0
adj Proof The adjoint matrix (∂z − LG (z)) exists, so that
π(∂z − LGaudin(z))Ψ(z) = 0, ⇒ adj π((∂z − L (z)) )π(∂z − L (z))Ψ(z) = 0, Gaudin Gaudin (15) hence π(det(∂z − LGaudin(z)))Id Ψ(z) = 0,
explicitly ∀i = 1, ..., n π(det(∂z − LGaudin(z)))Ψi (z) = 0. Manin matrices Main properties Further properties of MMs
The inverse of a Manin matrix M is again Manin. Schur’s formula for the determinant of block matrices holds: AB det = det(A)det(D − CA−1B)= CD det(D)det(A − BD−1C) .
The Cayley-Hamilton theorem: det(t − M)|t=M = 0 and Netwon identities hold Manin matrices Main properties Newton Identities
Consider the families of symmetric functions in n variables: 1 σ = λ , i = 1, ..., n, the k 1≤i1
Theorem The Newton identities between TrMk and the coefficients of the expansion of det(t + M) in powers of t hold for Manin matrices.. Manin matrices Main properties Talalaev’s Thm
Theorem (Talalaev,2004)
Let L(z) be the Lax matrix of the glr -Gaudin model, that is, let L(z) satisfy the r-matrix commutation relations
Π [L(z) ⊗ 1, 1 ⊗ L(u)] = [ , L(z) ⊗ 1 + 1 ⊗ L(u)]. (17) z − u Consider the differential operator in the variable z i det ∂z − L(z) = QHi (z)∂z ; Then: i=0,...,r X
∀i, j ∈ 0,..., r, and u, v ∈ C, [QHi (z)|z=u, QHj (z)|z=v ] = 0. (18) Manin matrices Main properties
Idea of the proof[Ta06]. The quantum determinant in the Yangian (RTT=TTR) case is basically the determinant of the matrix e−h∂z T (z). In the ”semiclassical” limit, we have
−h∂z 2 e T (z) − 1 = h(L(u) − ∂z )+ O(h ). Manin matrices Main properties Application: quantization of traces
As recalled above, traces of the powers of the Gaudin Lax matrix do not commute at the quantum level. A good strategy is not to consider these quantities, but rather the traces of the powers of the corresponding Manin matrices, i.e.,
k ˆ k k k−j Tr (∂z − L(z)) = (QTr)j (z)∂z , k = 1,..., r. (19) j X=0 Thanks to the Newton identities, these objects will commute among each other. Manin matrices Main properties
As it is easily seen, there is a recursion relation of the form k k+1 QTrj+1(z) ≃ QTrj (z), and hence, to obtain the expected number of independent quantities, we can consider simply the k coefficients QTrk , that is the coefficients of zeroth order of each differential ”polynomial” in (19). [n] These quantities are given by the traces of matrices Lˆk (z) , that can be called are ”quantum powers” of L(z), defined by the Fa`adi Bruno formula ∂ Lˆ[0](z)= Id, Lˆ[i](z)= Lˆ[i−1](z)Lˆ (z) − (Lˆ[i−1](z)). k k k k ∂z k Manin matrices Main properties Inversion properties
Theorem Let M be a Manin matrix, and assume that a two sided inverse matrix M−1 exists (i.e. M−1M = MM−1 = 1). Then M−1 is again a Manin matrix. Let us show that a theorem of Enriquez Rubtsov, and Babelon Talon about ”quantization” of separation relations – follows as a particular case from this theorem. Manin matrices Main properties BT+ER Theorem
Let {αi , βi }i=1,...,g be a set of quantum “separated” variables, i.e. satisfying the commutation relations
[αi , αj ] = 0, [βi , βj ] = 0, [αi , βj ]= f (αi , βi )δij , i, j, = 1 ..., g. satisfying a set of equations (i.e., quantum Jacobi separation relations) of the form g
Bj (αi , βi )Hj + B0(αi , βi ) = 0, i = 1,..., g, (20) j X=1 for a suitable set of quantum Hamiltonians H1,..., Hn. One assumes that the ordering in the expressions Ba, a = 0,... g between αi , βi has been chosen, and that the operators Hi are, as it is written above, on the right of the Bj . Then the statement is that the quantum operators H1,..., Hn fulfilling (20) commute among themselves). Manin matrices Main properties Proof via Manin property
The equations (20) can be compactly written, in matrix form, as
B · H = −V , with Bij = Rj (αi , βi ), Vi = B0(αi , βi ),
and thus one is lead to consider the g × (g + 1) matrix
V1 B1,1 ... B1,g A = ...... Vg Bg,1 ... Bg,g Thanks to the functional form of the Bij ’s and of the Vi ’s, this matrix is a ”Cartier-Foata” matrix (i.e., elements form different rows commute among each other), and hence, a fortiori a Manin matrix. Manin matrices Main properties
Given such a g × (g + 1) Cartier-Foata matrix A, we consider the (g + 1) × (g + 1) Cartier-Foata (and hence, Manin) matrix:
1 0 ... 0 V B ... B A˜ = 1 1,1 1,g ...... V B ... B g g,1 g,g Now it is obvious that the solutions Hi of quantum separation equation are the elements of the first column of the inverse of A˜, and namely −1 Hi = (A˜ )i+1,1, i = 1,..., g. Since A˜ is Cartier-Foata, its inverse is Manin, and thus the commutation of the Hi ’s can be obtained from the inversion theorem for Manin matrices.. Manin matrices Main properties Schur’s formula
Consider a Manin matrix M of size n, and denote its block as follows: A B M = k×k k×n−k (21) C D n−k×k n−k×n−k Assume that M, A, D are invertible. Then the same formulas as in the commutative case hold, i.e:
detc (M)= detc (A)detc (D −CA−1B)= detc (D)detc (A−BD−1C). (22) and the Schur’s complements, D − CA−1B, A − BD−1C are Manin. Manin matrices Main properties
Application 1:The Weinstein–Aronszajn formula for Manin matrices. Let A, B be n × k and k × n Manin matrices with pairwise commuting elements: ∀i, j, k, l : [Aij , Bkl ] = 0, then
col col det (1n×n − AB)= det (1k×k − BA). (23)
The matrix: 1 B k×k , is Manin. A 1 n×n Applying Schur’s formula one obtains the result. In particular, for
k n ∗ n Mn := 1n − xα ⊗ yβ xα, ∈ K , yβ ∈ K , α X=1 det(Mn) = det(1k − Sk ), where [Sk ]α,β = hxα, yβi. Manin matrices Main properties
Application 2 An identity by Mukhin, Tarasov, Varchenko. [MTV06]. Consider C[pi,j , qi,j ], i = 1, ..., n; j = 1, ..., k, endowed with the standard Poisson bracket: {pi,j , qk,l } = δi,k δj,l , {pi,j , pk,l } = {qi,j , qk,l } = 0. Consider their quantizationp ˆi,j , qˆi,j , i = 1, ..., n; j = 1, ..., k, with the relations [ˆpi,j , qˆk,l ]= δi,k δj,l , [ˆpi,j , pˆk,l ]=[ˆqi,j , qˆk,l ] = 0. Collect the variables in n × k-rectangular matrices Qcl , Pcl , Qˆ , Pˆ, and let K1, K2 be n × n, k × k matrices with elements in C. Let us introduce:
q −1 t L (z) = K1 + Qˆ (z − K2) Pˆ , (24) cl −1 t L (z) = K1 + Qcl (z − K2) Pcl (25)
These are Lax matrices of Gaudin type. Manin matrices Main properties
cl q Wick(det(λ − L (z))) = det(∂z − L (z)) (26)
Here we denote by Wick the linear map: C[λ, pi,j , qi,j ](z) → C[∂z , pˆi,j , qˆi,j ](z), defined as:
a cij bij a bij bij Wick(f (z)λ q pij )= f (z)∂z qˆij pˆij (27) ij ij ij ij Y Y Y Y ”Wick or normal ordering” product (w.r.t. the ”dynamical variables” {q, p} as well as the ”spectral” variables z, ∂z ). Manin matrices Main properties To show this we consider the following block matrix:
z − K Pˆ t MTV = 2 (28) Qˆ ∂ − K z 1 It is easy to see that MTV is a Manin matrix. Further – as it was observed by Mukhin, Tarasov and Varchenko –, the Lax matrix of the form above (24) appears as the Schur’s complement ′′D − CA−1B′′ of the matrix MTV:
−1 t q ∂z − K1 − Qˆ (z − K2) Pˆ = ∂z − L (z).
So by Schur’s theorem, we get
c c −1 t det (MTV )= det(z − K2)det (∂z − K2 − Qˆ (z − K2) Pˆ ). (29)
c Now remark that in det (MTV ) all variables z, qˆij stand on the left of the variables ∂z , pˆij . This is due to the column expansion of the determinant (e.g., z, qˆij stand in the first n-th columns of MTV). Manin matrices Main properties
A few ”No go” Facts Let M be a Manin matrix with elements in the associative ring K. Fact In general det(M) is not a central element of K. This should be compared with the quantum matrix group Funq(GLn), where detq is central. The reason why this property does not hold for Manin matrices is that their defining relations are half of those of quantum matrix groups. Fact In general [TrMk , TrMm] 6= 0, [TrM, det(M)] 6= 0. Actually, for this commutativity property one needs stronger conditions like 1 2 2 1 the Yang-Baxter relation RT T = T T R. Fact In general Mk , k = 2,..., is not a Manin matrix nor the sum of two Manin matricesis Manin. Fact Let M be a Manin matrix; then in general det(eM ) 6= eTr(M), log(det(M)) 6= Tr(log(M)). Manin matrices On Quantum separation of variables Quantum SoV
Let us briefly discuss some results and conjectures about the problem of separation of variables as formulated by E.Sklyanin. The program is not yet completed. The ultimate goal in this framework is to construct coordinates αi , βi such that a joint eigenfunction of all hamiltonians will be presented as product of functions of one variable:
1−particle Ψ(β1, β2, ...)= Ψ (βi ). i Y We consider this construction at the quantum level, trying to frame in this ”Manin matrices” realm some ideas of Sklyanin and others. Manin matrices On Quantum separation of variables
Let us remind that, in the classical case, the construction of separated variables for the systems we are considering goes, somewhat algorithmically1, as follows:
Step 1 One considers, for a gln model, along the Lax matrix L(z), the matrix M = λ − L(z) and its classical adjoint M∨. Step 2 One takes a vector ψ by means suitable linear combination of columns (or rows) of M∨; in the simplest case, one can take ψ to be one of the columns, say the last of M∨. One seeks for pairs (λi , zi ) that solve
ψi = 0, i = 1,..., n.
1We are herewith sweeping under the rug the problem known as “normalization of the Baker Akhiezer function”. Manin matrices On Quantum separation of variables
Step 3 To actually solve this problem, one proceeds as follows. Each component ψi of ψ is a polynomial of degree at most n − 1, one can form, out of ψ, the matrix Mψ defined as:
n−j−1 [Mψ]j,i = resλ=0ψi λ , i, j = 1,..., n.
Step 4 The separation coordinates are given by pairs (λi , zi ) where zi ’s are roots of Det(Mψ) and λi are the 2 corresponding values of λi , that can be obtained, e.g., via the Cramer’s rule. By construction , the Jacobi separation relations are the equation(s) of the spectral curve, Det(λ − L(z) = 0.
2In the quadratic R-matrix case, actually one has to take as canonical momenta, the logarithms of these λi . Manin matrices On Quantum separation of variables Quantum case: the Yangian
Let T (z) be a Lax matrix of the Yangian type, so (1 − e−∂z T (z)) is a Manin matrix and its adjoint matrix can be calculated by standard formulas. Let us denote by Mi,j(z) the matrix of the coefficients of expansion in left powers of e−∂z of the elements of the last column of (1 − e−∂z T (z))∨. I discuss in the case n = 3 how to results by Sklyanin can be framed in our picture ( the following arguments hold (=have been checked) for n = 2, 3, (4)). Fact Mi,j (z) is a Manin matrix. Manin matrices On Quantum separation of variables
Define B(z)= Detcolumn(M(z)); then
[B(z), B(u)] = 0. (30)
Consider any root β of the equation B(u) = 0. Then the system of 3 equations for the single variable α
2 M1,0(z)|z→β ... M1,2(z)|z→β α 0 ...... α = 0 M3,0(z)|z→β ... M3,2(z)|z→β 1 0 has a unique solution. Manin matrices On Quantum separation of variables
Consider all the roots βi of the equations B(u) = 0, and the corresponding variables αi . Then:
The variables αi , βi satisfy the following commutation relations:
[αi , βj ]= −αi δi,j , (31)
[αi , αj ] = 0, [βi , βj ] = 0, (32)
αi , βi satisfy the ”quantum characteristic equation”:
−∂ z −∂ ∀i : det(1 − e T (z))|z→βi ; e z →αi = 0 (33)
If T (z) is generic, then variables αi , βi are ”quantum coordinates” i.e. all elements of the algebra R can be expressed via αi , βi and the center (Casimirs) of the algebra R.