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``Manin'' Matrices and Quantum Spin Models

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``Manin'' Matrices and Quantum Spin Models “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).
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