Manin matrices and Talalaev's formula

We study special class of matrices with noncommutative entries and demonstrate their various applications in integrable systems theory. They appeared in Yu. Manin's works in 87-92 as linear homomorphisms between polynomial rings; more explicitly they read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: $[M_{ij}, M_{kl}]=[M_{kj}, M_{il}]$ (e.g. $[M_{11}, M_{22}]=[M_{21}, M_{12}]$). We claim that such matrices 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) holds true for them. On the other hand, we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability. For instance, Manin matrices (and their q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and the so--called Cartier-Foata matrices. Also, they enter Talalaev's hep-th/0404153 remarkable formulas: $det(\partial_z-L_{Gaudin}(z))$, det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show that theorems of linear algebra, after being established for such matrices, have various applications to quantum integrable systems and Lie algebras, e.g in the construction of new generators in $Z(U(\hat{gl_n}))$ (and, in general, in the construction of quantum conservation laws), in the Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also discuss applications to the separation of variables problem, new Capelli identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints e.g. in Newton id-s fixed, normal ordering convention turned to standard one, refs. adde

SU(3) Richardson-Gaudin models: three level systems

We present the exact solution of the Richardson-Gaudin models associated with the SU(3) Lie algebra. The derivation is based on a Gaudin algebra valid for any simple Lie algebra in the rational, trigonometric and hyperbolic cases. For the rational case additional cubic integrals of motion are obtained, whose number is added to that of the quadratic ones to match, as required from the integrability condition, the number of quantum degrees of freedom of the model. We discuss different SU(3) physical representations and elucidate the meaning of the parameters entering in the formalism. By considering a bosonic mapping limit of one of the SU(3) copies, we derive new integrable models for three level systems interacting with two bosons. These models include a generalized Tavis-Cummings model for three level atoms interacting with two modes of the quantized electric field.Comment: Revised version. To appear in Jour. Phys. A: Math. and Theo

Integrable Models From Twisted Half Loop Algebras

This paper is devoted to the construction of new integrable quantum mechanical models based on certain subalgebras of the half loop algebra of gl(N). Various results about these subalgebras are proven by presenting them in the notation of the St Petersburg school. These results are then used to demonstrate the integrability, and find the symmetries, of two types of physical system: twisted Gaudin magnets, and Calogero-type models of particles on several half-lines meeting at a point.Comment: 22 pages, 1 figure, Introduction improved, References adde

Spaces of quasi-exponentials and representations of gl_N

We consider the action of the Bethe algebra B_K on (\otimes_{s=1}^k L_{\lambda^{(s)}})_\lambda, the weight subspace of weight $\lambda$ of the tensor product of k polynomial irreducible gl_N-modules with highest weights \lambda^{(1)},...,\lambda^{(k)}, respectively. The Bethe algebra depends on N complex numbers K=(K_1,...,K_N). Under the assumption that K_1,...,K_N are distinct, we prove that the image of B_K in the endomorphisms of (\otimes_{s=1}^k L_{\lambda^{(s)}})_\lambda is isomorphic to the algebra of functions on the intersection of k suitable Schubert cycles in the Grassmannian of N-dimensional spaces of quasi-exponentials with exponents K. We also prove that the B_K-module (\otimes_{s=1}^k L_{\lambda^{(s)}})_\lambda is isomorphic to the coregular representation of that algebra of functions. We present a Bethe ansatz construction identifying the eigenvectors of the Bethe algebra with points of that intersection of Schubert cycles.Comment: Latex, 29 page