Twisted quantum affine superalgebra Uq[gl(m|n)(2)] and new Uq[osp(m|n)] invariant R-matrices

The minimal irreducible representations of U-q[gl(m|n)], i.e. those irreducible representations that are also irreducible under U-q[osp(m|n)] are investigated and shown to be affinizable to give irreducible representations of the twisted quantum affine superalgebra U-q[gl(m|n)((2))]. The U-q[osp(m|n)] invariant R-matrices corresponding to the tensor product of any two minimal representations are constructed, thus extending our twisted tensor product graph method to the supersymmetric case. These give new solutions to the spectral-dependent graded Yang-Baxter equation arising from U-q[gl(m|n)((2))], which exhibit novel features not previously seen in the untwisted or non-super cases.

Twisted quantum affine superalgebra U-q[sl(2/2)((2))], U-q[osp(2/2)] invariant R-matrices and a new integrable electronic model

We describe the twisted affine superalgebra sl(2\2)((2)) and its quantized version U-q[sl(2\2)((2))]. We investigate the tensor product representation of the four-dimensional grade star representation for the fixed-point sub superalgebra U-q[osp(2\2)]. We work out the tensor product decomposition explicitly and find that the decomposition is not completely reducible. Associated with this four-dimensional grade star representation we derive two U-q[osp(2\2)] invariant R-matrices: one of them corresponds to U-q [sl(2\2)(2)] and the other to U-q [osp(2\2)((1))]. Using the R-matrix for U-q[sl(2\2)((2))], we construct a new U-q[osp(2\2)] invariant strongly correlated electronic model, which is integrable in one dimension. Interestingly this model reduces in the q = 1 limit, to the one proposed by Essler et al which has a larger sl(2\2) symmetry.

The structure of quantum Lie algebras for the classical series, B-l, C-l and D-l

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at q = 1. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for adjoint x adjoint --> adjoint We present a practical method for the determination of these quantum Clebsch-Gordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum Lie algebras associated to the classical Lie algebras B-l, C-l and D-l. In the quantum case the structure constants of the Cartan subalgebra are non-zero and we observe that they are determined in terms of the simple quantum roots. We introduce an invariant Killing form on the quantum Lie algebras and find that it takes values which are simple q-deformations of the classical ones.

On super-RS algebra and Drinfeld realization of quantum affine superalgebras

We describe the realization of the super-Reshetikhin-Semenov-Tian-Shansky (RS) algebra in quantum affine superalgebras, thus generalizing the approach of Frenkel and Reshetikhin to the supersymmetric (and twisted) case. The algebraic homomorphism between the super-RS algebra and the Drinfeld current realization of quantum affine superalgebras is established by using the Gauss decomposition technique of Ding and Frenkel. As an application, we obtain Drinfeld realization of quantum affine superalgebra U-q [osp(1/2)((1))] and its degeneration - central extended super-Yangian double DY(h over bar) [osp(1/2)((1))].

Quantum Lie algebras, their existence, uniqueness and q-antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules L-h(g) of the quantized enveloping algebras U-h(g). On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra g an abstract quantum Lie algebra g(h) independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras L-h(g) are isomorphic to an abstract quantum Lie algebra g(h). In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras L-h(g) associated to the same g are isomorphic, 2) the quantum Lie product of any Ch(B) is q-antisymmetric. We also describe a construction of L-h(g) which establishes their existence.

Comments on the Drinfeld realization of the quantum affine superalgebra U-q[gl(m backslash n)(1)] and its Hopf algebra structure

By generalizing the Reshetikhin and Semenov-Tian-Shansky construction to supersymmetric cases, we obtain the Drinfeld current realization for the quantum affine superalgebra U-q[gl(m\n)((1))]. We find a simple coproduct for the quantum current generators and establish the Hopf algebra structure of this super current algebra.

Quantum V-algebras

The problem of the classification of the extensions of the Virasoro algebra is discussed. It is shown that all H-reduced G(r)-current algebras belong to one of the following basic algebraic structures: local quadratic W-algebras, rational U-algebras, nonlocal W-algebras, nonlocal quadratic WV-algebras and rational nonlocal UV-algebras. The main new features of the quantum Ir-algebras and their heighest weight representations are demonstrated on the example of the quantum V-3((1,1))-algebra.

Level-one highest weight representations of U-q[g1(1 vertical bar 1)] and associated vertex operators

We study the level-one irreducible highest weight representations of U-q[gl(1\1)] and associated q-vertex operators. We obtain the exchange relations satisfied by these vertex operators. The characters and supercharacters associated with these irreducible representations are calculated'. (C) 2000 Published by Elsevier Science B.V. All rights reserved.

U-q[(sl(2)over-cap vertical bar 1)] vertex operators, screen currents, and correlation functions at an arbitrary level

Bosonized q-vertex operators related to the four-dimensional evaluation modules of the quantum affine superalgebra U-q[sl((2) over cap\1)] are constructed for arbitrary level k=alpha, where alpha not equal 0,-1 is a complex parameter appearing in the four-dimensional evaluation representations. They are intertwiners among the level-alpha highest weight Fock-Wakimoto modules. Screen currents which commute with the action of U-q[sl((2) over cap/1)] up to total differences are presented. Integral formulas for N-point functions of type I and type II q-vertex operators are proposed. (C) 2000 American Institute of Physics. [S0022-2488(00)00608-3].

Level-one highest weight representation of U-q[sl((N)over-cap vertical bar 1)] and Bosonization of the multicomponent Super t-J model

We study the level-one irreducible highest weight representations of the quantum affine superalgebra U-q[sl((N) over cap\1)], and calculate their characters and supercharacters. We obtain bosonized q-vertex operators acting on the irreducible U-q[sl((N) over cap\1)] modules and derive the exchange relations satisfied by the vertex operators. We give the bosonization of the multicomponent super t-J model by using the bosonized vertex operators. (C) 2000 American Institute of Physics. [S0022- 2488(00)00508-9].

Izergin-Korepin model with a boundary

The Izergin-Korepin model on a semi-infinite lattice is diagonalized by using the level-one vertex operators of the twisted quantum affine algebra U-q[((2))(2)]. We give the bosonization of the vacuum state with zero particle content. Excitation states are given by the action of the vertex operators on the vacuum state. We derive the boundary S-matrix. We give an integral expression of the correlation functions of the boundary model, and derive the difference equations which they satisfy. (C) 2001 Elsevier Science B.V. All rights reserved.

Free field realizations of the elliptic affine Lie algebra sl(2, R) circle plus (Omega(R)/dR)

In this paper we construct two free field realizations of the elliptic affine Lie algebra sl(2, R) circle plus Omega(R)/dR where R = C[t. t(-1), u vertical bar u(2) = t(3) - 2bt(2) + t]. The first realization provides an analogue of Wakimoto`s construction for Affine Kac-Moody algebras, but in the setting of the elliptic affine Lie algebra. The second realization gives new types of representations analogous to Imaginary Verma modules in the Affine setting. (c) 2009 Elsevier B.V. All rights reserved.

Generalizations of Lie Algebras

The generalizations of Lie algebras appeared in the modern mathematics and mathematical physics. In this paper we consider recent developments and remaining open problems on the subject. Some of that developments have been influenced by lectures given by Professor Jaime Keller in his research seminar. The survey includes Lie superalgebras, color Lie algebras, Lie algebras in symmetric categories, free Lie tau-algebras, and some generalizations with non-associative enveloping algebras: tangent algebras to analytic loops, bialgebras and primitive elements, non-associative Hopf algebras.

Linear operators and their spectra with applications to quantum mechanics.

From lectures given at the New York university Institute for mathematica and mechanics, by R. Cournat and others.