Hidden Uq (sl(2)) Uq (sl(2)) Quantum Group Symmetry in Two Dimensional Gravity

In a previous paper, the quantum-group-covariant chiral vertex operators in the spin 1/2 representation were shown to act, by braiding with the other covariant primaries, as generators of the well known U_q(sl(2)) quantum group symmetry (for a single screening charge). Here, this structure is transformed to the Bloch wave/Coulomb gas operator basis, thereby establishing for the first time its quantum group symmetry properties. A U_q(sl(2)) U_q(sl(2)) symmetry of a novel type emerges: The two Cartan-generator eigenvalues are specified by the choice of matrix element (Vermamodules); the two Casimir eigenvalues are equal and specified by the Virasoro weight of the vertex operator considered; the co-product is defined with a matching condition dictated by the Hilbert space structure of the operator product. This hidden symmetry possesses a novel Hopf-like structure compatible with these conditions. At roots of unity it gives the right truncation. Its (non-linear) connection with the U_q(sl(2)) previously discussed is disentangled. Received: 25 April 1996/Accepted: 20 July 1996     

\hat sl(2) \oplus \hat sl(2)/\hat sl(2) coset theory is a Hamiltonian reduction of the \hat D(2|1;\alpha ) superalgebra

It is shown that $\hat sl(2)_{k_1 } \oplus \hat sl(2)_{k_2 } /\hat sl(2)_{k_1 + k_2 } $ coset theory is a quantum Hamiltonian reduction of the exceptional affine Lie superalgebra $\hat D(2|1;\alpha )$ . In addition, the W algebra of this theory is the commutant of the U _q D(2|1;a) quantum group.     

THE ADJOINT REPRESENTATION OF QUANTUM ALGEBRA Uq(sl(2))

Starting from any representation of the Lie algebra on the finite dimensional vector space V we can construct the representation on the space Aut(V). These representations are of the type of ad. That is one of the reasons, why it is important to study the adjoint representation of the Lie algebra on the universal enveloping algebra U(). A similar situation is for the quantum groups U_q(). In this paper, we study the adjoint representation for the simplest quantum algebra U_q(sl(2)) in the case that q is not a root of unity.     

Uq[sl(2)] Quantum Algebra in Quantum Hall Effect

For a two-dimensional system of electronsdescribed by a Hamiltonian involving two- and three-bodyinteractions and an external transverse magnetic field,we construct the U_q[sl(2)] quantum algebra,where the deformation parameter q is related to thefilling factor . We show that the Laughlin statesform a representation of this algebra.     

Heisenberg realization for U q (sl n ) on the flag manifold

We give the Heisenberg realization for the quantum algebra U _q (sl _n ), which is written by theq-difference operator on the flag manifold. We construct it from the action of U _q (sl _n ) on theq-symmetric algebraA _q (Mat _n ) by the Borel-Weil-like approach. Our realization is applicable to the construction of the free field realization for U _q [2].     

Some Tensor Product Representations of Uq(s $${\hat l}{\kern 1pt} $$ 2) at Roots of Unity

When q is a root of unity, a triangular decomposition of U _q(s ₂) is given and irreducibility conditions concerning some tensor product representations of U _q(s ₂) are presented. Their connection with physics is also pointed out.     

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 [osp(12)⁽¹⁾].     

The structure of U q (sl(2, C))

The structure of the deformation U _q (sl(2, C)) is discussed. The comultiplication, all commutation relations, and a conjugation follow in a clear way form the simple SL _q (2) structure. Fundamental and spin representation are given.     

The quantum super-Yangian and Casimir operators of Uq(gl(M |N))

TheZ ₂ graded Yangian Yq(gl(M |N)) associated with the Perk-SchultzR matrix is introduced. Its structural properties, the central algebra in particular, are studied. AZ ₂-graded associative algebra epimorphism Yq(gl(M |N)) Uq (gl(M |N)) is obtained in explicit form. Images of central elements of the quantum super-Yangian under this epimorphism yield the Casimir operators of the quantum supergroup Uq(gl(M |N)) constructed in an earlier publication.     

Quantum Affine Superalgebra Uq(gl(1|1)) and Its

Drinfeld's realization of quantum affine superalgebra U_q(gl(1|1)) is given based on the super version of RS construction method and Gauss decomposition.     

The <Emphasis Type="Italic">q</Emphasis>-boson-fermion realizations of the quantum superalgebra <Emphasis Type="Italic">U</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(<Emphasis Type="Italic">osp</Emphasis>(1/2))

We show that our construction of realizations for algebras and quantum algebras can be generalized to quantum superalgebras too. We studyan example of quantum superalgebra U _q (osp(1/2)) and give the boson-fermion realization with respect to one pair of q-boson operators and one pair of fermions.     

Embedding Uq(sl(2)) and Sine Algebras in Generalized Clifford Algebras

We establish the connection between certain quantum algebras and generalizedClifford algebras (GCA). To be precise, we embed the quantum tori Lie algebraand U _q (sl(2)) in GCA.     

Polynomial deformations of enveloping algebra U(sl(2,2102))

We define the polynomial deformation, U _t ^p, of enveloping algebra U(sl(2,)) as an analogue of quantum group and construct a nice set D _p ^even of parameters on which we described irreducible representations and classify the irreducible ones in terms of the highest weights. Also, we construct the Casimir elements and using them we see that every finite dimensional U _t ^p-module is semisimple.     

A supersymmetric Uq[osp(2|2)]-extended Hubbard model with boundary fields

A strongly correlated electron system associated with the quantum superalgebra U_q[osp(2|2)] is studied in the framework of the quantum inverse scattering method. By solving the graded reflection equation, two classes of boundary-reflection K-matrices leading to four kinds of possible boundary interaction terms are found. Performing the algebraic Bethe ansatz, we diagonalize the two-level transfer matrices which characterize the charge and the spin degrees of freedom, respectively. The Bethe-ansatz equations, the eigenvalues of the transfer matrices and the energy spectrum are presented explicitly. We also construct two impurities coupled to the boundaries. In the thermodynamic limit, the ground state properties and impurity effects are discussed.     

Generalization of the $$\mathcal{U} $$ q (gl(N)) Algebra and Staggered Models

We develop a technique for the construction of integrable models with a ₂ grading of both the auxiliary (chain) and quantum (time) spaces. These models have a staggered disposition of the anisotropy parameter. The corresponding Yang–Baxter equations are written down and their solution for the gl(N) case is found. We analyze in details the N = 2 case and find the corresponding quantum group behind this solution. It can be regarded as the quantum group , with a matrix deformation parameter q such that (q )² = q ². The symmetry behind these models can also be interpreted as the tensor product of the (–1)-Weyl algebra by an extension of _q (gl(N)) with a Cartan generator related to deformation parameter –1.     

En">The Second Cohomology of sl ( m | 1 ) with Coefficients in its Enveloping Algebra is Trivial

Using techniques developed in a recent article by the authors, it is proved that the 2-cohomology of the Lie superalgebra sl ( m | 1 ) m 2, with coefficients in its enveloping algebra is trivial. The obstacles in solving the analogous problem for sl ( 3 | 2 ) are also discussed.     

Irrational free field resolutions forW(sl(n)) and extended Sugawara construction

The existence of Miura-type free field realizations is established for the extended conformal algebrasW(sl(n)) at irrational values of the screening parameter. The problem of the closure of the algebra is reduced to a finite dimensional quantum group problem. The structure of the Fock space resolution and the character formula are obtained for the irreducible modules. As graded vector spaces these modules are shown to be isomorphic to the space ofsl(n) singlets in affine level 1 modules. The isomorphism is given by the free field realization of .     

Polynomial relations in the centre ofU q (sl(N))

When the parameter of deformationq is a root of unity, the centre ofU _q (sl(N)) contains, besides the usualq-deformed Casimirs, a set of new generators, which are basically themth powers of all the Cartan generators ofU _q (sl(N)). All these central elements are, however, not independent. In this Letter, generalizing the well-known case ofU _q (sl(2)), we explicitly write polynomial relations satisfied by the generators of the centre. Application to the parametrization of irreducible representations and to fusion rules are sketched.On leave from SPht, CE Saclay, 91191 Gif-sur-Yvette Cedex, France.