Deformations of enveloping algebra of Lie superalgebrasl(m,n)

Theq-deformations of the universal enveloping algebra ofsl(m, n) are considered, a Poincaré-Birkhoff-Witt type theorem is proved for these deformations, and the extra relations which are needed to definesl(m, n) as a contragredient algebra in addition to the Serre-type relations are identified with proof.     

Aq-difference analogue of U(g) and the Yang-Baxter equation

Aq-difference analogue of the universal enveloping algebra U(g) of a simple Lie algebra g is introduced. Its structure and representations are studied in the simplest case g=sl(2). It is then applied to determine the eigenvalues of the trigonometric solution of the Yang-Baxter equation related to sl(2) in an arbitrary finite-dimensional irreducible representation.     

Serre-type relations for special linear Lie superalgebras

It is pointed out that, for m, n 2, the naive Serre presentation corresponding to the simplest Cartan matrix of sl(m, n) does not define the Lie superalgebra sl(m, n) but a larger algebra s(m, n) of which sl(m, n) is a nontrivial quotient. The supplementary relations for the generators are found and the definition of the q-deformed universal enveloping algebra of sl(m, n) is modified accordingly.     

Construction of cyclic representations of quantum algebras at q p =1 from their regular representations

Cyclic representations of maximal dimension of the quantum algebra U _q L associated with any finite-dimensional simple Lie algebra L are studied from its regular representation at q ^p =1, which is proved to be a quotient module of itself as a left module with respect to some submodules. The general theory is given after an instructive example U _q sl(2) is studied. Another explicit example U _q sl(3) is also presented.This work is supported in part by the National Natural Science Foundation of China. Author Fu is also supported by the Jilin Provincial Science and Technology Foundation of China     

sl(N) Onsager's algebra and integrability

We define ansl(N) analog of Onsager's algebra through a finite set of relations that generalize the Dolan-Grady defining relations for the original Onsager's algebra. This infinite-dimensional Lie algebra is shown to be isomorphic to a fixed-point subalgebra ofsl(N) loop algebra with respect to a certain involution. As the consequence of the generalized Dolan-Grady relations a Hamiltonian linear in the generators ofsl(N) Onsager's algebra is shown to posses an infinite number of mutually commuting integrals of motion.     

A two-parameter deformation of the universal enveloping algebra ofsl(3, C)

Starting from a certain multi-parameter matrix that satisfies the quantum Yang-Baxter equation, a two-parameter deformation of the universal enveloping algebra of the simple Lie algebrasl(3, C) is derived. It is shown that this has same product relations and antipode as the standard one-parameter deformationU _q(sl(3, C)) but has a different coproduct. It is also shown that there exists a Hopf algebra whose product relations are merely the commutation relations ofsl(3, C) itself, but whose coproduct is different from the usual one for the universal enveloping algebra ofsl(3, C).     

Yangian and Quantum Universal Solutions¶of Gervais–Neveu–Felder Equations

We construct universal Drinfel'd twists defining deformations of Hopf algebra structures based upon simple Lie algebras and contragredient simple Lie superalgebras. In particular, we obtain deformed and dynamical double Yangians. Some explicit realisations as evaluation representations are given for sl _N , sl(1|2) and osp(1|2). Received: 11 May 2001 / Accepted: 16 October 2001     

New solvable lattice models in three dimensions

In this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models. The first is the Zamolodchikov model, which is the only nontrivial model on a three-dimen-sional lattice so far solved. The second is the chiral Potts model on the square lattice and its generalization associated with theU _q(sl(n)) algebra, which is of current interest due to its connections with high-genus algebraic curves and with representations of quantum groups at roots of unity. We show that this last sl(n)-generalized chiral Potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square-lattice layers with anN- valued (N2) spin at each site. Further, in theN=2 case this three-dimen-sional model reduces (after a modification of the boundary conditions) to the Zamolodchikov model we mentioned above.     

On the universalR-matrix ofU_q \widehat{sl}_2 at roots of unityat roots of unity

We show that the action of the universalR-matrix of the affine quantum algebra, whenq is a root of unity, can be renormalized by some scalar factor to give a well-defined nonsingular expression, satisfying the Yang-Baxter equation. It can be reduced to intertwining operators of representations, corresponding to Chiral Potts, if the parameters of these representations lie on the well-known algebraic curve.We also show that the affine forq is a root of unity from the autoquasitriangular Hopf algebra in the sense of Reshetikhin.This work is supported by NATO linkage grant LG 9303057.     

Irreducible Representations of Quantum sl(3) Enveloping Algebra

By referring to the construction method of finding representations of classical U(n) group, we developed a parallel method with which one can construct irreducible representations of quantum sl(3) enveloping algebra explicitly.     

Crystal algebra

We define the crystal algebra, an algebra which has a base of elements of crystal bases of a quantum group. The multiplication is defined by the tensor product rule of crystal bases. A universal n-colored crystal algebra is defined. We study the relation between those algebras and the tensor algebras of the crystal algebra of U _q (sl(2)) and give a presentation by generators and relations for the case of U _q (sl(n)).     

Representation theory of osp(1,2) q

We investigate the representations of the osp(1, 2) _q algebra. We derive all the finite-dimensional irreducible representations, whenq is not a root of unity. We also discuss the connection between those of osp(1, 2) _q and sl(2) _q .     

On affine yangians

A Yangian , a deformation of the universal enveloping algebra of the two-dimensional loop algebra sl(2) C [t ^–1,t;u], is constructed. This deformation is an analogue of a Yangian which was constructed by V. Drinfeld for any simple Lie algebra. The PBW theorem for is proved and some representations are constructed. Like usual Yangians, possesses a one-dimensional group of auto- morphisms and at zero level - a two-dimensional group of automorphisms. This observation allows one to conjecture that the representation theory of should give rise to new solutions of QYBE.Yangians of other affine algebras can be constructed similarly and they enjoy similar properties.     

Representations of the cyclically symmetric q-deformed algebra U q(so3)

An algebra homomorphism from the nonstandard q-deformed (cyclically symmetric) algebra U _q(so₃) to the extension Û _q(sl₂) of the Hopf algebra U _q(sl₂) is constructed. Not all irreducible representations (IR) of U _q(sl₂) can be extended to representations of Û _q(sl₂). Composing the homomorphism with irreducible representations of Û _q(sl₂) we obtain representations of U _q(so₃). Not all of these representations of U _q(so₃) are irreducible. Reducible representations of U _q(so₃) are decomposed into irreducible components. In this way we obtain all IR of U _q(so₃) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so₃ when q 1.     

The jordanian deformation of Su(2) and clebsch-gordan coefficients

Representation theory for the Jordanian quantum algebraU _h (sl(2)) is developed using a nonlinear relation between its generators and those of sl(2). Closed form expressions are given for the action of the generators ofU _h (sl(2)) on the basis vectors of finite dimensional irreducible representations. In the tensor product of two such representations, a new basis is constructed on which the generators ofU _h (sl(2)) have a simple action. Using this basis, a general formula is obtained for the Clebsch-Gordan coefficients ofU _h (sl(2)). Some remarkable properties of these Clebsch-Gordan coefficients are derived. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

The Lie algebra of the sl(2, ℂ)-valued automorphic functions on a torus

It is shown that the Lie algebra of the automorphic, meromorphic sl(2, )-valued functions on a torus is a geometric realization of a certain infinite-dimensional finitely generated Lie algebra. In the trigonometric limit, when the modular parameter of the torus goes to zero, the former Lie algebra goes over into the sl(2, )-valued loop algebra, while the latter goes into the Lie algebra (A ₁ ⁽¹⁾ )/(centre).     

Cremmer–Gervais <Emphasis Type="Italic">r</Emphasis>-Matrices and the Cherednik Algebras of Type <Emphasis Type="Italic">GL</Emphasis><Subscript>2</Subscript>

We give an interpretation of the Cremmer–Gervais r-matrices for \mathfraksl_n{\mathfrak{sl}_n} in terms of actions of elements in the rational and trigonometric Cherednik algebras of type GL ₂ on certain subspaces of their polynomial representations. This is used to compute the nilpotency index of the Jordanian r-matrices, thus answering a question of Gerstenhaber and Giaquinto. We also give an interpretation of the Cremmer–Gervais quantization in terms of the corresponding double affine Hecke algebra.     

The Clebsch-Gordan coefficients ofsl(2,R) in the parabolic basis

We realize the Lie algebra sl(2,R) in terms of second-order differential operators defined on a dense common domain of square-integrable functions on a two-chart space, where the self-adjoint extension(s) (families) lead to all (and only) self-adjoint irreducible representations of the algebra, single- as well as multi-valued over the group. This allows for a rather straight-forward evaluation of the Clebsch-Gordan coefficients of sl(2,R) in the parabolic subalgebra basis.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.I would like to thank the hospitality of Prof. J. A. de Azcárraga, at the Instituto de Fí'sica Teórica, Universidad de Valencia, where this review was written.     

On twist quantizations of <Emphasis Type="Italic">D</Emphasis> = 4 Lorentz and Poincare algebras

We use the decomposition of o(3, 1) = sl(2; ℂ)₁ ⊕sl(2; ℂ)₂ in order to describe nonstandard quantum deformation of o(3, 1) linked with Jordanian deformation of sl(2; ℂ). Using the twist quantization technique, we obtain the deformed coproducts and antipodes, which can be expressed in terms of real physical Lorentz generators. We describe the extension of the considered deformation of D = 4 Lorentz algebra to the twist deformation of D = 4 Poincare algebra with dimensionless deformation parameter. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.