Alexander Molev

Yangians and their applications

Publisher Summary This chapter discusses the Yangians theory and their applications. The discovery of the Yangians is motivated by quantum inverse scattering theory. The Yangians form a remarkable family of quantum groups related to rational solutions of the classical Yang–Baxter equation. For each simple finite-dimensional Lie algebra α over the field of complex numbers, the corresponding Yangian Y (α) is defined as a canonical deformation of the universal enveloping algebra U (α[ x ]) for the polynomial current Lie algebra α[x]. The deformation is considered in the class of Hopf algebras which guarantees its uniqueness under natural “homogeneity” conditions. For any simple Lie algebra α, the Yangian Y (α) contains the universal enveloping algebra U (α) as a subalgebra. The Lie algebra α is regarded as fixed point subalgebra of an involution σ of the appropriate general linear Lie algebra. The defining relations of the Yangian is written in a form of a single ternary (or RTT) relation on the matrix of generators. It originates from quantum inverse scattering theory. The Yangians are primarily regarded as a vehicle for producing rational solutions of the Yang-Baxter equation which plays a key role in the theory of integrable models.

Gelfand–Tsetlin Bases for Classical Lie Algebras

This is a review paper on the Gelfand-Tsetlin type bases for representations of the classical Lie algebras. Different approaches to construct the original Gelfand-Tsetlin bases for representations of the general linear Lie algebra are discussed. Weight basis constructions for representations of the orthogonal and symplectic Lie algebras are reviewed. These rely on the representation theory of the B,C,D type twisted Yangians

A Littlewood-Richardson rule for factorial Schur functions

We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.

COIDEAL SUBALGEBRAS IN QUANTUM AFFINE ALGEBRAS

We introduce two subalgebras in the type A quantum affine algebra which are coideals with respect to the Hopf algebra structure. In the classical limit q→1 each subalgebra specializes to the enveloping algebra , where is a fixed point subalgebra of the loop algebra with respect to a natural involution corresponding to the embedding of the orthogonal or symplectic Lie algebra into . We also give an equivalent presentation of these coideal subalgebras in terms of generators and defining relations which have the form of reflection-type equations. We provide evaluation homomorphisms from these algebras to the twisted quantized enveloping algebras introduced earlier by Gavrilik and Klimyk and by Noumi. We also construct an analog of the quantum determinant for each of the algebras and show that its coefficients belong to the center of the algebra. Their images under the evaluation homomorphism provide a family of central elements of the corresponding twisted quantized enveloping algebra.

Finite-dimensional irreducible representations of twisted Yangians

We study quantized enveloping algebras called twisted Yangians. They are analogs of the Yangian Y(gl(N)) for the classical Lie algebras of B, C, and D series. The twisted Yangians are subalgebras in Y(gl(N)) and coideals with respect to the coproduct in Y(gl(N)). We give a complete description of their finite-dimensional irreducible representations. Every such representation is highest weight and we give necessary and sufficient conditions for an irreducible highest weight representation to be finite dimensional. The result is analogous to Drinfeld’s theorem for the ordinary Yangians. Its detailed proof for the A series is also reproduced. For the simplest twisted Yangians we construct an explicit realization for each finite-dimensional irreducible representation in tensor products of representations of the corresponding Lie algebras.

A Basis for Representations of Symplectic Lie Algebras

Abstract:A basis for each finite-dimensional irreducible representation of the symplectic Lie algebra ??(2n) is constructed. The basis vectors are expressed in terms of the Mickelsson lowering operators. Explicit formulas for the matrix elements of generators of ??(2n) in this basis are given. The basis is natural from the viewpoint of the representation theory of the Yangians. The key role in the construction is played by the fact that the subspace of ??(2n− 2) highest vectors in any finite-dimensional irreducible representation of ??(2n) admits a natural structure of a representation of the Yangian Y(??(2)).

REPRESENTATIONS OF REFLECTION ALGEBRAS

We study a class of algebras B(n,l) associated with integrable models with boundaries. These algebras can be identified with coideal subalgebras in the Yangian for gl(n). We construct an analog of the quantum determinant and show that its coefficients generate the center of B(n,l). We develop an analog of Drinfelds highest weight theory for these algebras and give a complete description of their finite-dimensional irreducible representations.

On Higher-Order Sugawara Operators

The higher Sugawara operators acting on the Verma modules over the affine Kac-Moody algebra at the critical level are related to the higher Hamiltonians of the Gaudin model due to the work of Feigin, Frenkel, and Reshetikhin [8]. An explicit construction of the higher Hamiltonians in the case of was given recently by Talalaev [20]. We propose a new approach to these results from the viewpoint of the vertex algebra theory by proving directly the formulas for the higher-order Sugawara operators. The eigenvalues of the operators in the Wakimoto modules of critical level are also calculated.

A New Fusion Procedure for the Brauer Algebra and Evaluation Homomorphisms

We give a new fusion procedure for the Brauer algebra by showing that all primitive idempotents can be found by evaluating a rational function in several variables. The function takes values in the Brauer algebra and has the form of a product of R-matrix type factors. In particular, this provides a one-parameter version of the fusion procedure for the symmetric group. The R-matrices are solutions of the Yang–Baxter equation associated with the classical Lie algebras of types B, C, and D. Moreover, we construct an evaluation homomorphism from a reflection equation algebra to and show that the fusion procedure provides an equivalence between natural tensor representations of with the corresponding evaluation modules.

On the R-Matrix Realization of Yangians and their Representations

Abstract.We study the Yangians