Contents §0. Introduction §1. The Yangian §2. The quantum determinant and the centre of §3. The twisted Yangian §4. The Sklyanin determinant and the centre of §5. The quantum contraction and the quantum Liouville formula for the Yangian §6. The quantum contraction and the quantum Liouville formula for the twisted Yangian §7. The quantum determinant and the Sklyanin determinant of block matrices Bibliography

Yangians and 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

Gelfand–Tsetlin Bases for Classical Lie Algebras

We present the commutation and anticommutation relations, satisfied by the generators of the graded general linear, special linear and orthosymplectic Lie algebras, in canonical two‐index matrix form. Tensor operators are constructed in the enveloping algebra, including powers of the matrix of generators. Traces of the latter are shown to yield a sequence of Casimir invariants. The transformation properties of vector operators under these algebras are also exhibited. The eigenvalues of the quadratic Casimir invariants are given for the irreducible representations of ggl(m ‖ n), gsl(m ‖ n), and osp(m ‖ n) in terms of the highest‐weight vector. In such representations, characteristic polynomial identities of order (m+n), satisfied by the matrix of generators, are obtained in factorized form. These are used in each case to determine the number of independent Casimir invariants of the trace form.

Casimir invariants and characteristic identities for generators of the general linear, special linear and orthosymplectic graded Lie algebras

A method of computing eigenvalues of certain types of Casimir invariants has been developed for simple and classical Lie algebras. Especially these eigenvalues for algebrasA n , B n , C n , D n , and G 2 have been computed in closed terms. We also enumerate numbers and functional forms of all linearly independent vector operators in terms of generators in any irreducible representation of these algebras. Some polynomial identities among infinitesimal generators of these algebras are derived by means of the same technique.

Casimir invariants and vector operators in simple and classical Lie algebras

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.

Yangians and their applications

A hierarchy of tensor identities, satisfied by the generators of the general linear group GL(n), is obtained in terms of two different sets of invariants. An application to the identification of irreducible representations and the decomposition of reducible representations is described. Similar results are obtained for the generators of orthogonal, pseudo‐orthogonal, and symplectic groups.

H. S. Green

Papers

CHARACTERISTIC IDENTITIES FOR GENERATORS OF GL(n), O(n) AND Sp(n).