237 research outputs found

### A Remark on the Higher Capelli Identities

A simple proof of the higher Capelli identities is given.Comment: 5 pages, amste

### On the fusion procedure for the symmetric group

We give a new version of the fusion procedure for the symmetric group which
originated in the work of Jucys and was developed by Cherednik. We derive it
from the Jucys-Murphy formulas for the diagonal matrix units for the symmetric
group.Comment: 9 pages, reference to original work of Jucys (1971) was adde

### Yangians and transvector algebras

Olshanski's centralizer construction provides a realization of the Yangian
for the Lie algebra gl(n) as a subalgebra in the projective limit of a chain of
centralizers in the universal enveloping algebras. We give a modified version
of this construction based on a quantum analog of Sylvester's theorem. We then
use it to get an algebra homomorphism from the Yangian to the transvector
algebra associated with the general linear Lie algebras. The results are
applied to identify the elementary representations of the Yangian by
constructing their highest vectors explicitly in terms of elements of the
transvector algebra.Comment: Latex2e, 29 page

### 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 YangiansComment: 65 pages, bibliography is extended, minor corrections and changes are
mad

### Combinatorial bases for covariant representations of the Lie superalgebra gl(m|n)

Covariant tensor representations of gl(m|n) occur as irreducible components
of tensor powers of the natural (m+n)-dimensional representation. We construct
a basis of each covariant representation and give explicit formulas for the
action of the generators of gl(m|n) in this basis. The basis has the property
that the natural Lie subalgebras gl(m) and gl(n) act by the classical
Gelfand-Tsetlin formulas. The main role in the construction is played by the
fact that the subspace of gl(m)-highest vectors in any finite-dimensional
irreducible representation of gl(m|n) carries a structure of an irreducible
module over the Yangian Y(gl(n)). One consequence is a new proof of the
character formula for the covariant representations first found by Berele and
Regev and by Sergeev.Comment: 40 pages, minor corrections mad

### Irreducibility criterion for tensor products of Yangian evaluation modules

The evaluation homomorphisms from the Yangian Y(gl_n) to the universal
enveloping algebra U(gl_n) allow one to regard the irreducible
finite-dimensional representations of gl_n as Yangian modules. We give
necessary and sufficient conditions for irreducibility of tensor products of
such evaluation modules.Comment: 33 page

### A weight basis for representations of even orthogonal Lie algebras

A weight basis for each finite-dimensional irreducible representation of the
orthogonal Lie algebra o(2n) is constructed. The basis vectors are parametrized
by the D-type Gelfand--Tsetlin patterns. Explicit formulas for the matrix
elements of generators of o(2n) in this basis are given. The construction is
based on the representation theory of the Yangians and extends our previous
results for the symplectic Lie algebras.Comment: LaTeX2e, 21 page

### Pfaffian-type Sugawara operators

We show that the Pfaffian of a generator matrix for the affine Kac--Moody
algebra hat o_{2n} is a Segal--Sugawara vector. Together with our earlier
construction involving the symmetrizer in the Brauer algebra, this gives a
complete set of Segal--Sugawara vectors in type D.Comment: 4 page

### Center at the critical level for centralizers in type $A$

We consider the affine vertex algebra at the critical level associated with
the centralizer of a nilpotent element in the Lie algebra $\mathfrak{gl}_N$.
Due to a recent result of Arakawa and Premet, the center of this vertex algebra
is an algebra of polynomials. We construct a family of free generators of the
center in an explicit form. As a corollary, we obtain generators of the
corresponding quantum shift of argument subalgebras and recover free generators
of the center of the universal enveloping algebra of the centralizer produced
earlier by Brown and Brundan.Comment: 23 pages; revised version with more explicit formulas for generator

### Yangians and their applications

This is a review paper on the algebraic structure and representations of the
A type Yangian and the B, C, D types twisted Yangians. Some applications to
constructions of Casimir elements and characteristic identities for the
corresponding Lie algebras are also discussed.Comment: 55 page

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