A representation of the quantum toroidal algebra of type sl N is de ned on every integrable irreducible highest weight module of the the quantum a ne algebra of type gl N : The q-version of the level-rank duality giving the reciprocal decomposition of the q-Fock space with respect to mutually commutative actions of U 0 q ( b gl N ) of level L and U 0 q ( b sl L ) of level N is described.

Let A be an abelian variety over a number field, Tl the ladic Tate module, and Gl the image of the Galois action on Tl. Then Hi(Gl, Tl) is a finite l-group which vanishes for l ≫ 0. We apply this bound for i = 1 and i = 2 to show that ifK denotes the field generated by all torsion points of A, then A(K) is the direct sum of its torsion group and a free abelian group.

(1.2) Let k be an algebraically closed field of arbitrary characteristic. For a ring A, an A-module means a left A-module, unless otherwise specified. However, an ideal of A means a two-sided ideal, not a left ideal. Amod denotes the category of finitely generated A-modules. For a group G, a G-module means a kG-module, where kG is the group algebra of G over k. If V is a finite dimensional vect...

We first present a filtration on the ring Ln of Laurent polynomials such that direct sum decomposition its associated graded grLn agrees with grLn, as module over complex general linear Lie algebra gl(n), into simple submodules. Next, generalizing modules occurring in we give some explicit constructions weight multiplicity-free irreducible representations gl(n).

The paper is devoted to the generalization of Lusztig’s q-analog of weight multiplicities to the Lie superalgebras gl(n,m) and spo(2n,M). We define such qanalogs Kλ,μ(q) for the typical modules and for the irreducible covariant tensor gl(n,m)-modules of highest weight λ. For gl(n,m), the defined polynomials have nonnegative integer coefficients if the weight μ is dominant. For spo(2n,M), we sho...

Hence, it is easy to count the orbits of GL n q under the conjugation action of U n q but seems hard to do the same for the group U n q itself! We fix some further notation. Let V n q be the vector space of column vectors, a module for GL n q . Recall that a flag is a totally ordered set of n− 1 nonzero proper subspaces of V n q . For g in GL n q let f g be the number of flags fixed by g, so it...

The generalized Kazhdan-Lusztig polynomials for the finite dimensional irreducible representations of the general linear superalgebra are computed explicitly. Using the result we establish a one to one correspondence between the set of composition factors of an arbitrary r-fold atypical gl m|n-Kac-module and the set of composition factors of some r-fold atypical gl r|r-Kac-module. The result of...