Algebras and Representation Theory最新文献

We study local algebras, which are structures similar to (mathbb {Z})-graded algebras concentrated in degrees (-1,0,1), but without a product defined for pairs of elements at the same degree (pm 1). To any triple consisting of a Kac–Moody algebra ({mathfrak g}) with an invertible and symmetrisable Cartan matrix, a dominant integral weight of ({mathfrak g}) and an invariant symmetric bilinear form on ({mathfrak g}), we associate a local algebra satisfying a restricted version of associativity. From it, we derive a local Lie superalgebra by a commutator construction. Under certain conditions, we identify generators which we show satisfy the relations of the tensor hierarchy algebra W previously defined from the same data. The result suggests that an underlying structure satisfying such a restricted associativity may be useful in applications of tensor hierarchy algebras to extended geometry.

Let G be a connected simply connected semisimple complex algebraic group and (P, subset , G) a parabolic subgroup. We give a necessary and sufficient condition for a line bundle — on the blow-up of the generalized flag variety G/P along a smooth Schubert variety — to be ample (respectively, nef). Furthermore, it is shown that every such nef line bundle is actually globally generated. As a consequence, we are able to describe when such a blow-up is (weak) Fano.

This paper extends the concept of fundamental superalgebra, crucial in Kemer’s resolution of the Specht problem, to the framework of superalgebras equipped with a superinvolution. We aim to characterize the class of these special algebras and provide concrete examples. Some of them are developed by exploring connections with varieties of superalgebras with superinvolution which are minimal with respect to their corresponding exponent.

In this paper, using crystal theory, we establish the existence of a new family of irreducible components arising in the tensor product of two irreducible integrable highest weight modules over symmetrizable Kac–Moody algebras. This work is motivated by the Schur positivity conjecture, Kostant’s conjecture, and Wahl’s conjecture. Furthermore, we prove the Schur positivity conjecture in full generality for finite-dimensional simple Lie algebras under the assumption that (lambda>> mu ); that is, if (lambda ) and (mu ) are the dominant weights in the tensor product, then (lambda +wmu ) remains dominant for all w in the Weyl group.

Describing the decomposition of the Foulkes module (F_b^a) into irreducible Specht modules is an open problem when both (a,b > 3). In this article we provide a new approach for the Generalized Foulkes module (F_{nu }^a) (with arbitrary partition (nu ) of b) through its restriction to a maximal Young subgroup ({S_b times S_{ab -b}}).

In this article, we study the decomposition into irreducible components of the fixed point locus under the action of (Gamma ) a finite subgroup of (textrm{SL}_2(mathbb {C})) of the smooth Nakajima quiver variety of the Jordan quiver. The quiver variety associated with the Jordan quiver is either isomorphic to the punctual Hilbert scheme of (mathbb {C}^2) or to the Calogero-Moser space. We describe the irreducible components using quiver varieties over the McKay’s quiver associated with the finite subgroup (Gamma ). We moreover give a general combinatorial model of the indexing set of these irreducible components in terms of certain elements of the root lattice of the affine Lie algebra associated with (Gamma ). Finally, we prove that for every projective, symplectic resolution of a wreath product singularity, there exists an irreducible component of the fixed point locus of the punctual Hilbert scheme of the plane that is isomorphic to the resolution.

In this paper, we count the number of aCM vector bundles with a toric structure on the Veronese surface, up to a twist by hyperplane divisor class. The main ingredients are the equivalence introduced by A. A. Klyachko between toric vector bundles and certain decreasing filtrations, together with a combinatorial criterion for the vanishing of cohomology derived from this framework.

In this paper, we study the mirabolically induced modules for the loop and affine algebras of (mathfrak {sl}_m). Among the main results, we give a free field realization of all irreducible mirabolically induced modules, and obtain a character formula for such modules with finite dimensional weight spaces.

We define the twisted affine Yangian of type C and construct surjective homomorphisms from twisted affine Yangians of type C to the universal enveloping algebra of the rectangular W-algebra associated with (mathfrak {so}(ln)) and a nilpotent element whose Jordan form corresponds to the partition ((l^n)) in the case when l and n are even.