Algebras and Representation Theory最新文献

This paper is about the highest weight module theory for affine superalgebra (widetilde{mathfrak g}) of ({mathfrak g}={mathfrak {osp}_{1|2}(mathbb C)}) and loop superalgebra ({mathfrak g}{otimes }{mathbb {C}}[t,t^{-1}]). Among the main results, we obtain (i) a necessary and sufficient condition for Verma type (ell )-highest weight (widetilde{mathfrak g})-modules to be irreducible; (ii) a free field(-like) realization of all irreducible (ell )-highest weight (widetilde{mathfrak g})-modules; (iii) a character formula for all irreducible (ell )-highest weight (widetilde{mathfrak g})-modules with finite dimensional weight spaces. We also obtain three similar results for highest weight ({mathfrak g}{otimes }{mathbb {C}}[t,t^{-1}])-modules.

We first construct an action of the extended double affine braid group (mathcal {ddot{B}}) on the quantum toroidal algebra (U_{q}(mathfrak {g}_{textrm{tor}})) in untwisted and twisted types. As a crucial step in the proof, we obtain a finite Drinfeld new style presentation for a broad class of quantum affinizations. In the simply laced cases, using our action and certain involutions of (mathcal {ddot{B}}) we produce automorphisms and anti-involutions of (U_{q}(mathfrak {g}_{textrm{tor}})) which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements C and (k_{0}^{a_{0}}dots k_{n}^{a_{n}}) up to inverse. This can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups used by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type A due to Miki which have been instrumental in the study of the structure and representation theory of (U_{q}(mathfrak {sl}_{n+1,textrm{tor}})).

The polynomial ideals studied by A. Garsia and C. Procesi play an important role in the theory of Kostka polynomials. We give multiparameter flat deformations of these ideals and define an action of the extended affine symmetric group on the corresponding quotient algebras multiplied by the sign representation. We show that the images of these modules under the affine Schur-Weyl duality are dual to the local Weyl modules for the loop algebra (mathfrak {sl}_{n+1}[t^{pm 1}]).

We use a unified elementary approach to prove the second part of classical, mixed, super, and mixed super Schur-Weyl dualities for general linear groups and supergroups over an infinite ground field of arbitrary characteristic. These dualities describe the endomorphism algebras of the tensor space and mixed tensor space, respectively, over the group algebra of the symmetric group and the Brauer wall algebra, respectively. Our main new results are the second part of the mixed Schur-Weyl dualities and mixed super Schur-Weyl dualities over an infinite ground field of positive characteristic.

Let (mathscr {A}) be a connected cochain DG algebra such that (H(mathscr {A})) is a Noetherian graded algebra. We give some criteria for (mathscr {A}) to be homologically smooth in terms of the singularity category, the cone length of the canonical module k and the global dimension of (mathscr {A}). For any cohomologically finite DG (mathscr {A})-module M, we show that it is compact when (mathscr {A}) is homologically smooth. If (mathscr {A}) is in addition Gorenstein, we get

where (textrm{CMreg}M) is the Castelnuovo-Mumford regularity of M, (textrm{depth}_{mathscr {A}}mathscr {A}) is the depth of (mathscr {A}) and ( mathrm {Ext.reg}, M) is the Ext-regularity of M.

We find new presentations of the modified Ariki-Koike algebra (known also as Shoji’s algebra) (mathcal {H}_{n,r}) over an integral domain R associated with a set of parameters (q,u_1,ldots ,u_r) in R. It turns out that the algebra (mathcal {H}_{n,r}) has a set of generators (t_1,ldots ,t_n) and (g_1,ldots g_{n-1}) subject to some defining relations similar to the relations of Yokonuma-Hecke algebra. We also obtain a presentation of (mathcal {H}_{n,r}) which is independent of the choice of (u_1,ldots u_r). As applications of the presentations, we find an explicit and direct isomorphism between the modified Ariki-Koike algebras with different choices of parameters ((u_1,ldots ,u_r)). We also find an explicit trace form on the algebra (mathcal {H}_{n,r}) which is symmetrizing provided the parameters (u_1,ldots , u_r) are invertible in R. We show that the symmetric group (mathfrak {S}(r)) acts on the algebra (mathcal H_{n,r}), and find a basis and a set of generators of the fixed subalgebra (mathcal H_{n,r}^{mathfrak {S}(r)}).

We show that the order type of the simplest version of a hammock for string algebras lies in the class of finite description linear orders–the smallest class of linear orders containing (textbf{0}), (textbf{1}), and that is closed under isomorphisms, finite order sum, anti-lexicographic product with (omega ) and (omega ^*), and shuffle of finite subsets–using condensation (localization) of linear orders as a tool. We also introduce two finite subsets of the set of bands and use them to describe the location of left (mathbb {N})-strings in the completion of hammocks.

To every Hopf heap or quantum cotorsor of Grunspan a Hopf algebra of translations is associated. This translation Hopf algebra acts on the Hopf heap making it a Hopf-Galois co-object. Conversely, any Hopf-Galois co-object has the natural structure of a Hopf heap with the translation Hopf algebra isomorphic to the acting Hopf algebra. It is then shown that this assignment establishes an equivalence between categories of Hopf heaps and Hopf-Galois co-objects.

Let H be a finite-dimensional Hopf algebra over an algebraically closed field (Bbbk ) with the dual Chevalley property. We prove that H is of finite corepresentation type if and only if it is coNakayama, if and only if the link quiver (textrm{Q}(H)) of H is a disjoint union of basic cycles, if and only if the link-indecomposable component (H_{(1)}) containing (Bbbk 1) is a pointed Hopf algebra and the link quiver of (H_{(1)}) is a basic cycle.

We prove that the duals of the quantum Frobenius morphisms and their splittings by Lusztig are compatible with quantum cluster monomials. After specialization, we deduce that the canonical Frobenius splittings on flag varieties are compatible with cluster algebra structures on Schubert cells.