Algebras and Representation Theory最新文献

We introduce a new elliptic quantum toroidal algebra (U_{q,kappa ,p}({mathfrak {g}}_{tor})) associated with an arbitrary toroidal algebra ({mathfrak {g}}_{tor}). We show that (U_{q,kappa ,p}({mathfrak {g}}_{tor})) contains two elliptic quantum algebras associated with a corresponding affine Lie algebra (widehat{mathfrak {g}}) as subalgebras. They are analogue of the horizontal and the vertical subalgebras in the quantum toroidal algebra (U_{q,kappa }({mathfrak {g}}_{tor})). A Hopf algebroid structure is introduced as a co-algebra structure of (U_{q,kappa ,p}({mathfrak {g}}_{tor})) using the Drinfeld comultiplication. We also investigate the Z-algebra structure of (U_{q,kappa ,p}({mathfrak {g}}_{tor})) and show that the Z-algebra governs the irreducibility of the level ((k (ne 0),l))-infinite dimensional (U_{q,kappa ,p}({mathfrak {g}}_{tor}))-modules in the same way as in the elliptic quantum group (U_{q,p}(widehat{mathfrak {g}})). As an example, we construct the level (1, l) irreducible representation of (U_{q,kappa ,p}({mathfrak {g}}_{tor})) for the simply laced ({mathfrak {g}}_{tor}). We also construct the level (0, 1) representation of (U_{q,kappa ,p}({mathfrak {gl}}_{N,tor})) and discuss a conjecture on its geometric interpretation as an action on the torus equivariant elliptic cohomology of the affine (A_{N-1}) quiver variety.

Coloured quiver mutation, introduced by Buan, A.B., Thomas, H (Adv. Math. 222(3), 971–995 2009), gives a combinatorial interpretation of tilting in higher cluster categories. In type A work of Baur, K., Marsh, B. (Trans. Am. Math. Soc. 360(11), 5789-5803 2008) shows that m-coloured quivers and m-coloured quiver mutations have a nice geometrical description, given in terms of ((m+2))-angulations of a regular polygon, and rotations of an m-diagonal. In this paper, using such correspondence, we describe presentations of braid groups of type A arising from coloured quivers of mutation type A.

Let (mathfrak g) be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra (mathfrak h). We prove a unique factorization property for tensor products of parabolic Verma modules. More generally, we prove unique factorization for products of characters of parabolic Verma modules when restricted to certain subalgebras of (mathfrak h). These include fixed point subalgebras of (mathfrak h) under subgroups of diagram automorphisms of (mathfrak g) and twisted graph automorphisms in the affine case.

For a field (varvec{K}) and a finite group (varvec{G}), we study the lattice of preradicals over the group algebra (varvec{KG}), denoted by (varvec{KG})-(varvec{pr}). We show that if (varvec{KG}) is a semisimple algebra, then (varvec{KG})-(varvec{pr}) is completely described, and we establish conditions for counting the number of its atoms in some specific cases. If (varvec{KG}) is an algebra of finite representation type, but not a semisimple one, we completely describe (varvec{KG})-(varvec{pr}) when the characteristic of (varvec{K}) is a prime (varvec{p}) and (varvec{G}) is a cyclic (varvec{p})-group. For group algebras of infinite representation type, we show that the lattices of preradicals over two representative families of such algebras are not sets (in which case, we say the algebras are (varvec{mathfrak {p}})-large). Besides, we provide new examples of (varvec{mathfrak {p}})-large algebras. Finally, we prove the main theorem of this paper which characterizes the representation type of group algebras (varvec{KG}) in terms of their lattice of preradicals.

We propose a quantization algebra of the Loday-Ronco Hopf algebra (k[Y^infty ]), based on the Topological Recursion formula of Eynard and Orantin. We have shown in previous works that the Loday-Ronco Hopf algebra of planar binary trees is a space of solutions for the genus 0 version of Topological Recursion, and that an extension of the Loday Ronco Hopf algebra as to include some new graphs with loops is the correct setting to find a solution space for arbitrary genus. Here we show that this new algebra (k[Y^infty ]_h) is still a Hopf algebra that can be seen in some sense to be made precise in the text as a quantization of the Hopf algebra of planar binary trees, and that the solution space of Topological Recursion (mathcal {A}^h_{text {TopRec}}) is a subalgebra of a quotient algebra (mathcal {A}_{text {Reg}}^h) obtained from (k[Y^infty ]_h) that nevertheless doesn’t inherit the Hopf algebra structure. We end the paper with a discussion on the cohomology of (mathcal {A}^h_{text {TopRec}}) in low degree.

We study minimal triangular structures on abelian extensions. In particular, we construct a family of minimal triangular semisimple Hopf algebras and prove that the Hopf algebra (H_{b:y}) in the semisimple Hopf algebras of dimension 16 classified by Y. Kashina in 2000 is minimal triangular Hopf algebra with smallest dimension among non-trivial semisimple triangular Hopf algebras (i.e. not group algebras or their dual).

Let (Lambda = left[ begin{array}{cc} A &{} 0 M &{} B end{array}right] ) be an Artin algebra and (_BM_A) a B-A-bimodule. We prove that there is a triangle equivalence (D_{sg}(Lambda ) cong D_{sg}(A)coprod D_{sg}(B)) between the corresponding singularity categories if (_BM) is semi-simple and (M_A) is projective. As a result, we obtain a new method for describing the singularity categories of certain bounded quiver algebras.

Kuznetsov and Polishchuk provided a general algorithm to construct exceptional collections of maximal length for homogeneous varieties of type A, B, C, D. We consider the case of the spinor tenfold and we prove that the corresponding collection is full, i.e. it generates the whole derived category of coherent sheaves. We also verify strongness and purity of such collection. As a step of the proof, we construct some resolutions of homogeneous vector bundles which might be of independent interest.

In this paper we study the asymptotic behavior of the number of summands in tensor products of finite dimensional representations of affine (semi)group (super)schemes and related objects.