On the homogeneized Weyl Algebra (original) (raw)
Related papers
On the homogenized Weyl Algebra
arXiv (Cornell University), 2012
The aim of this paper is to give relations between the category of finetely generated graded modules over the homogeneized Weyl algebra Bn, the finetely generated modules over the Weyl algebra An and the finetely generated graded modules over the Yoneda algebra B ! n of Bn. We will give these relations both at the level of the categories of modules and at the level of the derived categories.
A categorical approach to Weyl modules
Transformation Groups, 2010
Global and local Weyl Modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in were obtained. In this paper, we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.
On homomorphisms between global Weyl modules, Represent. Theory 15
2011
Let g be a simple finite dimensional Lie algebra and let A be a commutative associative algebra with unity. Global Weyl modules for the generalized loop algebra g ⊗ A were defined in [6, 7] for any dominant integral weight λ of g by generators and relations and further studied in [4]. They are expected to play a role similar to that of Verma modules in the study of categories of representations of g ⊗ A. One of the fundamental properties of Verma modules is that the space of morphisms between two Verma modules is either zero or one-dimensional and also that any non-zero morphism is injective. The aim of this paper is to establish an analogue of this property for global Weyl modules. This is done under certain restrictions on g, λ and A. A crucial tool is the construction of fundamental global Weyl modules in terms of fundamental local Weyl modules given in Section 3.
Communications in Algebra, 2019
We study structural properties of truncated Weyl modules. A truncated Weyl module WN (λ) is a local Weyl module for g[t]N = g ⊗ C[t] t N C[t] , where g is a finite-dimensional simple Lie algebra. It has been conjectured that, if N is sufficiently small with respect to λ, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when λ is a multiple of certain fundamental weights, including all minuscule ones for simply laced g. We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov-Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari-Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that g = sl2 related to Demazure flags and chains of inclusions of truncated Weyl modules.
On certain submodules of Weyl modules
2013
For k = 1, 2, ..., n−1 let V k = V (λ k) be the Weyl module for the special orthogonal group G = SO(2n + 1, F) with respect to the k-th fundamental dominant weight λ k of the root system of type Bn and put Vn = V (2λn). It is well known that all of these modules are irreducible when char(F) = 2 while when char(F) = 2 they admit many proper submodules. In this paper, assuming that char(F) = 2, we prove that V k admits a chain of submodules V k = M k ⊃ M k−1 ⊃ ... ⊃ M1 ⊃ M0 ⊃ M−1 = 0 where Mi ∼ = Vi for 1, ..., k−1 and M0 is the trivial 1-dimensional module. We also show that for i = 1, 2, ..., k the quotient Mi/Mi−2 is isomorphic to the so called i-th Grassmann module for G. Resting on this fact we can give a geometric description of Mi−1/Mi−2 as a submodule of the i-th Grassmann module. When F is perfect G ∼ = Sp(2n, F) and Mi/Mi−1 is isomorphic to the Weyl module for Sp(2n, F) relative to the i-th fundamental dominant weight of the root system of type Cn. All irreducible sections of the latter modules are known. Thus, when F is perfect, all irreducible sections of V k are known as well.
Applied Categorical Structures, 2009
Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there is always a naturally associated Koszul theory. To obtain this, the notions of Koszul algebras, linear modules and Koszul duality are extended to additive (graded) categories over a field. The main focus of this paper is to provide these generalizations and the necessary preliminaries.
Weyl modules for the hyperspecial
2014
We develop the theory of global and local Weyl modules for the hyperspecial maximal parabolic subalgebra of type A (2) 2n. We prove that the dimension of a local Weyl module depends only on its highest weight, thus establishing a freeness result for global Weyl modules. Furthermore, we show that the graded local Weyl modules are level one Demazure modules for the corresponding affine Lie algebra. In the last section we derive the same results for the special maximal parabolic subalgebras of the twisted affine Lie algebras not of type A (2) 2n .