103,140 research outputs found

    Local Weyl modules for equivariant map algebras with free abelian group actions

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    Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted) multiloop algebras. Local Weyl modules play an important role in the theory of finite-dimensional representations of loop algebras and quantum affine algebras. In the current paper, we extend the definition of local Weyl modules (previously defined only for generalized current algebras and twisted loop algebras) to the setting of equivariant map algebras where g is semisimple, X is affine of finite type, and the group is abelian and acts freely on X. We do so by defining twisting and untwisting functors, which are isomorphisms between certain categories of representations of equivariant map algebras and their untwisted analogues. We also show that other properties of local Weyl modules (e.g. their characterization by homological properties and a tensor product property) extend to the more general setting considered in the current paper.Comment: 18 pages. v2: Minor correction

    Degree cones and monomial bases of Lie algebras and quantum groups

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    We provide N\mathbb{N}-filtrations on the negative part Uq(n−)U_q(\mathfrak{n}^-) of the quantum group associated to a finite-dimensional simple Lie algebra g\mathfrak{g}, such that the associated graded algebra is a skew-polynomial algebra on n−\mathfrak{n}^-. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an N\mathbb{N}-filtration on any finite dimensional simple g\mathfrak{g}-module. We prove for type An\tt{A}_n, Cn\tt{C}_n, B3\tt{B}_3, D4\tt{D}_4 and G2\tt{G}_2 that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any g\mathfrak{g}.Comment: 26 pages, an inaccuracy correcte

    Conjugacy classes in Weyl groups and q-W algebras

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    We define noncommutative deformations Wqs(G)W_q^s(G) of algebras of functions on certain (finite coverings of) transversal slices to the set of conjugacy classes in an algebraic group GG which play the role of Slodowy slices in algebraic group theory. The algebras Wqs(G)W_q^s(G) called q-W algebras are labeled by (conjugacy classes of) elements ss of the Weyl group of GG. The algebra Wqs(G)W_q^s(G) is a quantization of a Poisson structure defined on the corresponding transversal slice in GG with the help of Poisson reduction of a Poisson bracket associated to a Poisson-Lie group G∗G^* dual to a quasitriangular Poisson-Lie group. The algebras Wqs(G)W_q^s(G) can be regarded as quantum group counterparts of W-algebras. However, in general they are not deformations of the usual W-algebras.Comment: 48 pages; some arguments in the proof of Proposition 12.2 are clarifie

    Differential Geometry of Group Lattices

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    In a series of publications we developed "differential geometry" on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first order differential calculi (over the algebra of functions) on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set. A particular class of digraphs are Cayley graphs, also known as group lattices. They are determined by a discrete group G and a finite subset S. There is a distinguished subclass of "bicovariant" Cayley graphs with the property that ad(S)S is contained in S. We explore the properties of differential calculi which arise from Cayley graphs via the above correspondence. The first order calculi extend to higher orders and then allow to introduce further differential geometric structures. Furthermore, we explore the properties of "discrete" vector fields which describe deterministic flows on group lattices. A Lie derivative with respect to a discrete vector field and an inner product with forms is defined. The Lie-Cartan identity then holds on all forms for a certain subclass of discrete vector fields. We develop elements of gauge theory and construct an analogue of the lattice gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear connections are considered and a simple geometric interpretation of the torsion is established. By taking a quotient with respect to some subgroup of the discrete group, generalized differential calculi associated with so-called Schreier diagrams are obtained.Comment: 51 pages, 11 figure

    Semidirekte Produkte endlicher Gruppenschemata: Gabriel-Köcher und Auslander-Reiten-Komponenten

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    Let G be a connected algebraic group over an algebraically closed field k. If char(k)=0, then there is a strong correspondence between representations of G and those of its Lie algebra \g. This changes dramatically in the situation char(k)=p>0 of positive characteristic. In this case, it turned out to be useful to approximate G by the ascending sequence (G_r)_{r\geq 1} consisting of its so-called Frobenius kernels. Each G_r is an infinitesimal group scheme, it is not uniquely determined by its group of k-rational points anymore; in fact, the latter are trivial. Its representation theory is equivalent to that of the dual Hopf algebra kG_r:=k[G_r]^* of its finite-dimensional coordinate ring k[G_r]. The Hopf algebra kG_1 is isomorphic to the restricted universal enveloping algebra \U_0(\g) of \g which shows that the representation theory of G_1 is equivalent to that of \g as a restricted Lie algebra. The representation theory of \g itself can be approximated by studying the family \{\Uchi(\g):\chi\in\g^*\} of its reduced enveloping algebras. Many results are known in case of reductive groups. In general, every group G is an extension of a reductive group H by a unipotent group U. If U is non-trivial, then the 'next best' case is that this extension splits, which in turn leads to a semidirect product G=U\rtimes H. Since the functor G\mapsto G_r is left exact, the rth Frobenius kernel of G is then the semidirect product U_r\rtimes H_r of the Frobenius kernels of U and H. It is exactly this point of view on which this thesis is build upon. Simple G_r-modules correspond to simple H_r-modules via the inflation functor mod(H_r)\to mod(G_r) defined by pullback along the projection G_r\to H_r and the principal indecomposable G_r-modules are induced by principal indecomposable H_r-modules. We will also establish a formula for the Gabriel quiver of the Hopf algebra kG_r and analyze the behaviour of the inflation functor in terms of Auslander-Reiten sequences. Furthermore, we will show that the stable Auslander-Reiten quiver of kG_r does not admit components of Euclidean type. We will formulate these results more general for finite group schemes and certain reduced enveloping algebras of restricted Lie algebras. As a major example, we will consider the Schrödinger group S:=H\rtimes SL(2), the semidirect product of the reductive group SL(2) with the Heisenberg group H\subseteq \SL(3), along with its quotient \Sbar\cong \G_a^2\rtimes SL(2) by the center. The group S has already been considered over the field of complex numbers and is related to physics. We will show that the Gabriel quivers of the Hopf algebras of the Frobenius kernels of S and \Sbar are all connected and take a closer look at reduced enveloping algebras U_\chi(\sbar) of the restricted Lie algebra \sbar=\Lie(\Sbar)

    On dimension growth of modular irreducible representations of semisimple Lie algebras

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    In this paper we investigate the growth with respect to pp of dimensions of irreducible representations of a semisimple Lie algebra g\mathfrak{g} over F‾p\overline{\mathbb{F}}_p. More precisely, it is known that for p≫0p\gg 0, the irreducibles with a regular rational central character λ\lambda and pp-character χ\chi are indexed by a certain canonical basis in the K0K_0 of the Springer fiber of χ\chi. This basis is independent of pp. For a basis element, the dimension of the corresponding module is a polynomial in pp. We show that the canonical basis is compatible with the two-sided cell filtration for a parabolic subgroup in the affine Weyl group defined by λ\lambda. We also explain how to read the degree of the dimension polynomial from a filtration component of the basis element. We use these results to establish conjectures of the second author and Ostrik on a classification of the finite dimensional irreducible representations of W-algebras, as well as a strengthening of a result by the first author with Anno and Mirkovic on real variations of stabilities for the derived category of the Springer resolution.Comment: 24 pages, v2 acknowledgements added; v3 references adde

    Unzerlegbare Moduln und AR-Komponenten von domestischen endlichen Gruppenschemata

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    In representation theory one studies modules to get an insight of the linear structures in a given algebraic object. Thanks to the theorem of Krull-Remak-Schmidt, any finite-dimensional module over a finite-dimensional algebra can be decomposed in a unique way into indecomposable modules. In this way, one reduces this problem to the study of indecomposable modules. In this work we are interested in representation theory of the group algebra of a finite group scheme. Examples of these algebras are given by group algebras of ordinary groups or by universal enveloping algebras of Lie algebras. Our main interest lies in the finite group schemes of domestic representation type. One of the main results in this work provides a full classification of the indecomposable modules for a certain subclass of the domestic finite group schemes. Based on this classification we will make some observations regarding the Auslander-Reiten quiver and geometric invariants which lead us to more general results. The shape of these quivers is well understood for the algebras we are investigating in this work. We will give a concrete description of the Euclidean components with respect to the McKay quiver of a certain binary polyhedral group scheme. An important fact for this work is that the module category of a group scheme is closed under taking tensor products. Friedlander and Suslin proved that the even cohomology ring of a finite group scheme is a finitely generated commutative algebra. The variety defined by this algebra is the cohomological support variety of the group scheme. These varieties contain many interesting information about the representation theory of the group schemes they are assigned to. In this work we will study the ramification index of a morphism between two support varieties. As we will see, this number has a connection to the ranks of the tubes in the Auslander-Reiten quivers
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