On the representation theory of the vertex algebra L−5/2(sl(4))

We study the representation theory of non-admissible simple affine vertex algebra [Formula: see text]. We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra [Formula: see text], and show that it generates the maximal ideal in [Formula: see text]. We classify irreducible [Formula: see text]-modules in the category [Formula: see text], and determine the fusion rules between irreducible modules in the category of ordinary modules [Formula: see text]. It turns out that this fusion algebra is isomorphic to the fusion algebra of [Formula: see text]. We also prove that [Formula: see text] is a semi-simple, rigid braided tensor category. In our proofs, we use the notion of collapsing level for the affine [Formula: see text]-algebra, and the properties of conformal embedding [Formula: see text] at level [Formula: see text] from D. Adamovic et al. [Finite vs infinite decompositions in conformal embeddings, Comm. Math. Phys. 348 (2016) 445–473.]. We show that [Formula: see text] is a collapsing level with respect to the subregular nilpotent element [Formula: see text], meaning that the simple quotient of the affine [Formula: see text]-algebra [Formula: see text] is isomorphic to the Heisenberg vertex algebra [Formula: see text]. We prove certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction functor [Formula: see text]. It turns out that the properties of [Formula: see text] are more subtle than in the case of minimal reduction.

Cohomology of simple modules for sl3(k) in characteristic 3

In this paper we calculate cohomology of a classical Lie algebra of type A2 over an algebraically field k of characteristic p = 3 with coefficients in simple modules. To describe their structure we will consider them as modules over an algebraic group SL3(k). In the case of characteristic p = 3, there are only two peculiar simple modules: a simple that module isomorphic to the quotient module of the adjoint module by the center, and a one-dimensional trivial module. The results on the cohomology of simple nontrivial module are used for calculating the cohomology of the adjoint module. We also calculate cohomology of the simple quotient algebra Lie of A2 by the center.

Permutation orbifolds of 𝔰𝔩2 vertex operator algebras

We analyze two types of permutation orbifolds: (i) S2-orbifolds of the universal level k vertex operator algebra Vk(𝔰𝔩2) and of its simple quotient Lk(𝔰𝔩2), and (ii) the S3-orbifold of the level one simple vertex operator algebra L1(𝔰𝔩2). We determine their structures and discuss related W-algebras.

Maximal Weight Composition Factors for Weyl Modules

Fix an irreducible (finite) root system R and a choice of positive roots. For any algebraically closed field k consider the almost simple, simply connected algebraic group Gk over k with root system k. One associates with any dominant weight λ for R two Gk-modules with highest weight λ, the Weyl module V( λ)k and its simple quotient L(µ)k . Let λ and μ be dominant weights with μ < j such that μ is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists k such that L(μ)k is a composition factor of V(j)k , and they exhibit an example in type E8 where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact: one that uses a classiffication of the possible pairs (λ, μ), and another that relies only on the classiûcation of root systems.

Squeakr: An Exact and Approximate k-mer Counting System

Motivationk-mer-based algorithms have become increasingly popular in the processing of high-throughput sequencing (HTS) data. These algorithms span the gamut of the analysis pipeline from k-mer counting (e.g., for estimating assembly parameters), to error correction, genome and transcriptome assembly, and even transcript quantification. Yet, these tasks often use very different k-mer representations and data structures. In this paper, we set forth the fundamental operations for maintaining multisets of k-mers and classify existing systems from a data-structural perspective. We then show how to build a k-mer-counting and multiset-representation system using the counting quotient filter (CQF), a feature-rich approximate membership query (AMQ) data structure. We introduce the k-mer-counting/querying system Squeakr (Simple Quotient filter-based Exact and Approximate Kmer Representation), which is based on the CQF. This off-the-shelf data structure turns out to be an efficient (approximate or exact) representation for sets or multisets of k-mers.ResultsSqueakr takes 2×−3;4.3× less time than the state-of-the-art to count and perform a random-point-query workload. Squeakr is memory-efficient, consuming 1.5X–4.3X less memory than the state-of-the-art. It offers competitive counting performance, and answers point queries (i.e. queries for the abundance of a particular k-mer) over an order-of-magnitude faster than other systems. The Squeakr representation of the k-mer multiset turns out to be immediately useful for downstream processing (e.g., de Bruijn graph traversal) because it supports fast queries and dynamic k-mer insertion, deletion, and modification.Availabilityhttps://github.com/splatlab/[email protected]

The structure of the C*-algebra of a locally injective surjection

In this paper we investigate the ideal structure of the C*-algebra of a locally injective surjection introduced by the second-named author. Our main result is that a simple quotient of the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is either a full matrix algebra, a crossed product of a minimal homeomorphism of a compact metric space of finite covering dimension, or it is purely infinite and hence covered by the classification result of Kirchberg and Phillips. It follows in particular that if the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is simple, then it is automatically purely infinite, unless the map in question is a homeomorphism. A corollary of this result is that if the C*-algebra of a one-sided subshift is simple, then it is also purely infinite.

Decomposing locally compact groups into simple pieces

We present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple.Two appendices introduce results and examples around the concept of quasi-product.

KOSTANT'S PROBLEM AND PARABOLIC SUBGROUPS

Let be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I ⊆ S let I be the corresponding semi-simple subalgebra of . Denote by WI the Weyl group of I and let w○ and wI○ be the longest elements of W and WI, respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjects onto the space of all ad-finite linear transformations of a given module, is the same for the simple highest weight I-module LI(x) of highest weight x ⋅ 0, x ∈ WI, as the answer for the simple highest weight -module L(xwI○w○) of highest weight xwI○w○ ⋅ 0. We also give a new description of the unique quasi-simple quotient of the Verma module Δ(e) with the same annihilator as L(y), y ∈ W.

Finite normal edge-transitive Cayley graphs

An approach to analysing the family of Cayley graphs for a finite group G is given which identifies normal edge-transitive Cayley graphs as a sub-family of central importance. These are the Cayley graphs for G for which a subgroup of automorphisms exists which both normalises G and acts transitively on edges. It is shown that, for a nontrivial group G, each normal edge-transitive Cayley graph for G has at least one homomorphic image which is a normal edge-transitive Cayley graph for a characteristically simple quotient group of G. Moreover, given a normal edge-transitive Cayley graph ΓH for a quotient group G/H, necessary and sufficient conditions are obtained for the existence of a normal edge-transitive Cayley graph Γ for G which has ΓH as a homomorphic image, and a method for obtaining all such graphs Γ is given.