(1.2) Let k be an algebraically closed field of arbitrary characteristic. For a ring A, an A-module means a left A-module, unless otherwise specified. However, an ideal of A means a two-sided ideal, not a left ideal. Amod denotes the category of finitely generated A-modules. For a group G, a G-module means a kG-module, where kG is the group algebra of G over k. If V is a finite dimensional vect...

We give an analogue for vertex operator algebras and superalgebras of the notion of endomorphism ring of a vector space by means of a notion of “local system of vertex operators” for a (super) vector space. We first prove that any local system of vertex operators on a (super) vector space M has a natural vertex (super)algebra structure with M as a module. Then we prove that for a vertex (operat...

1. [10 points] Determine whether the following statements are true or false (you have to include proofs/counterexamples): (a) Let R be an integral domain, F – a free R-module of finite rank, and M – a torsion R-module. Then there is no injective homomorphism from F to M . Solution: True. Suppose there was an injective homomorpism φ : F → M . Then let N = φ(F ); N is a submodule of M , and there...

We prove the existence of a homomorphism vertex algebras, from vacuum Verma module over loop algebra an untwisted affine algebra, whose construction is analogous to that Feigin-Frenkel at critical level algebra.

The objective of this research is to develop computer software that can recognize the Ancient Tamil handwritten characters by using the genetic algorithm technique (RATHCPM). The system consists of 5 main modules, which are: 1) image acquisition module, 2) image preprocessing module, 3) feature extraction module, 4) character recognition module, and 5) display result module. Each module has the...

Simplicial toric stack bundles are smooth Deligne-Mumford stacks over smooth varieties with fibre a toric Deligne-Mumford stack. We compute the Grothendieck K-theory of simplicial toric stack bundles and study the Chern character homomorphism.

Definition 1.2. The module of relative differential forms of B over A is a Bmodule ΩB/A, together with an A-derivation d : B → ΩB/A satisfying the following universal property: for any B-module M and for any A-derivation d : B → M , there exists a unique B-module homomorphism f : ΩB/A → M such that d ′ = f ◦d. If DerA(B,M) denotes the set of all A-derivations from B into M , then we have a natu...

Let A and B be commutative rings such that B is an A-algebra. We define ΩB/A to be the B-module of regular differentials over A. By definition, this B-module ΩB/A is a unique B-module which has the following property. For a B-module M we denote by DerA(B,M ) the set of all A-derivations (an A-homomorphism φ:B → M is called an A-derivation if φ(xy) = xφ(y) + yφ(x) and φ(x) = 0 for any x ∈ A ). T...