scholarly journals RATIONAL REPRESENTATIONS OF GL2

2010 ◽  
Vol 53 (2) ◽  
pp. 257-275 ◽  
Author(s):  
VANESSA MIEMIETZ ◽  
WILL TURNER
Keyword(s):  
Representation Theory ◽  
Basic Algebra ◽  
Combinatorial Structure ◽  
Closed Field ◽  
Rational Representation ◽  
Algebraically Closed Field ◽  
Infinite Dimensional

AbstractLet F be an algebraically closed field of characteristic p. We fashion an infinite dimensional basic algebra ←p(F), with a transparent combinatorial structure, which controls the rational representation theory of GL2(F).

scholarly journals On Some Infinite Dimensional Representations of Semi-Simple Lie Algebras

1965 ◽  
Vol 25 ◽  
pp. 211-220 ◽  
Author(s):  
Hiroshi Kimura
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional

Let g be a semi-simple Lie algebra over an algebraically closed field K of characteristic 0. For finite dimensional representations of g, the following important results are known; 1) H1(g, V) = 0 for any finite dimensional g space V. This is equivalent to the complete reducibility of all the finite dimensional representations,2) Determination of all irreducible representations in connection with their highest weights.3) Weyl’s formula for the character of irreducible representations [9].4) Kostant’s formula for the multiplicity of weights of irreducible representations [6],5) The law of the decomposition of the tensor product of two irreducible representations [1].

On restrictions of generic modules of tame algebras

2013 ◽  
Vol 11 (3) ◽  
Author(s):  
Raymundo Bautista ◽  
Efrén Pérez ◽  
Leonardo Salmerón
Keyword(s):  
Basic Algebra ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Generic Modules ◽  
Tame Algebras

AbstractGiven a convex algebra ∧0 in the tame finite-dimensional basic algebra ∧, over an algebraically closed field, we describe a special type of restriction of the generic ∧-modules.

scholarly journals A nine-dimensional algebra which is not a block of a finite group

2020 ◽  
Author(s):  
Markus Linckelmann ◽  
William Murphy
Keyword(s):  
Finite Group ◽  
Basic Algebra ◽  
Group Algebras ◽  
Closed Field ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
A Finite Group ◽  
Rule Out

Abstract We rule out a certain nine-dimensional algebra over an algebraically closed field to be the basic algebra of a block of a finite group, thereby completing the classification of basic algebras of dimension at most 12 of blocks of finite group algebras.

scholarly journals JACOBSON RADICAL ALGEBRAS WITH QUADRATIC GROWTH

2013 ◽  
Vol 55 (A) ◽  
pp. 135-147 ◽  
Author(s):  
AGATA SMOKTUNOWICZ ◽  
ALEXANDER A. YOUNG
Keyword(s):  
Jacobson Radical ◽  
Quadratic Growth ◽  
Closed Field ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Infinite Dimensional

AbstractWe show that over every countable algebraically closed field $\mathbb{K}$ there exists a finitely generated $\mathbb{K}$-algebra that is Jacobson radical, infinite-dimensional, generated by two elements, graded and has quadratic growth. We also propose a way of constructing examples of algebras with quadratic growth that satisfy special types of relations.

scholarly journals Representations of rank one algebraic monoids

1988 ◽  
Vol 30 (2) ◽  
pp. 237-241
Author(s):  
Lex E. Renner
Keyword(s):  
Representation Theory ◽  
Algebraic Variety ◽  
Semisimple Group ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Dominant Weights ◽  
Rank One ◽  
Algebraic Monoid ◽  
Algebraic Monoids

One of the fundamental results of representation theory is the identification of the irreducible representations of a semisimple group by their dominant weights [3]. The purpose of this paper is to establish similar results for a class of reductive algebraic monoids.Let k be an algebraically closed field. An algebraic monoid is an affine algebraic variety M defined over k, together with an associative morphism m:M × M → M and a two-sided unit 1 ∈ M for m.

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

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