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.

CARTAN SUBMONOIDS OF ALGEBRAIC MONOIDS

1995 ◽  
Vol 05 (03) ◽  
pp. 367-377 ◽  
Author(s):  
WENXUE HUANG
Keyword(s):  
Maximal Torus ◽  
Unit Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Completely Regular ◽  
Algebraic Monoid ◽  
Algebraic Monoids

Let M be an irreducible linear algebraic monoid defined over an algebraically closed field K with idempotent set E(M), T a maximal torus of the unit group G of M. We call CM(T)c a Cartan submonoid of M. The following are proved: (1) If M is reductive with zero or completely regular, then CM(T) is irreducible and regular and [Formula: see text]; (2) If M is regular, then M is solvable iff NM(CM(T))=CM(T), in which case, CM(T) is irreducible and regular; (3) If M is regular, then [Formula: see text].

scholarly journals Commutative algebraic monoid structures on affine spaces

2019 ◽  
Vol 22 (08) ◽  
pp. 1950064
Author(s):  
Ivan Arzhantsev ◽  
Sergey Bragin ◽  
Yulia Zaitseva
Keyword(s):  
Characteristic Zero ◽  
Affine Space ◽  
Arbitrary Dimension ◽  
Toric Varieties ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Affine Spaces ◽  
Algebraic Monoid ◽  
Algebraic Monoids

We study commutative associative polynomial operations [Formula: see text] with unit on the affine space [Formula: see text] over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.

scholarly journals A remark on the Grothendieck-Lefschetz theorem about the Picard group

1978 ◽  
Vol 71 ◽  
pp. 169-179 ◽  
Author(s):  
Lucian Bădescu
Keyword(s):  
Algebraic Variety ◽  
Picard Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Lefschetz Theorem

Let K be an algebraically closed field of arbitrary characteristic. The term “variety” always means here an irreducible algebraic variety over K. The notations and the terminology are borrowed in general from EGA [4].

scholarly journals A Krull-Schmdit type theorem for coherent sheaves

2014 ◽  
Vol 22 (2) ◽  
pp. 51-56
Author(s):  
A. S. Argáez
Keyword(s):  
Finite Group ◽  
Projective Variety ◽  
Type Theorem ◽  
Coherent Sheaf ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Coherent Sheaves ◽  
A Finite Group ◽  
Natural Decomposition

AbstractLet X be projective variety over an algebraically closed field k and G be a finite group with g.c.d.(char(k), |G|) = 1. We prove that any representations of G on a coherent sheaf, ρ : G → End(ℰ), has a natural decomposition ℰ ≃ ⊕ V ⊗k ℱV, where G acts trivially on ℱV and the sum run over all irreducible representations of G over k.

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

scholarly journals Corrections to “Seminormal rings and weakly normal varieties”

1987 ◽  
Vol 107 ◽  
pp. 147-157 ◽  
Author(s):  
Marie A. Vitulli
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Regular Function ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Regular Functions ◽  
Normal Varieties

In “Seminormal rings and weakly normal varieties” we introduced the notion of a c-regular function on an algebraic variety defined over an algebraically closed field of characteristic zero. Our intention was to describe those k-valued functions on a variety X that become regular functions when lifted to the normalization of X, but without any reference to the normalization in the definition. That is, we aspired to identify the c-regular functions on X with the regular functions on the weak normalization of X

scholarly journals IRREDUCIBLE REPRESENTATIONS OF THE HAMILTONIAN ALGEBRAH(2r;n)

2011 ◽  
Vol 90 (3) ◽  
pp. 403-430 ◽  
Author(s):  
YU-FENG YAO ◽  
BIN SHU
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Irreducible Representations ◽  
Full Description ◽  
Closed Field ◽  
Maximal Subalgebra ◽  
Algebraically Closed Field ◽  
Cartan Type ◽  
Restricted Lie Algebra ◽  
Graded Lie Algebra

AbstractLetL=H(2r;n) be a graded Lie algebra of Hamiltonian type in the Cartan type series over an algebraically closed field of characteristicp>2. In the generalized restricted Lie algebra setup, any irreducible representation ofLcorresponds uniquely to a (generalized)p-characterχ. When the height ofχis no more than min {pni−pni−1∣i=1,2,…,2r}−2, the corresponding irreducible representations are proved to be induced from irreducible representations of the distinguished maximal subalgebraL0with the aid of an analogy of Skryabin’s category ℭ for the generalized Jacobson–Witt algebras and modulo finitely many exceptional cases. Since the exceptional simple modules have been classified, we can then give a full description of the irreducible representations withp-characters of height below this number.

Subregular nilpotent representations of [sscr ][Iscr ]n and [sscr ][oscr ]2n+1

1999 ◽  
Vol 126 (2) ◽  
pp. 223-257 ◽  
Author(s):  
JENS CARSTEN JANTZEN
Keyword(s):  
Lie Algebras ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
First Case ◽  
Nilpotent Representations

Let [gscr ] be one of the Lie algebras [sscr ][Iscr ]n(K) or [sscr ][oscr ]2n+1(K) over an algebraically closed field K of characteristic p>0. Suppose in the first case that n∉Zp and in the second case that p≠2. This assumption implies that [gscr ] is simple. In this paper I study certain irreducible representations of [gscr ].

scholarly journals Algebraic Barth-Lefschetz theorems

1996 ◽  
Vol 142 ◽  
pp. 17-38 ◽  
Author(s):  
Lucian Bădescu
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Algebraic Variety ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Basic Properties ◽  
Positive Integers ◽  
Lefschetz Theorems ◽  
Algebraic Scheme

We shall work over a fixed algebraically closed field k of arbitrary characteristic. By an algebraic variety over k we shall mean a reduced algebraic scheme over k. Fix a positive integer n and e = (e0, el,…, en) a system of n + 1 weights (i.e. n + 1 positive integers e0, el,…, en). If k[T0, Tl,…, Tn] is the polynomial k-algebra in n + 1 variables, graded by the conditions deg(Ti) = ei i = 0, 1,…, n, denote by Pn(e) = Proj(k[T0, T1,…, Tn]) the n-dimensional weighted projective space over k of weights e. We refer the reader to [3] for the basic properties of weighted projective spaces.

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