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].

Affinely spanned quasi-stochastic algebraic monoids

2017 ◽  
Vol 27 (08) ◽  
pp. 1061-1072
Author(s):  
W. Huang ◽  
J. Li
Keyword(s):  
Matrix Element ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Unit Group ◽  
Initial Study ◽  
Zariski Closure ◽  
Closed Field ◽  
Basic Class ◽  
Algebraic Monoid ◽  
Algebraic Monoids

A linear algebraic monoid over an algebraically closed field [Formula: see text] of characteristic zero is called (row) quasi-stochastic if each row of each matrix element is of sum one. Any linear algebraic monoid over [Formula: see text] can be embedded as an algebraic submonoid of the maximum affinely spanned quasi-stochastic monoid of some degree [Formula: see text]. The affinely spanned quasi-stochastic algebraic monoids form a basic class of quasi-stochastic algebraic monoids. An initial study of structure of affinely spanned quasi-stochastic algebraic monoids is conducted. Among other things, it is proved that the Zariski closure of a parabolic subgroup of the unit group of an affinely spanned quasi-stochastic algebraic monoid is affinely spanned.

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 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.

THE LOEWY STRUCTURE OF -VERMA MODULES OF SINGULAR HIGHEST WEIGHTS

2015 ◽  
Vol 16 (4) ◽  
pp. 887-898
Author(s):  
Noriyuki Abe ◽  
Masaharu Kaneda
Keyword(s):  
Algebraic Group ◽  
Maximal Torus ◽  
Positive Characteristic ◽  
Closed Field ◽  
Verma Modules ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Loewy Length ◽  
Frobenius Kernel ◽  
Field Of Positive Characteristic

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

Conjugacy Classes in Algebraic Monoids II

1994 ◽  
Vol 46 (3) ◽  
pp. 648-661 ◽  
Author(s):  
Mohan S. Putcha
Keyword(s):  
Unit Group ◽  
Conjugacy Classes ◽  
Reductive Groups ◽  
Bijective Correspondence ◽  
Twisted Conjugacy ◽  
Algebraic Monoid ◽  
Algebraic Monoids

AbstractLet M be a connected linear algebraic monoid with zero and a reductive unit group. We show that there exist reductive groups G1,..., Gt, each with an automorphism, such that the conjugacy classes of M are in a natural bijective correspondence with the twisted conjugacy classes of Gi, i = 1,..., t.

scholarly journals Products of idempotents in algebraic monoids

2006 ◽  
Vol 80 (2) ◽  
pp. 193-203 ◽  
Author(s):  
Mohan S. Putcha
Keyword(s):  
Sufficient Conditions ◽  
Unit Group ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Monoid ◽  
Necessary And Sufficient ◽  
Algebraic Monoids

AbstractLet M be a reductive algebraic monoid with zero and unit group G. We obtain a description of the submonoid generated by the idempotents of M. In particular, we find necessary and sufficient conditions for M\G to be idempotent generated.

AN ALGEBRAIC MONOID APPROACH TO LINEAR ASSOCIATIVE ALGEBRAS, II

1996 ◽  
Vol 06 (05) ◽  
pp. 623-634 ◽  
Author(s):  
WENXUE HUANG
Keyword(s):  
Lie Algebra ◽  
Jacobson Radical ◽  
Borel Subgroup ◽  
Unit Group ◽  
Zariski Closure ◽  
Closed Field ◽  
Associative Algebras ◽  
Lie Bracket ◽  
Algebraic Monoid ◽  
Relationship Of

Let A be a linear associative K-algebra with unity (LAA), K an algebraically closed field. A K–subalgebra of A containing the unity of A is referred to as a sub-LAA of A. A with respect to the multiplication (respectively, Lie bracket) is an algebraic monoid (respectively, Lie algebra) over K, denoted by AM (respectively, AL). Let G(A) and J(A) denote the unit group of AM and Jacobson radical of A, respectively. The following are proved in this paper. (i) If BM is a Borel submonoid of AM, the Zariski closure of a Borel subgroup of G(A), then B is a sub-LAA of A and [Formula: see text], where [Formula: see text] is the set of Borel submonoids of AM. (ii) If dim A≠3, then G(A) is nilpotent iff for every proper sub-LAA N of A, G(N) is nilpotent. In addition, a relationship of Borel submonoids of AM and Borel subalgebras of AL is revealed.

scholarly journals Commutative algebraic monoid structures on affine surfaces

2020 ◽  
Vol 33 (1) ◽  
pp. 177-191
Author(s):  
Sergey Dzhunusov ◽  
Yulia Zaitseva
Keyword(s):  
Characteristic Zero ◽  
Closed Field ◽  
Affine Variety ◽  
Commutative Monoid ◽  
General Classification ◽  
Algebraically Closed Field ◽  
Algebraic Monoid

Abstract We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n - 1 n-1 or 𝑛 on a normal affine variety of dimension 𝑛.

On splitting of the normalizer of a maximal torus in linear groups

2015 ◽  
Vol 14 (07) ◽  
pp. 1550114 ◽  
Author(s):  
Alexey Galt
Keyword(s):  
Finite Field ◽  
Maximal Torus ◽  
Linear Groups ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Maximal Tori

We describe linear groups over an algebraically closed field in which the normalizer of a maximal torus splits over the torus. We describe linear groups over a finite field and their maximal tori in which the normalizer of the maximal torus splits over the torus.

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.

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