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.

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].

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 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.

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.

Linear Operators Preserving Similarity Classes and Related Results

1994 ◽  
Vol 37 (3) ◽  
pp. 374-383 ◽  
Author(s):  
Chi-Kwong Li ◽  
Stephen Pierce
Keyword(s):  
Positive Integer ◽  
Characteristic Zero ◽  
Linear Operators ◽  
Partial Answer ◽  
Zariski Closure ◽  
Closed Field ◽  
Algebraically Closed Field

AbstractLet Mn be the algebra of n × n matrices over an algebraically closed field of characteristic zero. For A ∊ Mn, denote by the collection of all matrices in Mn that are similar to A. In this paper we characterize those invertible linear operators ϕ on Mn that satisfy , where for some given A1,..., Ak ∊ Mn and denotes the (Zariski) closure of S. Our theorem covers a result of Howard on linear operators mapping the set of matrices annihilated by a given polynomial into itself, and extends a result of Chan and Lim on linear operators commuting with the function f(x) = xk for a given positive integer k ≥ 2. The possibility of weakening the invertibility assumption in our theorem is considered, a partial answer to a conjecture of Howard is given, and some extensions of our result to arbitrary fields are discussed.

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.

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.

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 A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

Tangent bundle of hypersurfaces in G/P

2008 ◽  
Vol 4 (1) ◽  
pp. 91-100
Author(s):  
Indranil Biswas ◽  
Georg Schumacher
Keyword(s):  
Line Bundle ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Tangent Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Smooth Hypersurface ◽  
Canonical Line Bundle

AbstractLet G be a simple linear algebraic group defined over an algebraically closed field k of characteristic p ≥ 0, and let P be a maximal proper parabolic subgroup of G. If p > 0, then we will assume that dimG/P ≤ p. Let ι : H ↪ G/P be a reduced smooth hypersurface in G/P of degree d. We will assume that the pullback homomorphism is an isomorphism (this assumption is automatically satisfied when dimH ≥ 3). We prove that the tangent bundle of H is stable if the two conditions τ(G/P) ≠ d and hold; here n = dimH, and τ(G/P) ∈ is the index of G/P which is defined by the identity = where L is the ample generator of Pic(G/P) and is the anti–canonical line bundle of G/P. If d = τ(G/P), then the tangent bundle TH is proved to be semistable. If p > 0, and then TH is strongly stable. If p > 0, and d = τ(G/P), then TH is strongly semistable.

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