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 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.

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

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].

On the classification of finite quasi-quantum groups

2017 ◽  
Vol 29 (10) ◽  
pp. 1730003 ◽  
Author(s):  
Mamta Balodi ◽  
Hua-Lin Huang ◽  
Shiv Datt Kumar
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Tools And Methods

We give an overview of the classification results obtained so far for finite quasi-quantum groups over an algebraically closed field of characteristic zero. The main classification results on basic quasi-Hopf algebras are obtained by Etingof, Gelaki, Nikshych, and Ostrik, and on dual quasi-Hopf algebras by Huang, Liu and Ye. The objective of this survey is to help in understanding the tools and methods used for the classification.

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 Rings of differential operators on toric varieties

1994 ◽  
Vol 37 (1) ◽  
pp. 143-160 ◽  
Author(s):  
A. G. Jones
Keyword(s):  
Line Bundle ◽  
Toric Variety ◽  
Characteristic Zero ◽  
Differential Operators ◽  
Toric Varieties ◽  
Closed Field ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Rings Of Differential Operators

Let be a finite dimensional toric variety over an algebraically closed field of characteristic zero, k. Let be the sheaf of differential operators on . We show that the ring of global sections, is a finitely generated Noetherian k-algebra and that its generators can be explicitly found. We prove a similar result for the sheaf of differential operators with coefficients in a line bundle.

scholarly journals Classification of Nilpotent Lie Superalgebras of Multiplier-Rank Less than or Equal to 6

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Shuang Lang ◽  
Jizhu Nan ◽  
Wende Liu
Keyword(s):  
Characteristic Zero ◽  
Lie Superalgebras ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

In this paper, we classify all the finite-dimensional nilpotent Lie superalgebras of multiplier-rank less than or equal to 6 over an algebraically closed field of characteristic zero. We also determine the covers of all the nilpotent Lie superalgebras mentioned above.

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 Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

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