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

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.

HOPF ALGEBRAS OF DIMENSION 30

Journal of Algebra and Its Applications ◽

10.1142/s0219498810003902 ◽

2010 ◽

Vol 09 (01) ◽

pp. 11-15 ◽

Cited By ~ 2

Author(s):

DAIJIRO FUKUDA

Keyword(s):

Hopf Algebra ◽

Group Algebra ◽

Hopf Algebras ◽

Characteristic Zero ◽

Closed Field ◽

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.

Involutions on finite-dimensional algebras over real closed fields

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700010193 ◽

2004 ◽

Vol 77 (1) ◽

pp. 123-128 ◽

Cited By ~ 1

Author(s):

W. D. Munn

Keyword(s):

Characteristic Zero ◽

Real Closed Field ◽

Closed Field ◽

Finite Dimensional Algebra ◽

Dimensional Algebra ◽

Real Closed Fields ◽

Finite Dimensional ◽

Closed Fields ◽

Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

On the Cohomology and Extensions of n-ary Multiplicative Hom-Nambu-Lie Superalgebras

Advances in Mathematical Physics ◽

10.1155/2020/1961836 ◽

2020 ◽

Vol 2020 ◽

pp. 1-17

Author(s):

Lijun Tian ◽

Baoling Guan ◽

Yao Ma

Keyword(s):

Lie Algebras ◽

Lie Superalgebra ◽

Lie Superalgebras ◽

Closed Field ◽

In this paper, we discuss the representations of n-ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notion of representations for n-ary multiplicative Hom-Nambu-Lie algebras. We also give the cohomology of an n-ary multiplicative Hom-Nambu-Lie superalgebra and obtain a relation between extensions of an n-ary multiplicative Hom-Nambu-Lie superalgebra b by an abelian one a and Z1b,a0¯. We also introduce the notion of T∗-extensions of n-ary multiplicative Hom-Nambu-Lie superalgebras and prove that every finite-dimensional nilpotent metric n-ary multiplicative Hom-Nambu-Lie superalgebra over an algebraically closed field of characteristic not 2 in the case α is a surjection is isometric to a suitable T∗-extension.

On the Existence of the Graded Exponent for Finite Dimensional ℤp-graded Algebras

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-104-9 ◽

2012 ◽

Vol 55 (2) ◽

pp. 271-284 ◽

Cited By ~ 3

Author(s):

Onofrio M. Di Vincenzo ◽

Vincenzo Nardozza

Keyword(s):

Characteristic Zero ◽

Prime Order ◽

Closed Field ◽

Graded Algebras ◽

AbstractLet F be an algebraically closed field of characteristic zero, and let A be an associative unitary F-algebra graded by a group of prime order. We prove that if A is finite dimensional then the graded exponent of A exists and is an integer.

On the classification of finite quasi-quantum groups

Reviews in Mathematical Physics ◽

10.1142/s0129055x17300035 ◽

2017 ◽

Vol 29 (10) ◽

pp. 1730003 ◽

Cited By ~ 1

Author(s):

Mamta Balodi ◽

Hua-Lin Huang ◽

Shiv Datt Kumar

Keyword(s):

Quantum Groups ◽

Hopf Algebras ◽

Characteristic Zero ◽

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.

Generators of Simple Modular Lie Superalgebras

Algebra Colloquium ◽

10.1142/s1005386716000365 ◽

2016 ◽

Vol 23 (02) ◽

pp. 347-360

Author(s):

Liming Tang ◽

Wende Liu

Keyword(s):

Lie Superalgebras ◽

Closed Field ◽

Characteristic P ◽

Cartan Type ◽

Finite Dimensional ◽

Modular Lie Superalgebras

Let X be one of the finite-dimensional graded simple Lie superalgebras of Cartan type W, S, H, K, HO, KO, SHO or SKO over an algebraically closed field of characteristic p > 3. In this paper we prove that X can be generated by one element except the ones of type W, HO, KO or SKO in certain exceptional cases in which X can be generated by two elements. As a subsidiary result, we prove that certain classical Lie superalgebras or their relatives can be generated by one or two elements.

Group Gradings on Finite Dimensional Lie Algebras

Algebra Colloquium ◽

10.1142/s1005386713000540 ◽

2013 ◽

Vol 20 (04) ◽

pp. 573-578 ◽

Cited By ~ 4

Author(s):

Dušan Pagon ◽

Dušan Repovš ◽

Mikhail Zaicev

Keyword(s):

Finite Group ◽

Lie Algebras ◽

Characteristic Zero ◽

Closed Field ◽

Solvable Radical ◽

Finite Dimensional ◽

Levi Subalgebra ◽

Group Gradings ◽

Finite Dimensional Lie Algebras

We study gradings by non-commutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if L is graded by a non-abelian finite group G, then the solvable radical R of L is G-graded and there exists a Levi subalgebra B=H1⊕ ⋯ ⊕ Hm homogeneous in G-grading with graded simple summands H1,…,Hm. All Supp Hi (i=1,…,m) are commutative subsets of G.

Commutative algebraic monoid structures on affine spaces

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500640 ◽

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 ◽

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.

Cohomology of model filiform Lie superalgebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498818500743 ◽

2018 ◽

Vol 17 (04) ◽

pp. 1850074 ◽

Cited By ~ 2

Author(s):

Wende Liu ◽

Yong Yang

Keyword(s):

Characteristic Zero ◽

Cohomology Group ◽

Lie Superalgebra ◽

Lie Superalgebras ◽

Closed Field ◽

Algebra Structure ◽

Ground Field ◽

One Dimensional ◽

First Cohomology Group

Suppose the ground field [Formula: see text] is an algebraically closed field of characteristic zero. By means of spectral sequences, the computation of the first cohomology group of the model filiform Lie superalgebra [Formula: see text] with coefficients in the adjoint module is reduced to the computation of the first cohomology group of an Abel ideal and a one-dimensional subalgebra. Then, by calculating the outer derivations, the algebra structure of the first cohomology group of [Formula: see text] is completely characterized.

Représentations banales de

Compositio Mathematica ◽

10.1112/s0010437x12000590 ◽

2013 ◽

Vol 149 (4) ◽

pp. 679-704 ◽

Cited By ~ 6

Author(s):

Alberto Mínguez ◽

Vincent Sécherre

Keyword(s):

Complex Numbers ◽

Closed Field ◽

Locally Compact ◽

Central Division ◽

Finite Dimensional ◽

Compact Field ◽

Residue Characteristic

AbstractLet${\rm F}$be a non-Archimedean locally compact field of residue characteristic$p$, let${\rm D}$be a finite-dimensional central division${\rm F}$-algebra and let${\rm R}$be an algebraically closed field of characteristic different from$p$. We definebanalirreducible${\rm R}$-representations of the group${\rm G}={\rm GL}_{m}({\rm D})$. This notion involves a condition on the cuspidal support of the representation depending on the characteristic of${\rm R}$. When this characteristic is banal with respect to${\rm G}$, in particular when${\rm R}$is the field of complex numbers, any irreducible${\rm R}$-representation of${\rm G}$is banal. In this article, we give a classification of all banal irreducible${\rm R}$-representations of${\rm G}$in terms of certain multisegments, called banal. When${\rm R}$is the field of complex numbers, our method provides a new proof, entirely local, of Tadić’s classification of irreducible complex smooth representations of${\rm G}$.