scholarly journals Central Extension of Left Alternative Algebras and the Second Cohomology Group

2021 ◽  
Vol 2021 ◽  
pp. 1-3
Author(s):  
Said Boulmane
Keyword(s):  
Central Extension ◽  
Cohomology Group ◽  
Alternative Algebra ◽  
Closed Field ◽  
Central Extensions ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Alternative Algebras

The purpose of this paper is to prove that the second cohomology group H 2 A , F of a left alternative algebra A over an algebraically closed field F of characteristic 0 can be interpreted as the set of equivalent classes of one-dimensional central extensions of A .

Cohomology of model filiform Lie superalgebras

2018 ◽  
Vol 17 (04) ◽  
pp. 1850074 ◽  
Author(s):  
Wende Liu ◽  
Yong Yang
Keyword(s):  
Characteristic Zero ◽  
Cohomology Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Closed Field ◽  
Algebra Structure ◽  
Ground Field ◽  
Algebraically Closed 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.

NOETHER SETTINGS FOR CENTRAL EXTENSIONS OF GROUPS WITH ZERO SCHUR MULTIPLIER

2002 ◽  
Vol 01 (01) ◽  
pp. 107-112 ◽  
Author(s):  
ESTHER BENEISH
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Cyclic Group ◽  
Central Extension ◽  
Closed Field ◽  
Schur Multiplier ◽  
Central Extensions ◽  
Algebraically Closed Field ◽  
Rational Extension ◽  
A Finite Group

Let G be a finite group, and let F be an algebraically closed field. The rational extension of F, F(G) = F(xg : g ∈ G) with G action given by hxg = xhg for g, h ∈ G, is referred to as the Noether setting for G. We show that if G is a group with zero Schur multiplier, then for any central extension G′ of G, F(G′)G′ and F(G)G are stably equivalent over F. That is, F(G)G is stably rational over F if and only if F(G′)G′ is stably rational over F. In particular, if G is a cyclic group or an abelian group with zero Schur multiplier, then F(G′) is stably rational over F.

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.

Second Cohomology of the Modular Lie Superalgebra of Cartan Type K

2009 ◽  
Vol 16 (02) ◽  
pp. 309-324 ◽  
Author(s):  
Wenjuan Xie ◽  
Yongzheng Zhang
Keyword(s):  
Cohomology Group ◽  
Lie Superalgebra ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Cartan Type ◽  
Modular Lie Superalgebra ◽  
Finite Dimensional ◽  
Type K

Let 𝔽 be an algebraically closed field and char 𝔽 = p > 3. In this paper, we determine the second cohomology group of the finite-dimensional Contact superalgebra K(m,n,t).

On Simple Connectedness of Minimal Representation-Infinite Algebras

2015 ◽  
Vol 22 (04) ◽  
pp. 639-654
Author(s):  
Hailou Yao ◽  
Guoqiang Han
Keyword(s):  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Closed Field ◽  
Minimal Representation ◽  
Algebraically Closed Field ◽  
Hochschild Cohomology Group ◽  
Simple Connectedness ◽  
Simply Connected ◽  
Equivalent Conditions ◽  
Infinite Algebras

Let A be a connected minimal representation-infinite algebra over an algebraically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group H1(A) is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.

scholarly journals Invariant Hopf 2-Cocycles for Affine Algebraic Groups

2018 ◽  
Vol 2020 (2) ◽  
pp. 344-366
Author(s):  
Pavel Etingof ◽  
Shlomo Gelaki
Keyword(s):  
Finite Groups ◽  
Cohomology Group ◽  
Algebraic Groups ◽  
Closed Field ◽  
Algebraically Closed Field

Abstract We generalize the theory of the second invariant cohomology group $H^{2}_{\textrm{inv}}(G)$ for finite groups G, developed in [3, 4, 14], to the case of affine algebraic groups G, using the methods of [9, 10, 12]. In particular, we show that for connected affine algebraic groups G over an algebraically closed field of characteristic 0, the map Θ from [14] is bijective (unlike for some finite groups, as shown in [14]). This allows us to compute $H^{2}_{\textrm{inv}}(G)$ in this case, and in particular show that this group is commutative (while for finite groups it can be noncommutative, as shown in [14]).

scholarly journals Twisted geometric Satake equivalence

2010 ◽  
Vol 9 (4) ◽  
pp. 719-739 ◽  
Author(s):  
Michael Finkelberg ◽  
Sergey Lysenko
Keyword(s):  
Positive Integer ◽  
Algebraic Group ◽  
Tensor Category ◽  
Reductive Group ◽  
Closed Field ◽  
Primitive Character ◽  
Perverse Sheaves ◽  
Central Extensions ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field

AbstractLet k be an algebraically closed field and O = k[[t]] ⊂ F = k((t)). For an almost simple algebraic group G we classify central extensions 1 → $\mathbb{G}_m\to E\to G(\bm{F})\$m → E → G(F) → 1; any such extension splits canonically over G(O). Fix a positive integer N and a primitive character ζ : μN(K) →$\mathbb{Q}}_\ell^*}$ (under some assumption on the characteristic of k). Consider the category of G(O)-bi-invariant perverse sheaves on E with $\mathbb{G}_m$m-monodromy ζ. We show that this is a tensor category, which is tensor equivalent to the category of representations of a reductive group ǦE,N. We compute the root datum of ǦE,N.

Projective Representations of Minimum Degree of Group Extensions

1978 ◽  
Vol 30 (5) ◽  
pp. 1092-1102 ◽  
Author(s):  
Walter Feit ◽  
Jacques Tits
Keyword(s):  
Simple Group ◽  
Central Extension ◽  
Finite Simple Group ◽  
Minimum Degree ◽  
Projective Representation ◽  
Closed Field ◽  
Natural Question ◽  
Central Extensions ◽  
Ordinary Representation ◽  
Projective Representations

Let G be a finite simple group and let F be an algebraically closed field. A faithful projective F-representation of G of smallest possible degree often cannot be lifted to an ordinary representation of G, though it can of course be lifted to an ordinary representation of some central extension of G. It is a natural question to ask whether by considering non-central extensions, it is possible in some cases to decrease the smallest degree of a faithful projective representation.

The Derivation Algebra of M48(C)

1974 ◽  
Vol 17 (3) ◽  
pp. 375-378 ◽  
Author(s):  
Edgar G. Goodaire ◽  
Roy C. Snell
Keyword(s):  
Alternative Algebra ◽  
Multiplication Table ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Derivation Algebra ◽  
Dickson Algebra

Let C by a Cayley-Dickson algebra over an algebraically closed field F of characteristic 0. A multiplication table for a basis of this 8-dimensional alternative algebra can be found in [3], page 137, where we take α = β = γ = — 1.

scholarly journals Lie Algebras with Abelian Centralizers

2010 ◽  
Vol 17 (04) ◽  
pp. 629-636 ◽  
Author(s):  
Igor Klep ◽  
Primož Moravec
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Closed Field ◽  
Split Extension ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Solvable Case ◽  
Finite Dimensional ◽  
Finite Dimensional Lie Algebras

We classify all finite-dimensional Lie algebras over an algebraically closed field of characteristic 0, whose nonzero elements have abelian centralizers. These algebras are either simple or solvable, where the only simple such Lie algebra is [Formula: see text]. In the solvable case, they are either abelian or a one-dimensional split extension of an abelian Lie algebra.

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