Geodesic
Lie Algebras and Algebras of Associative Type
Matematičeskie zametki, Tome 88 (2010) no. 1, pp. 43-52 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type $A=\bigoplus_{\alpha\in G}A_\alpha$ graded by some group $G$, over a field of characteristic zero, has a nonzero component $A_1$ (where 1 stands for the identity element of $G$), and $A_1$ is a semisimple associative algebra. Let $B=\bigoplus_{\alpha\in G}B_\alpha$ be a finite-dimensional semisimple Lie algebra over a prime field $F_p$, and let $B$ be graded by a commutative group $G$. If $B=F_p\otimes_{\mathbb Z}A_L$, where $A_L$ is the commutator algebra of a $\mathbb Z$-algebra $A=\bigoplus_{\alpha\in G}A_\alpha$; if $\mathbb Q\otimes_{\mathbb Z}A$ is an algebra of associative type, then the 1-component of the algebra $K\otimes_{\mathbb Z}B$, where $K$ stands for the algebraic closure of the field $F_p$, is the sum of some algebras of the form $\operatorname{gl}(n_i,K)$.
Keywords: algebra of associative type, graded algebra, finite-dimensional semisimple Lie algebra, finite-dimensional semisimple algebra of associative type. 
@article{MZM_2010_88_1_a3,
     author = {N. A. Koreshkov},
     title = {Lie {Algebras} and {Algebras} of {Associative} {Type}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {43--52},
     year = {2010},
     volume = {88},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_1_a3/}
}
TY  - JOUR
AU  - N. A. Koreshkov
TI  - Lie Algebras and Algebras of Associative Type
JO  - Matematičeskie zametki
PY  - 2010
SP  - 43
EP  - 52
VL  - 88
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_2010_88_1_a3/
LA  - ru
ID  - MZM_2010_88_1_a3
ER  - 
%0 Journal Article
%A N. A. Koreshkov
%T Lie Algebras and Algebras of Associative Type
%J Matematičeskie zametki
%D 2010
%P 43-52
%V 88
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_2010_88_1_a3/
%G ru
%F MZM_2010_88_1_a3
N. A. Koreshkov. Lie Algebras and Algebras of Associative Type. Matematičeskie zametki, Tome 88 (2010) no. 1, pp. 43-52. http://geodesic.mathdoc.fr/item/MZM_2010_88_1_a3/

[1] N. A. Koreshkov, “O nilpotentnosti i razlozhenii algebr assotsiativnogo tipa”, Izv. vuzov. Matem., 2006, no. 9, 34–42 | MR

[2] Y. A. Bahturin, M. V. Zaicev, “Identities of graded algebras”, J. Algebra, 205:1 (1998), 1–12 | DOI | MR | Zbl

[3] Yu. A. Bakhturin, M. V. Zaitsev, S. K. Segal, “$G$-tozhdestva neassotsiativnykh algebr”, Matem. sb., 190:11 (1999), 3–14 | MR | Zbl

[4] M. V. Zaitsev, S. K. Segal, “Konechnye graduirovki prostykh artinovykh kolets”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2001, no. 3, 21–24 | MR | Zbl