On Codimension Growth of Graded Pi-Algebras
DOI:
https://doi.org/10.5644/SJM.12.3.10Keywords:
Graded algebra, bicharacter, color Lie superalgebra, simple algebra, polynomial identity, n-th codimension, graded PI-exponentAbstract
Two finite dimensional simple Lie algebras over an algebraically closed field are isomorphic if and only if they satisfy the same polynomial identities (A. Koshkulei, Y. Razmyslov, 1983). As an alternative approach to the characterization of finite dimensional simple Lie algebras, some numerical invariants of the algebra identities can be used. We associate with a finite dimensional Lie algebra $L$ a sequence of integers $c_n(L)$, called the $n$-th codimensions of $L$. It appears that these quantities grow asymptotically like $k^n$, for some nonnegative integer $k\leq$ dim $L$ (A. Giambruno, M. Zaicev, 1999). Moreover, $k =$ dim $L$ iff algebra $L$ is simple.
* This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Croatia, September, 22-24, 2016.
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