Star-fundamental algebras: polynomial identities and asymptotics,Transactions of the American Mathematical Society

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Star-fundamental algebras: polynomial identities and asymptotics
Transactions of the American Mathematical Society ( IF 1.2 ) Pub Date : 2020-08-28 , DOI: 10.1090/tran/8182
Antonio Giambruno , Daniela La Mattina , Cesar Polcino Milies

Abstract:We introduce the notion of star-fundamental algebra over a field of characteristic zero. We prove that in the framework of the theory of polynomial identities, these algebras are the building blocks of a finite dimensional algebra with involution

. To any star-algebra

is attached a numerical sequence

, $n\ge 1$ , called the sequence of

-codimensions of

. Its asymptotic is an invariant giving a measure of the

-polynomial identities satisfied by

. It is well known that for a PI-algebra such a sequence is exponentially bounded and $\exp ^*(A)=\lim _{n\to \infty }\sqrt [n]{c_n^*(A)}$ can be explicitly computed. Here we prove that if

is a star-fundamental algebra,

$\displaystyle C_1n^t\exp ^*(A)^n\le c_n^*(A)\le C_2n^t \exp ^*(A)^n,$

where

are constants and

is explicitly computed as a linear function of the dimension of the skew semisimple part of

and the nilpotency index of the Jacobson radical of

. We also prove that any finite dimensional star-algebra has the same

-identities as a finite direct sum of star-fundamental algebras. As a consequence, by the main result in [J. Algebra 383 (2013), pp. 144-167] we get that if

is any finitely generated star-algebra satisfying a polynomial identity, then the above still holds and, so, $\lim _{n\to \infty }\log _n \frac {c_n^*(A)}{\exp ^*(A)^n}$ exists and is an integer or half an integer.

[1]

中文翻译：

星基代数：多项式恒等式和渐近性

摘要：我们介绍了特征零域上的恒星基本代数的概念。我们证明，在多项式恒等式理论的框架内，这些代数是具有对合的有限维代数的构造块

。到任何星形代数

安装一个数字序列

， $n \ ge 1$ 被称为序列

的-codimensions

。它的渐近是一个不变量，它给出

满足的多项式恒等式

。众所周知，对于PI代数，这样的序列是指数有界的并且可以被显式地计算。在这里，我们证明如果是恒星基本代数， $\\ exp ^ *（A）= \ lim _ {n \ to \ infty} \ sqrt [n] {c_n ^ *（A）}$

$\ displaystyle C_1n ^ t \ exp ^ *（A）^ n \ le c_n ^ *（A）\ le C_2n ^ t \ exp ^ *（A）^ n，$

其中为常数，并明确地计算为的偏半简单部分的尺寸和的Jacobson根的幂指数的线性函数。我们还证明，任何有限维星形代数与恒星基本代数的有限直接和具有相同的恒等式。结果，[J。代数383（2013），第144-167页]我们得到的结论是，如果是满足多项式恒等式的有限生成恒星代数，则上述仍然成立，因此是存在的，并且是整数或整数的一半。

$\ lim _ {n \ to \ infty} \ log _n \ frac {c_n ^ *（A）} {\ exp ^ *（A）^ n}$