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Semigroup graded algebras and graded PI-exponent. (arXiv:1511.01860v2 [math.RA] UPDATED)
来源于:arXiv
Let $S$ be a finite semigroup and let $A$ be a finite dimensional $S$-graded
algebra. We investigate the exponential rate of growth of the sequence of
graded codimensions $c_n^S(A)$ of $A$, i.e $\lim\limits_{n \rightarrow \infty}
\sqrt[n]{c_n^S(A)}$. For group gradings this is always an integer. Recently in
[20] the first example of an algebra with a non-integer growth rate was found.
We present a large class of algebras for which we prove that their growth rate
can be equal to arbitrarily large non-integers. An explicit formula is given.
Surprisingly, this class consists of an infinite family of algebras simple as
an $S$-graded algebra. This is in strong contrast to the group graded case for
which the growth rate of such algebras always equals $\dim (A)$. In light of
the previous, we also handle the problem of classification of all $S$-graded
simple algebras, which is of independent interest. We achieve this goal for an
important class of semigroups that is crucial for a solution of t 查看全文>>