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Register a new userThe Polynomial Part of the Codimension Growth of Affine PI Algebras
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Let $F$ be a field of characteristic zero and $W$ be an associative affine $F$-algebra satisfying a polynomial identity (PI). The codimension sequence associated to $W$, $c_n(W)$, is known to be of the form $Theta (c n^t d^n)$, where $d$ is the well known (PI) exponent of $W$. In this paper we establish an algebraic interpretation of the polynomial part (the constant $t$) by means of Kemers theory. In particular, we show that in case $W$ is a basic algebra, then $t = frac{d-q}{2} + s$, where $q$ is the number of simple component in $W/J(W)$ and $s+1$ is the nilpotency degree of $J(W)$. Thus proving a conjecture of Giambruno.
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We present a proof of Kemers representability theorem for affine PI algebras over a field of characteristic zero.
Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of $B$. Then $A$ is isomorphic to $B$ as an $S$-graded algebra.
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