The Gelfand-Kirillov dimension of quadratic algebras satisfying the cyclic condition

Cedó, Ferran; Jespers, Eric; Okniński, Jan

doi:10.1090/S0002-9939-05-08003-2

The Gelfand-Kirillov dimension of quadratic algebras satisfying the cyclic condition
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by Ferran Cedó, Eric Jespers and Jan Okniński PDF

Proc. Amer. Math. Soc. 134 (2006), 653-663 Request permission

Abstract:

We consider algebras over a field $K$ presented by generators $x_1,\dots ,$ $x_n$ and subject to $n\choose 2$ square-free relations of the form $x_{i}x_{j}=x_{k}x_{l}$ with every monomial $x_{i}x_{j}, i\neq j$, appearing in one of the relations. It is shown that for $n>1$ the Gelfand-Kirillov dimension of such an algebra is at least two if the algebra satisfies the so-called cyclic condition. It is known that this dimension is an integer not exceeding $n$. For $n\geq 4$, we construct a family of examples of Gelfand-Kirillov dimension two. We prove that an algebra with the cyclic condition with generators $x_1,\dots ,x_n$ has Gelfand-Kirillov dimension $n$ if and only if it is of $I$-type, and this occurs if and only if the multiplicative submonoid generated by $x_1,\dots ,x_n$ is cancellative.

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