Zariski Closed Algebras in Varieties of Universal Algebra

What is it about?

An algebra $A$ is called representable if it is a subalgebra of a finite-dimensional algebra $B$ over a (larger) base field. In the associative theory, $B$ is usually taken to be a matrix algebra, but since matrix algebras are not nearly as well behaved in the nonassociative theory (for example, the matrices over an alternative algebra are not necessarily alternative), the authors use this other description. Representable algebras are a far broader class than finite-dimensional algebras; for example, any relatively free associative affine PI-algebra is representable, and linear Lie or Jordan algebras are representable, by definition. When representing an algebra $A$ inside the finite-dimensional algebra $B$, one is led naturally to consider its closure in $B$ under the Zariski topology, which is an algebra in the same variety. The Zariski closure of an associative algebra was utilized in [A. Kanel-Belov, L. H. Rowen and U. Vishne, Trans. Amer. Math. Soc. 362 (2010), no. 9, 4695–4734; MR2645047] in the first step of the proof of the affirmation of Specht's question for associative affine PI-algebras over an arbitrary commutative Noetherian base ring (in particular, over a finite field). The reason Zariski-closed algebras are important in PI-theory is that any representable algebra is PI-equivalent to its Zariski closure, so questions about polynomial identities of representable PI-algebras reduce at once to the case of Zariski-closed algebras, which is much more amenable. The purpose of the paper is to develop (in the general context of universal algebra) the theory of the Zariski closure of a representable algebra (not necessarily associative) over an arbitrary integral domain, with the application that the codimension sequence of any representable Zariski-closed PI-algebra over an arbitrary field is exponentially bounded. Much sharper estimates exist in characteristic zero for special classes such as associative PI-algebras. Reviewed by Leonid M. Martynov

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