The geometry of polynomial identities.

*(English. Russian original)*Zbl 1393.16017
Izv. Math. 80, No. 5, 910-953 (2016); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 80, No. 5, 103-152 (2016).

In [Transl., Ser. 2, Am. Math. Soc. 148, 65–71 (1991; Zbl 0736.16013)] A. R. Kemer developed his structure theory of T-ideals. This allowed him to solve and understand the famous Specht problem for associative algebras over a field \(F\) of characteristic 0, whether the polynomial identities of an algebra are equivalent to a finite system. One of the key features of the theory is that any finitely generated PI-algebra has the same identities as (or is PI-equivalent to) a finite dimensional algebra. If the PI-algebra is not finitely generated then it is PI-equivalent to the Grassmann envelope of a finite dimensional superalgebra. Section 1 is introductory. It contains basic definitions and the description of the results. Section 2 gives a quick overview of the required theory of Kemer. The author stresses the role of fundamental algebras which are the building blocks of the PI-equivalence classes of finite-dimensional algebras. Section 3 contains consequences of the theory of Kemer. The main objects to study are the finitely generated relatively free algebras which are homomorphic images \(F\langle X\rangle/I\) of the free algebra \(F\langle X\rangle\) of finite rank modulo the ideal \(I\) of the polynomial identities of a given algebra. This allows to restrict the considerations to ordinary algebras only (without dealing with superalgebras). The first goal of the paper is to show that for a given relatively free algebra \(F\langle X\rangle/I\) there is a canonical finite filtration by T-ideals \(K_i\) such that each quotient \(K_i/K_{i-1}\) has a natural structure of a finitely generated module over a finitely generated commutative algebra \({\mathcal T}_i\). In fact \(K_i/K_{i-1}\) is a two-sided ideal in a special trace algebra which is finitely generated as a module over \({\mathcal T}_i\). The algebras \({\mathcal T}_i\) are coordinate rings of certain representation varieties and the word geometry, appearing in the title, refers to the geometric description of the algebraic varieties supporting the various modules \(K_i/K_{i-1}\). At the next step, the author shows that these varieties are natural quotient varieties parametrizing semisimple representations. Then, based on explicit information on these varieties and using general facts of invariant theory, the author deduces a number of corollaries. He computes the Gelfand-Kirillov dimension of the relatively free algebras and the growth of the cocharacter sequence. Section 4 indicates a method to classify finite-dimensional algebras up to PI-equivalence. Here, the author discusses the problem of choosing a canonical fundamental algebra in each PI-equivalence class. Although the main results are given in characteristic 0, the author connects them with results of S. Donkin [Invent. Math. 110, No. 2, 389–401 (1992; Zbl 0826.20036)]
and A. N. Zubkov [Algebra Logika 35, No. 4, 433–457 (1996; Zbl 0941.16012); translation in Algebra Logic 35, No. 4, 241–254 (1996)] on the first and second fundamental theorems of invariant theory of matrices in positive characteristic. This allows to see what happens in PI-theory also in this case. The paper is very well written and contains a long list of references. It is not only an interesting survey on PI-algebras but contains also important new results. This paper allows to see the theory from completely another point of view.

Reviewer: Vesselin Drensky (Sofia)

##### MSC:

16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |

15A24 | Matrix equations and identities |

16R30 | Trace rings and invariant theory (associative rings and algebras) |