Derived Equivalence Classification of Symmetric Algebras of Polynomial Growth

We complete the derived equivalence classification of all symmetric algebras of polynomial growth, by solving the subtle problem of distinguishing the standard and nonstandard nondomestic symmetric algebras of polynomial growth up to derived equivalence. Introduction and the main result Throughout the article, K will denote a fixed algebraically closed field. By an algebra is meant an associative, finite-dimensional K-algebra with an identity. For an algebra A, we denote by modA the category of finite dimensional right A-modules and by D the standard duality HomK(−,K) on modA. An algebra A is called selfinjective if AA is an injective A-module, or equivalently, the projective Amodules are injective. Prominent classes of selfinjective algebras are formed by the Frobenius algebras A for which there exists an associative, nondegenerate, K-bilinear form (−,−) : A × A → K, and the symmetric algebras A for which there exists an associative, symmetric, nondegenerate, K-bilinear form (−,−) : A × A → K. By the classical theorems of T. Nakayama [30], [31], an algebra A is Frobenius (respectively, symmetric) if and only if A ∼= D(A) in modA (respectively, as A-A-bimodules). We also mention that every selfinjective algebra A is Morita equivalent to a Frobenius algebra, namely to its basic algebra. Moreover, for every algebra B, the trivial extension T(B) = BnD(B) of B by the B-B-bimodule D(B) is a symmetric algebra, and B is a factor algebra of T(B). It follows also from a result of T. Nakayama [31] that the left socle and the right socle of a selfinjective algebra A coincide, and we denote them by soc(A). Two selfinjective algebras A and Λ are said to be socle equivalent if the factor algebras A/ soc(A) and Λ/ soc(Λ) are isomorphic. According to the remarkable Tame and Wild Theorem of Y.A. Drozd [12] the class of (finitedimensional) K-algebras over K may be divided into two disjoint classes. One class consists of the tame algebras for which the indecomposable modules occur, in each dimension d, in a finite number of discrete and a finite number of one-parameter families. The second class consists of the wild algebras for which the representation theory comprises the representation theories of all finitedimensional algebras over K (see [37, Chapter XIX]). Hence a classification of finite-dimensional modules is only feasible for tame algebras. More precisely, following Y.A. Drozd [12], an algebra A is said to be tame, if for any positive integer d, there exists a finite number of K[x]-A-bimodules Mi, 1 ≤ i ≤ nd, which are finitely generated and free as left K[x]-modules (K[x] is the polynomial algebra in one variable over K) and all but finitely many isomorphism classes of indecomposable modules of dimension d in modA are of the form K[x]/(x−λ)⊗K[x]Mi for some λ ∈ K and some i ∈ {1, . . . , nd}. Let μA(d) be the least number of K[X]-A-bimodules satisfying the above condition for d. Then A is said to be of polynomial growth (respectively, domestic) if there exists a positive integer m such that μA(d) ≤ d (respectively, μA(d) ≤ m) for all d ≥ 1. Moreover, from the validity of the second Brauer–Thrall conjecture, μA(d) = 0 for all d ≥ 1 if and only if A is representation-finite (there are only finitely many isomorphism classes of indecomposable modules in modA). One central problem of modern representation theory is the determination of the module categories modA of tame selfinjective algebras A. Recently, the module categories of all selfinjective algebras of polynomial growth have been described completely. It has been proved by the second named author [40] that a nonsimple basic connected selfinjective algebra A is of polynomial growth if and only if A is socle equivalent to an orbit algebra B̂/G, where B̂ is the repetitive category of an algebra 2000 Mathematics Subject Classification. 16G10, 18E30, 16D50, 16G60.