Counting Finite-Dimensional Algebras Over Finite Field
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Counting Finite-Dimensional Algebras Over Finite Field Nikolaas D. Verhulst Abstract. In this paper, we describe an elementary method for counting the number of non-isomorphic algebras of a fixed, finite dimension over a given finite field. We show how this method works in the case of 2dimensional algebras over the field F2 .
Introduction Classifying finite-dimensional algebras over a given field is usually a very hard problem. The first general result was a classification by Hendersson and Searle of 2-dimensional algebras over the base field R, which appeared in 1992 ([1]). This was generalised in 2000 by Petersson ([3]), who managed to give a full classification of 2-dimensional algebras over an arbitrary base field. The methods employed in these papers are quite involved and rely on a large amount of previous work by many illustrious authors. Our aim in this paper is to give perhaps not a classification but at least a way to compute the exact number of non-isomorphic n-dimensional algebras over a fixed finite field by elementary means. Indeed, nothing more complicated than linear algebra and some very basic results about group actions will be needed: we describe isomorphism classes of n-dimensional K-algebras as orbits of a certain GLn (K)-action on Matn (K)n and use a basic result about group actions to count these orbits. In the first three sections, we give a proof based on concrete calculations, while Sect. 4 is dedicated to a more abstract alternative which avoids all computations. In Sect. 5, we work out the concrete example n = 2, K = F2 . 0123456789().: V,-vol
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Results Math
1. Notation and Basics Fix a field K. In this article, an algebra is understood to be a K-vector space A equipped with a multiplication, i.e. a bilinear map A × A → A. If a, b are in A, we will write ab for the image of (a, b) under this map. We do not assume algebras to have a unit or to be associative. By the dimension of an algebra we mean its dimension as a K-vector space. Two algebras A and A will be called isomorphic if there exists a K-linear bijection f : A → A with f (ab) = f (a)f (b) for all a, b in A. The isomorphism class of an algebra A will be denoted by [A]. For n ∈ N, we define Algn (K) to be the set of isomorphism classes of n-dimensional algebras. Given a vector M = (Mi )i=1,...,n of n (n × n)-matrices over K, we can define an algebra alg(M) which is K n as a K-vector space and for which multiplication is defined to be the unique bilinear map K n × K n → K n with ei ej = (Mi )kj ek k
where the ei are the canonical basis vectors of K n . Intuitively, this means that multiplying an element a ∈ alg(M) on the left with ei is multiplying the coordinate vector of a (with respect to the canonical basis) with Mi and interpreting the result again as a coordinate vector (with respect to the canonical basis). This allows us to define the map [alg] : Matn (K)n → Algn (K), M → [alg(M)] which will play an important role in this paper. Lemma 1.1. The map [alg] defined above is surje
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