FUNDAMENTAL THEOREM OF FINITE DIMENSIONAL Z 2 -GRADED ASSOCIATIVE ALGEBRAS LINDSAY BRUNSHIDLE, ALICE FIALOWSKI, JOSH FRINAK, MICHAEL PENKAVA, AND DAN WACKWITZ Abstract. In this paper, we give an extension of the Fundamental Theorem of finite dimensional algebras to the case of Z 2 -graded algebras. Essentially the results are the same as in the classical case, except that the notion of a Z 2 - graded division algebra needs to be modified. We classify all finite dimensional Z 2 -graded division algebras over C and R. 1. Introduction Super Lie algebras, or Z 2 -graded Lie algebras, have been studied for a long time, and have many applications in mathematics and physics. The notion of a Z 2 - graded associative algebra is not so well known, but these algebras are examples of Z 2 -graded A ∞ algebras, and thus they arise naturally in the study of A ∞ algebras. In the case of Lie algebras, the Z 2 -graded Jacobi identity picks up some signs that depend on the parity of the elements being bracketed, but the associativity relation for Z 2 -graded associative algebras does not pick up any signs. So it may seem at first glance that there are no new features which arise in the study of Z 2 -graded associative algebras. The moduli space of equivalence classes of Z 2 -graded associative algebras on a vector space of dimension m|n differs from the moduli space of associative structures on the same space ignoring the grading in two important ways. First, a Z 2 -graded algebra structure is required to be an even map, which means that not all associative algebra structures are allowed in the Z 2 -graded case. Secondly, the moduli space is given by equivalence classes of algebra structures under an action by the group of linear automorphisms of the vector space. For the Z 2 -graded case, we only allow even automorphisms, which means that the equivalence classes are potentially smaller in the Z 2 -graded case. Since there are fewer allowable Z 2 -graded algebra structures, but also fewer equivalences between them, it is not obvious whether the moduli space of Z 2 -graded associative algebras on a Z 2 -graded vector space is larger or smaller than the moduli space of all associative algebra structures on the vector space. What is true is that there is a map between the moduli space of Z 2 -graded algebra structures on a Z 2 -graded vector space and the moduli space of all algebra structures on the underlying space. This map in general is neither injective nor surjective. Date : August 8, 2010. Key words and phrases. Associative Algebras, Graded algebras. Research of these authors was partially supported by OTKA grants K77757 and NK72523, and grants from the the University of Wisconsin-Eau Claire. 1