Polynomial invariants and moduli of generic two-dimensional commutative algebras

Abstract

Let V be a two-dimensional vector space over a field F of characteristic not 2 or 3. We show there is a canonical surjection ν from the set of suitably generic commutative algebra structures on V modulo the action of GL(V ) onto the plane F 2 . In these coordinates, which are quotients of invariant quartic polynomials, properties such as associativity and the existence of zero divisors are described by simple algebraic conditions. The map ν is a bijection over the complement of a degenerate elliptic curve Γ and over Γ we give an explicit parametrisation of the fibre in terms of Galois extensions of F. Algebras in ν −1 (Γ) are exactly those which admit non-trivial automorphisms. We show how ν can be lifted to a map from the SL(V )−moduli space to an algebraic hypersurface Γ ′ in a four-dimensional vector space whose equation is essentially the classical Eisenstein equation for the covariants of a binary cubic. This map is the restriction of a surjective map from the set of stable commutative algebras on V modulo the action of SL(V ) onto Γ ′ .

Varieties of bicommutative algebras

“…(See Grattan-Guinness [21] for the contributions of Peirce.) Starting in 2000 with the paper by Petersson [26] (which contains also the history of the classification) and the paper by Anan'in and Mironov [3] there are several papers containing different kinds of classification of two-dimensional algebras -by Goze and Remm [20], Ahmed, Bekbaev, and Rakhimov [1], Rausch de Traubenberg and Slupinski [27], Kaygorodov and Volkov [22]. Concerning the polynomial identities of two-dimensional algebras, Giambruno, Mishchenko, and Zaicev [15] proved that the growth of the codimension sequence c n (A) of such an algebra A over a field of characteristic 0 is either linear (and bounded by n + 1) or grows exponentially as 2 n .…”