The Moduli Space of 3-Dimensional Associative Algebras

Abstract

Abstract. In this paper, we give a classification of the 3-dimensional associative algebras over the complex numbers, including a construction of the moduli space, using versal deformations to determine how the space is glued together.

Classifying complements for associative algebras

“…Denote by f r the linear isomorphism from X to X induced by f r . We will prove that f r is an algebra morphism if we consider X endowed with the multiplication given by (6). For all x, y ∈ X we have: Therefore, X r is an algebra and the proof is now finished.…”

“…r(x), x r(y), y By applying r to the second part of (8) it follows that r is a deformation map of the matched pair (A, X, ⊲, ⊳, ↼, ⇀). Furthermore, (6) and (8) show that v : X r → X is also an algebra map. The proof is now finished.…”

“…Let A ⊂ E be a given extension of algebras. If X is a given A-complement of E then Theorem 2.3 provides the description of all complements of A in E: any A-complement of E is isomorphic to an r-deformation of X, as defined by (6). In other words, exactly as in the case of Hopf algebras, Lie algebras or Leibniz algebras, given X an A-complement of E all the other A-complements of E are deformations of the algebra X by certain maps r : X → A associated with the canonical matched pair which arises from the factorization E = A + X.…”

Extensions of associative algebras

“…In this paper, we will illustrate how to use the presentation of the main results by giving examples of the construction of moduli spaces of extensions. In [11,12], the ideas presented here were used to construct moduli spaces of low-dimensional complex associative algebras.…”

“…First, note that δ is a D µ -cocycle, and λ must be a D µ -cocycle by condition (5), so they determine D µ -cohomology classesδ andλ in H µ . If λ, ψ determine an extension, and λ ∈λ, then λ , ψ determine an equivalent extension, where λ and ψ are given by the formulas (11) and (12). Moreover, condition (4) yields the MC formula…”