Abstract
An algebraic scheme for constructing deformations of structure constants for associative algebras generated by deformation driving algebras (DDAs) is discussed. An ideal of left divisors of zero plays a central role in this construction. Deformations of associative three‐dimensional algebras with the DDA being a three‐dimensional Lie algebra and their connection with integrable systems are studied.
Highlights
An idea to study deformations of structure constants for associative algebras goes back to the classical works of Gerstenhaber 1, 2
The interpretation and formalization of the WDVV equation in terms of Frobenius manifolds proposed by Dubrovin 5, 6 provides us with a method to describe class of deformations of the so-called Frobenius algebras
The structure constants Cjlk x are said to define deformations of the algebra A generated by given DDA if all fjk are left zero divisors with common right zero divisor
Summary
An idea to study deformations of structure constants for associative algebras goes back to the classical works of Gerstenhaber 1, 2. The interpretation and formalization of the WDVV equation in terms of Frobenius manifolds proposed by Dubrovin 5, 6 provides us with a method to describe class of deformations of the so-called Frobenius algebras An extension of this approach to general algebras and corresponding F-manifolds has been given by Hertling and Manin 7. This requirement gives rise to the central system which governs deformations generated by DDA This central system of equations for structure constants differs, in general, from the associativity condition. We will study the deformations of the structure constants for the threedimensional algebra in the case when the DDA is given by one of the three-dimensional Lie algebras. Deformations driven by the nilpotent and solvable DDAs are considered in Sections 6 and 7, respectively
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