Deformations of Modules of Differential Forms

Abstract

We study non-trivial deformations of the natural action of the Lie algebra Vect(R n ) on the space of differential forms on R n . We calculate abstractions for integrability of infinitesimal multi-parameter deformations and determine the commutative associative algebra corresponding to the miniversal deformation in the sense of [3].

Деформация вложений супералгебры Ли контактных векторных полей на $S^{1\mid 2}$ в супералгебру Ли псевдодифференциальных операторов на $S^{1\mid 2}$

“…i,j,k уравнения (3.9) определено с точностью до 1-коцикла, и в рабо-тах [17], [19], [20] было показано, что разные представители одного класса решений уравнения (3.9) соответствуют эквивалентным деформациям. Таким образом, все-гда можно избавиться от ρ (2) i,j,k .…”

sl(2)-Trivial Deformations of VectPol(R)-Modules of Symbols

“…Deformation theory of Lie algebra homomorphisms was first considered with only one-parameter deformation [7,12,15]. Recently, deformations of Lie (super)algebras with multi-parameters were intensively studied (see, e.g., [1,2,13,14]). Here we give an outline of this theory.…”

“…Of course, any t λ,λ+k appears in the expressions of L (1) or L (2) if and only if δ − λ and k are integers satisfying δ − n ≤ λ, λ + k ≤ δ. Theorem 2 is proved.…”

“…Of course, there are many formal deformations with less then 11 independent parameters. The deformation L = L + L (1) + L (2) , is the miniversal sl(2)-trivial deformation of S 7 λ+7 with base A = C[t]/R, where t is the family of all parameters given in the expression of L (1) and R is the ideal generated by the polynomials b t λ,λ+3 t λ+3,λ+7 − a t λ,λ+4 t λ+4,λ+7 and ω λ,λ+5 (t)t λ+5,λ+7 .…”

“…We use the framework of Fialowski [6] (see also [1] and [2]) and consider (multi-parameter) deformations over complete local algebras. We construct the miniversal deformation of this action and define the complete local algebra related to this deformation.…”