Deformation of vect(1)-modules of symbols

Abstract

We consider the action of the Lie algebra of polynomial vector fields, vect(1), by the Lie derivative on the space of symbols S n δ = n j=0 F δ−j . We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space H 2 diff (vect(1), D ν,µ ) where D ν,µ is the space of differential operators from F ν to F µ . Necessary second-order integrability conditions of any infinitesimal deformations of S n δ are given. We describe completely the formal deformations for some spaces S n δ and we give concrete examples of non trivial deformations.

Cohomology of the Lie Superalgebra of Contact Vector Fields on 𝕂1|1 and Deformations of the Superspace of Symbols

“…Using the same arguments as in proof of Proposition 6.2 together with Lemma 6.7, Proposition 6.2 and Proposition 6.3, we get the necessary integrability conditions for L (5) . Under these conditions, it can be easily checked that δ(L (m) ) = 0 for m = 5, 6, 7, 8.…”

“…Recently, deformations of Lie (super)algebras with multi-parameters were intensively studied (see, e.g., [1,3,5,6,24,25]). Here we give an outline of this theory.…”

“…Later Fialowski and Fuchs, using this framework, gave a construction for a versal deformation. Formal deformations of the vect(1)-module S n β were studied in [1,5]. Moreover, the formal deformations that become trivial once the action is restricted to sl(2) were completely described in [6].…”

“…, 5,λ = − 3(2λ + 9)(2λ + 3) 2 2(2λ + 7)(2λ + 1)(2λ 2 + 13λ + 17) , 6,λ = − 3(2λ + 7) 2(2λ 2 + 13λ + 17)…”

Deformation of $\mathfrak{sl}\, (2)$ and $\mathfrak{osp}\, (1|2)$ -modules of symbols

“…Deformation theory of Lie algebra homomorphisms was first considered with only one-parameter of deformation [7,10,14]. Recently, deformations of Lie (super)algebras with multi-parameters were intensively studied ( see, e.g., [1,2,4,5,6,11,12,13]).…”