Structures of W(2.2) Lie conformal algebra

Abstract

Abstract:The purpose of this paper is to study W .2; 2/ Lie conformal algebra, which has a free COE@ -basisIn this paper, we study conformal derivations, central extensions and conformal modules for this Lie conformal algebra. Also, we compute the cohomology of this Lie conformal algebra with coefficients in its modules. In particular, we determine its cohomology with trivial coefficients both for the basic and reduced complexes.

Classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra W(a,b)

“…Therefore, h 1 (λ, ∂) = 0 and c = 0. By (28), g 1 (λ, ∂) = 0. By comparing the coefficients of λ in (24), we can assume that k 1 (λ, ∂) = k 1 (∂), Then, according to (22), one can obtain…”

“…Subcase c2: when b = 0, a = 1 and c = b, h 1 (µ, ∂) = (µ + ∂ + c)d 0 . By (28), one can get g 1 (λ, ∂) = (c − λ)d 0 . According to (25), we have…”

“…W(2, 0) is just the W (2, 2) Lie conformal algebra. The study of its structure and representation can refer to [28,25]. In this paper, our aim is to study compatible left-symmetric conformal algebraic structures on W(a, b) = C[∂]L ⊕ C[∂]W .…”

“…Taking h(λ) = h 1 λ into (48), we get h 1 (λ, ∂) = (λ + ∂)h 1 . According to(28), one can have g 1 (λ, ∂) = −h 1 λ. Then(20) holds.…”

Extensions of modules over a class of Lie conformal algebras 𝒲(b)

“…The other one is W (−1), which is exactly the W (2, 2) Lie conformal algebra. Its conformal derivations and central extensions were studied in [10].…”