θ-Unitary representations for theW-algebraW(2,2)

Abstract

In this article, conjugate-linear anti-involutions and -unitary HarishChandra modules over the W-algebra W(2,2) are studied. It is proved that there are only two classes that conjugate-linear anti-involutions over W(2,2). The main result of this article is that a -unitary Harish-Chandra module over the W-algebra W(2,2) is simply a -unitary Harish-Chandra module over the Virasoro algebra and hence is either a simple highest or lowest weight module or a simple module from the intermediate series of the form A a,b for some a 2 R, b 2 1 2 þ ffiffiffiffiffiffi ffi À1 p R:

Representations of Super $W(2,2)$ algebra $\mathfrak{L}$

“…It possesses a basis {L m , I m |m ∈ Z} as a vector space over the complex field C, with the Lie brackets [L m , L n ] = (m − n)L m+n , [L m , I n ] = (m − n)I m+n , [I m , I n ] = 0. Structures and representations of W (2, 2) are extensively investigated in many references, such as [2], [4], [5], [7], [8] and [15]. Some Lie superalgebras with W -algebra W (2, 2) as their even parts were constructed in [9] as an application of the classification of Balinsky-Novikov super-algebras with dimension 2|2.In this paper we consider the infinite dimensional Lie super W (2, 2)-algebra over the algebraic closed field C (for convenience, we denote it L), with the following non-vanishing brackets:…”

Derivation algebras and automorphism groups of a class of deformative super $W$-algebras $W^s_λ(2,2)$

“…It has a basis {L m , I m | m ∈ Z} as a vector space over the complex field C, with the non-trivial Lie brackets [L m , L n ] = (m − n)L m+n and [L m , I n ] = (m−n)I m+n . Structures and representations of W (2, 2) are extensively investigated in the known references, such as [2], [7], [8], [10], [11], [18], [19] and the corresponding references. Some Lie superalgebras whose even parts are the W (2, 2) Lie algebras are constructed in [16].…”