The Algebraic Structures And Representations Of W-Algebra W(2,2)

This thesis study the algebraic strctures and modules of W-algebra W(2,2). At first, we introduce definitions and properties of central extensions and 2-cohomology groups, and compute the universal central extension of Witt algebra, i.e. Virasoro alge-bra. Thus we obtain a corollary of the dimension of 2-homology group of W(2,2) and realized it by the central extension of W’(α,β) in general, which is also the universal central extension. Then we calculate the derivations of W’(O, -1) by exact sequences, and its automorphism group by semidirect product. Hence we get the derivations and automorphism group of W(2,2) according to the properties of universal central exten-sion. Finally, simple W(2,2)-modules with specific properties are constructed through finite quotients of some subalgebras and relative characteristics are disscussed in the end.