Structures And Representations Of A Class Of Not-finitely Z-graded Lie Algebras Of Block Type

In this thesis, we study the structure theory and representation theory of a class of infinite-dimensional Z2-graded but not-finitely Z-graded Block type Lie algebras B(q), where q is a nonzero complex parameter.In Chapter1, we introduce the research background and the main results of this thesis. Block type Lie algebras were firstly introduced by Richard Earl Block [7] in the late50s, which can be viewed as analogous of the Zassenhaus algebras in characteristic zero case. There exist intimate relations between the Lie algebras B(q) and the well-known Virasoro algebra and W1+∞algebra. The Lie algebra B(q) can be also viewed as some special cases of (generalized) Cartan-type Lie algebras. It is well-known that the Virasoro algebra and W1+∞algebra play very important roles in various physical theories, such as conformal field theory, the theory of the quantum Hall effect, etc. The representation theory of Cartan type Lie algebras is far from being well developed. So, it is very interesting to study the structure theory and representation theory of B(q).In Chapter2, we study the structure theory of B(q). Suppose that the parameter q are positive integers. By studying the locally finite elements and locally nilpotent elements of B(q), we firstly characterize the automorphism group of B(q) and give the isomorphic classification of B(q). Since B(q) is not finitely-generated Z-graded Lie algebra, we employ a technique developed in [61] to characterize the structure of the derivation algebra of B(q). In addition, we also uniformly characterize the central extensions of B(q). At last, we discuss the structure theory of B(q) when the parameter q are not positive integers.In Chapter3, we study the representation theory of B(q). Since B(q) contains a Virasoro subalgebra, motivated by Olivier Mathieu’s classification on Harish-Chandra modules over the Virasoro algebra [39], we give a rough classification of the quasifinite irreducible modules of B(q) in this chapter. The notion of general quasifinite modules over infinite dimensional graded Lie algebras was firstly proposed by Victor Kac and Andrey Radul [29] when studying the representation theory of the W1+∞algebra. Kac et al.[6,17,29,31] pointed out that the quasifinite representation of this kind of infinite-dimensional not-finitely Z-graded Lie algebras is a highly nontrivial problem. In addition, we also classify the quasifinite irreducible highest weight B(g)-modules and the irreducible B(q)-modules of the intermediate series.In Chapter4, we continue to study the representation theory of B(q). Based on the classification of the quasifinite irreducible highest weight B(q)-modules in Chapter3, together with the classification of the highest weight unitary modules over Vira-soro algebra, we completely classify the quasifinite irreducible highest weight unitary modules over B(q). This classification indicates that the quasifinite irreducible highest weight unitary modules over B(q) can be almost viewed as a single-parameter gener-alization of the irreducible highest weight unitary modules over Virasoro algebra.