Structures And Representations Of 2 Dim Affine-Virasoro Algebra

Virasoro algebra is the simplest and important algebra over the infinite di-mensional Lie algebra structures and representations.Recently,Virasoro algebra has been attracted more attention and has been successfully applied to the field-s of physical and mathematical,such as conformal field theory and the system of quantum integrability.2 dim affine-Virasoro algebra is the generalizaton of Virasoro algebra,it can be considered as the Lie algebra of the semidirect product of the Virasoro algebra and two Heisenberg algebras.Structures and representations of 2 dim affine-Virasoro algebra were studied in this paper,we obtain its automorphism group and determine its irreducible module,more details as follows:First we study structures of 2 dim affine-Virasoro algebra,we get the deriva-tion algebra,proof that the derivation is inner derivation,further we obtain that(?)is an infinite dimensional complete Lie algebra.Besides,by series of Lie operation and some special observations and methods,we get the second cohomology group of(?),and determine three 2-cocycles,namely that dim H2(g,C)=3,where H2(g,C)=C2(g,C)/B2(g,C)is called the second cohomology group of g,g is Lie algebra.What's more,we show that the universal central extension of(?)in the category of Leibniz algebras is the same as in the category of Lie algebras.Lastly,we discuss automorphism group.Then we discuss unitary representation of 2 dim affine-Virasoro algebra with center c,combine with lemma 2.1 and lemma 2.2 of[34],we obtain Lemma 2.7,and get a unitary weight module over the 2-dim affine-Virasoro algebra with center c is completely determined by a unitary weight module over the Virasoro algebra,further an irreducible unitary weight module V over the Lie algebra(?)is the highest weight module (?) or the lowest module (?) or V is isomorphic to Aa,b'.At last,we give condition of isomorphism between two generalized 2 dim affine-Virasoro algebras,which similar to two generalized Virasoro algebras.Then we discuss automorphism group.At last,Verma module over generalized 2 dim affine-Virasoro algebra is defined and its irreducibility is completely determined,more details as follows:(?)The Verma module V(c,h)is an irreducible (?)[G]module if c ?0.(?)If c = 0,h = 0,then the Verma module V(0,h)contains a unique maximal proper submodule N(0,h)^ where N(0,h)is generated by {L-kvh,M-kuh,N-kuh|k?G+} if h= 0,by {M-kuh,N-kvh|k ? G+} if h ? 0.