Representations Over Current Lie Algebras And Yangians

Let g be a complex finite-dimensional simple Lie algebra,h be a fixed Cartan subalgebra of g and S be a unital commutative associative algebra over C.The current Lie algebras g[t]:=g(?)C[t]are related to Yangians,the main results of this paper are studying representations over current Lie algebras and as an application we apply it into Yangians.Firstly,we construct a class of infinite-dimensional modules W(f(u),g(u))called Weyl modules of infinite type over the current Lie algebras g[t],which have some maximal properties in the category O(g[t])over g[t].More explicitly,for any highest weight module V ∈ O(g[t]),there exists a Weyl module of infinite type W(f(u),g(u))∈O(g[t])such that V is isomorphic to the quotient of W(f(u),g(u)).Moreover,we give necessary and sufficient conditions on the decomposition of tensor product W((f1(u),g1(u))*(f2(u),g2(u)))≌(f1(u),g1(u))(?)W(f2(u),g2(u)).Next,we study a class of non-weight modules over the map Lie algebras g(?)S,i.e.,restricted to U(h)are isomorphic to the regular modules.We show that these modules consist of evaluation modules,we also give irreducibility criteria.Moreover,we give some new irreducible modules constructed by tensor product of irreducible modules.As a corollary,we reobtain the results in Nilsson[91,92].We classify all U(h)-free modules of rank 1 over map Witt algebras Wn(?)S by using the embedding of Lie algebra of type An into Wn,and we show that these modules are all evaluation modules.We also study the rank 1 free modules over map Virasoro algebras Vir(?)S and W(2,2)(?)S.For the former,we show that these modules consist of evaluation modules.However,there exists a non-evaluation rank 1 free module for the latter.Finally,as an application we study a class of polynomial modules over Yangian Y(gln+1),which have central characters.Moreover,we construct a weight functor in the category of Y(gln+1)-module,which maps any Y(gln+1)-module to a weight module over Y(gln+1),and we also give the central character formula in the case of n=1.