Integrable Representations For Toroidal Extended Affine Lie Algebras

Extended affine Lie algebras(EALAs)are natural generalizations of the finite di-mensional simple Lie algebras and the affine Kac-Moody algebras.In the study of the representation theory of EALAs,one of the most fundamental problems is to give its clas-sification of irreducible integrable modules with finite dimensional weight spaces.Around 1986,V.Chari and A.N.Pressley classified the irreducible integrable modules with finite dimensional weight spaces for nullity 1 EALAs(i.e.,affine Kac-Moody algebras).And such modules for nullity 0 EALAs(i.e.,finite dimensional simple Lie algebras)are fi-nite dimensional simple modules.Toroidal EALAs are a class of the most fundamental EALAs.In this thesis,we give a complete classification of irreducible integrable modules with finite dimensional weight spaces for nullity 2 toroidal EALAs such that the action of the core of the nullity 2 toroidal EALA is non-trivial.When we deal with those modules,we need some results about a class of loop modules for the nullity 2 toroidal EALA and the Harish-Chandra modules for rank 2 Heisenberg-Virasoro algebra(See Chapter 3 and Chapter 4,respectively).As far as we know,the representation theory of EALAs with nullity greater than or equal to 2 is much less understood.The known irreducible modules for toroidal EALAs were those constructed in[B]by applying the theory of vertex operator algebras.Moti-vated by Chari-Pressley's loop module construction[CP1]for affine Kac-Moody algebras and Billig's construction[B]for toroidal EALAs,we constructed a new class of irreducible integrable modules for the nullity 2 toroidal EALAs,which indeed generalizes Billig's construction,and we proved that those modules are complete reducible under certain conditions.See Chapter 3.The classification of irreducible integrable modules with finite dimensional weight spaces for nullity 2 toroidal EALAs such that the action of the core of nullity 2 toroidal EALAs is non-trivial is a very interesting and important problem.When the canonical central elements act non-trivial,such a module is an irreducible quotient of the induced module from some Z-graded irreducible module with zero central charge for a Heisenberg subalgebra.Otherwise,such a module is an irreducible quotient of the induced module from some Z2-graded irreducible module for the semi-product of the Laurent polynomial ring in two variables with its skew derivation Lie algebra.We note that in the case of the full toroidal Lie algebra,such a module is always an irreducible quotient of the induced module from some graded irreducible module for the semi-product between the Laurent polynomial ring with its derivation Lie algebra(See[RJ]for details).In the case that the action of the core of the nullity 2 EALA is trivial,such a module becomes an irreducible module for centerless Virasoro-like algebra with finite dimensional graded weight spaces and vice versa.In Chapter 4,we prove that a Harish-Chandra module for rank 2 Heisenberg-Virasoro algebra is either a uniformly bounded module,or a generalized highest weight module.Moreover,we classify all the generalized highest weight Harish-Chandra modules.In Chapter 5,we study Verma modules for rank 2 Heisenberg-Virasoro algebra.