Infinite Order Matrix Algebra I Accompanied Lee Portrayed Representation

The depiction of the dual space structure of the Lie algebra is a basic problem in the representation theory. The dual space of the subalgebra structure for the general linear Lie algebra gl(n) can be implemented on a subspace. And for the Lie superalgebra, the dual space of its subalgebra structure can also be implemented on a subspace. In this paper, the results are generalized to the Lie algebra of infinite matrix. Let M(∞) be the vector space of all the infinite matrices on C, gl(∞) is a special subspace of M(∞). Product and bracket are defined on gl(∞), and it is proved that g(∞) is a Lie Algebra over C. Let g be the Lie subalgebra of gl(∞), g*be the dual space of g and g+be the limited dual space of g. In this paper, the coadjoint action of g on g*is defined, making g*into g-module, and g+is a g-submodule of g*. It is proved that there exits a subspace W of gl (∞), which is a g-module and isomorphic to g+. This conclusion has a certain significance to the study of lie algebra of infinite matrix. Concretely, this artical contains several parts:In Chapter1, recall some basic knowledge about Lie algebra and give an example.In Chapter2, define the infinite matrix and constructed the Lie algebra of infinite matrix.In Chapter3, The dual space of the subalgebra structure for the Lie algebra of infinite matrix can also be implemented on a subspace.