无限阶矩阵李代数余伴随表示的刻画

Description on Coadjoint Representations of an Infinite Matrix Algebra

设M∞()是C上所有无限阶矩阵构成的向量空间,gl(∞)是M∞()的一个特殊子空间,关于括积运算gl(∞)是一个李代数。对gl(∞)的李子代数g,令g*是g的对偶空间,g+是g的受限对偶空间。定义了g在g*上的余伴随作用,使其成为g-模,g+是g*的g-子模。证明了gl(∞)中存在子空间W,作为g-模,它与g+同构。

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 gl（∞） 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.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＋.