An Infinite-dimensional Representation Of A Semisimple Lie Algebra

The finite-dimensional complex semisimple Lie algebra is an important type of Lie algebra,and its structure and finite-dimensional representation theory have been studied very clearly.In recent years,people have paid more attention to the research on the infinite-dimensional representation theory of finite-dimensional complex semisimple Lie algebras,but due to the number of dimensions,the research is very difficult,and it is even very difficult to give an example of a certain type of infinite-dimensional irreducible representation.In addition,Heisenberg-Virasoro algebra is an important type of infinite-dimensional Lie algebra,it has important applications in mathematics and physics,and its structure and representation theory have attracted many scholar's attention.This paper mainly studies the infinite dimensional representation of the A-type Lie algebra and the centroid of the twisted deformative Heisenberg-Virasoro alge-bra.First,a class of infinite-dimensional representations of the A2 type Lie algebra with five parameters is constructed from the intermediate sequence module of the Witt algebra,and then it is proved that this type of representation is irreducible under certain conditions.Thus,a class of infinite dimensional representation of A2 type semisimple Lie algebra is given.Finally,using the grading of the twisted deformative Heisenberg-Virasoro algebra and the ad-semisimple element,we deter-mine the centroid of the twisted deformative Heisenberg-Virasoro algebra with three parameters.