Typical Lie Algebra Of The General Linear Group Representation Theory

As is well-known, study the representation of an algebraic can study fromwithin the algebra (that is, research on the algebraic representation space), canstudy from the outside of the algebra (that is, Study the algebra as a representationspace). This paper will study representation of the general linear lie algebra gl(3, C)from outside. This paper will get general linear lie algebra gl(3, C) in general lineargroups GL(2, C) all kinds of said structure. This not only studying the theories ofthe general linear lie algebras gl(3, C), will also give some realize of9dimensionalrepresentation of the general linear group GL(2, C).Firstly, use properties of conjugate representation to keep track and multipli-cation of partitioned matrix methods is given all irreducible submodules of gl(3; C).By the space of knowledge can be proved that the conjugate representation is com-pletely reducible.Secondly, because of the symmetric matrix and anti-symmetric matrix in thetransposed representation can constitute a submodule of gl(3; C). According to thischaracteristic can be resolve GL(2; C)-module gl(3; C) into the directly of irreduciblerepresentation.This paper not only promoted the development of algebra and physics in be-tween, you can also help people with physical and other aspects of the mathematicsknowledge to solve problems, and has a guiding role in solving practical problems.