Gauss decomposition

of a topological group

A representation of an everywhere-dense subset in the form , where is an Abelian subgroup of , and and are nilpotent groups of , normalized by . If is the group of non-singular real matrices of order , is the subgroup of diagonal matrices, (respectively, ) is the subgroup of lower-triangular (upper-triangular) matrices with unit elements on the principal diagonal, and is the subset of matrices in whose principal minors are non-zero, then the decomposition is known as the Gauss decomposition of the general linear group and is directly connected with the Gauss method for the solution of systems of linear equations: If , where , , , is the non-singular coefficient matrix of the system of linear equations , then it may be converted by the Gauss method into the triangular form by multiplying it from the left by the lower-triangular matrix , . A rigorous definition of the Gauss decomposition necessitates the introduction of the following terms. Let be a topological group, let be a subgroup of it, and let and be nilpotent subgroups in , normalized by . The subgroup is called a triangular truncation of if: 1) , , where is the commutator subgroup of the group and and are connected solvable subgroups of ; and 2) the set is everywhere dense in , and the decomposition is unique. The decomposition is called a triangular decomposition in . If is an Abelian group, this decomposition is called a completely-triangular decomposition or a Gauss decomposition. In such a case the subgroups , are solvable. Let be an irreducible (continuous) representation of in a finite-dimensional vector space , and let be the subspace of all vectors in which are fixed with respect to ; will then be invariant with respect to , while the representation of on will be irreducible. The representation unambiguously defines , up to an equivalence. Let also denote the representation of on restricting to on and being trivial on . Let denote the representation of on the vector space induced by this . Then is contained (as an invariant part) in , and the space is one-dimensional. If is an Abelian subgroup, then is one-dimensional and is a character of the group . The following examples of triangular decompositions of Lie groups are known. 1) Let be a reductive connected complex Lie group with Cartan subalgebra and let be a reductive connected subgroup in containing . The subgroup is then a triangular truncation of . 2) Let be a reductive connected linear Lie group; will then contain a triangular truncation , where is a simply-connected Abelian subgroup in (generated by the non-compact roots in the Lie algebra of ), and is the centralizer of in the maximal compact subgroup . 3) In particular, any reductive connected complex Lie group permits a Gauss decomposition , where is the Cartan subgroup of and (respectively ) is an analytic subgroup in whose Lie algebra is spanned by all root vectors , (respectively ), with denoting the roots with respect to , i.e. and are opposite Borel subgroups (cf. Borel subgroup). In examples 1)–3) the subgroups and are simply connected, is open in in the Zariski topology, while the mapping , , is an isomorphism of algebraic varieties (and, in particular, a homeomorphism). This implies that the algebraic variety is rational.

References

[1]	D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)