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In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group.
The Jordan–Chevalley decomposition of an element in algebraic group as a product of semisimple and unipotent elements; The Bruhat decomposition = of a semisimple algebraic group into double cosets of a Borel subgroup can be regarded as a generalization of the principle of Gauss–Jordan elimination, which generically writes a matrix as the product of an upper triangular matrix with a lower ...
When F is the real numbers, (,) is the Lie algebra of the general linear group (,) , the group of invertible n x n real matrices (or equivalently, matrices with ...
In other words, a linear Lie algebra is the image of a Lie algebra representation. Any Lie algebra is a linear Lie algebra in the sense that there is always a faithful representation of g {\displaystyle {\mathfrak {g}}} (in fact, on a finite-dimensional vector space by Ado's theorem if g {\displaystyle {\mathfrak {g}}} is itself finite ...
Over a field of characteristic zero, a connected subgroup H of a linear algebraic group G is uniquely determined by its Lie algebra . [7] But not every Lie subalgebra of g {\displaystyle {\mathfrak {g}}} corresponds to an algebraic subgroup of G , as one sees in the example of the torus G = ( G m ) 2 over C .
Suppose G is a closed subgroup of GL(n;C), and thus a Lie group, by the closed subgroups theorem.Then the Lie algebra of G may be computed as [2] [3] = {(;)}. For example, one can use the criterion to establish the correspondence for classical compact groups (cf. the table in "compact Lie groups" below.)
A real Lie algebra is usually complexified enabling analysis in an algebraically closed field. Working over the complex numbers in addition admits nicer bases. The following theorem applies: A real-linear finite-dimensional representation of a real Lie algebra extends to a complex-linear representation of its complexification.
In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, [1] over an algebraically closed field of characteristic zero, if : is a finite-dimensional representation of a solvable Lie algebra, then there's a flag = = of invariant subspaces of () with =, meaning that () for each and i.
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