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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.
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 ...
Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. [20] On the other hand, Lie groups with isomorphic Lie algebras need not be isomorphic. Furthermore, this result remains true even if we assume the groups are connected.
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 ...
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 ...
The most basic example is the Lie algebra of matrices with the commutator as Lie bracket, or more abstractly as the endomorphism algebra of an n-dimensional vector space, (). This is the Lie algebra of the general linear group GL( n ), and is reductive as it decomposes as g l n = s l n ⊕ k , {\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl ...
One setting in which the Lie algebra representation is well understood is that of semisimple (or reductive) Lie groups, where the associated Lie algebra representation forms a (g,K)-module. Examples of unitary representations arise in quantum mechanics and quantum field theory, but also in Fourier analysis as shown in the following example.
In Lie theoretic terms, () is the compact real form of (,), which means that complex linear, continuous irreducible representations of the latter are in one-to-one correspondence with complex linear, algebraic irreps of the former, via the inclusion () (,).
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