Ads
related to: general linear lie algebra
Search results
Results from the WOW.Com Content Network
The infinite general linear group or stable general linear group is the direct limit of the inclusions GL(n, F) → GL(n + 1, F) as the upper left block matrix. It is denoted by either GL( F ) or GL(∞, F ) , and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places.
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 ...
The Lie algebra of the general linear group GL(n, C) of invertible matrices is the vector space M(n, C) of square matrices with the Lie bracket given by
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 .
A Lie algebra representation also arises in nature. If : G → H is a homomorphism of (real or complex) Lie groups, and and are the Lie algebras of G and H respectively, then the differential: on tangent spaces at the identities is a Lie algebra homomorphism.
This article gives a table of some common Lie groups and their associated Lie algebras.. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple).
A special case of Lie correspondence is a correspondence between finite-dimensional representations of a Lie group and representations of the associated Lie algebra. The general linear group () is a (real) Lie group and any Lie group homomorphism : ()
Ads
related to: general linear lie algebra