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quaternions with zero real part, with Lie bracket the commutator; isomorphic to real 3-vectors, with Lie bracket the cross product; also isomorphic to su(2) and to so(3,R) Yes Yes 3 M(n,R) n×n matrices, with Lie bracket the commutator n 2: sl(n,R) square matrices with trace 0, with Lie bracket the commutator Yes Yes n 2 −1 so(n)
The first result in this direction is Lie's third theorem, which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem, which says every finite-dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every ...
Here are some matrix Lie groups and their Lie algebras. [14] ... the 1-dimensional subspace is an ideal in the 2-dimensional Lie algebra , by the formula [,] = ...
The Lie algebra of the compact form is 14-dimensional. The associated Lie group has no outer automorphisms, no center, and is simply connected and compact. The Lie algebra of the non-compact (split) form has dimension 14. The associated simple Lie group has fundamental group of order 2 and its outer automorphism group is the trivial group.
The group SL(2, R) acts on its Lie algebra sl(2, R) by conjugation (remember that the Lie algebra elements are also 2 × 2 matrices), yielding a faithful 3-dimensional linear representation of PSL(2, R). This can alternatively be described as the action of PSL(2, R) on the space of quadratic forms on R 2. The result is the following representation:
Type VI 0: This Lie algebra is the semidirect product of R 2 by R, with R where the matrix M has non-zero distinct real eigenvalues with zero sum. It is solvable and unimodular. It is the Lie algebra of the 2-dimensional Poincaré group, the group of isometries of 2-dimensional Minkowski space.
For certain types of Lie groups, namely compact [2] and semisimple [3] groups, every finite-dimensional representation decomposes as a direct sum of irreducible representations, a property known as complete reducibility. For such groups, a typical goal of representation theory is to classify all finite-dimensional irreducible representations of ...
A Lie algebra is said to be reductive if the adjoint representation is semisimple. Certainly, every (finite-dimensional) semisimple Lie algebra is reductive, since every representation of is completely reducible, as we have just noted. In the other direction, the definition of a reductive Lie algebra means that it decomposes as a direct sum of ...