Search results
Results from the WOW.Com Content Network
In the case of the complex special linear group SL(n,C), the compact real form is the special unitary group SU(n) and the split real form is the real special linear group SL(n,R). The classification of real forms of semisimple Lie algebras was accomplished by Élie Cartan in the context of Riemannian symmetric spaces. In general, there may be ...
For a linear algebraic group G over the real numbers R, the group of real points G(R) is a Lie group, essentially because real polynomials, which describe the multiplication on G, are smooth functions. Likewise, for a linear algebraic group G over C, G(C) is a complex Lie group. Much of the theory of algebraic groups was developed by analogy ...
The underlying real Lie algebra of the complex Lie algebra G 2 has dimension 28. It has complex conjugation as an outer automorphism and is simply connected. The maximal compact subgroup of its associated group is the compact form of G 2. The Lie algebra of the compact form is 14-dimensional.
The compact classical groups are the compact real forms of the complex classical groups. These are, in turn, SU(n), SO(n) and Sp(n). One characterization of the compact real form is in terms of the Lie algebra g. If g = u + iu, the complexification of u, and if the connected group K generated by {exp(X): X ∈ u} is compact, then K is a compact ...
If G is connected with Lie algebra 𝖌, then its universal covering group G is simply connected. Let G C be the simply connected complex Lie group with Lie algebra 𝖌 C = 𝖌 ⊗ C, let Φ: G → G C be the natural homomorphism (the unique morphism such that Φ *: 𝖌 ↪ 𝖌 ⊗ C is the canonical inclusion) and suppose π: G → G is the universal covering map, so that ker π is the ...
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.
Given a complex Lie algebra , a real Lie algebra is said to be a real form of if the complexification is isomorphic to .. A real form is abelian (resp. nilpotent, solvable, semisimple) if and only if is abelian (resp. nilpotent, solvable, semisimple). [2]
Thus, the span of these three operators forms a Lie algebra, which is isomorphic to the Lie algebra so(3) of the rotation group SO(3). [2] Then if V {\displaystyle V} is any subspace of the quantum Hilbert space that is invariant under the angular momentum operators, V {\displaystyle V} will constitute a representation of the Lie algebra so(3).