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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 ...
In abstract algebra, an automorphism of a Lie algebra is an isomorphism from to itself, that is, a bijective linear map preserving the Lie bracket. The set of automorphisms of g {\displaystyle {\mathfrak {g}}} are denoted Aut ( g ) {\displaystyle {\text{Aut}}({\mathfrak {g}})} , the automorphism group of g {\displaystyle {\mathfrak {g}}} .
In physics, a vector space basis of the Lie algebra of a Lie group G may be called a set of generators for G. (They are "infinitesimal generators" for G , so to speak.) In mathematics, a set S of generators for a Lie algebra g {\displaystyle {\mathfrak {g}}} means a subset of g {\displaystyle {\mathfrak {g}}} such that any Lie subalgebra (as ...
This is made precise in the article on the tensor algebra: the tensor algebra has a Hopf algebra structure on it, and because the Lie bracket is consistent with (obeys the consistency conditions for) that Hopf structure, it is inherited by the universal enveloping algebra. Given a Lie group G, one can construct the vector space C(G) of ...
Wentworth wrote a series of textbooks on mathematics, of which The Boston Globe noted in 1886, "his Complete Algebra and Elements of Geometry are used extensively by many of the more important schools in America, and doubtless will, with but very few changes, be employed as standard works for a half century to come."
Let be the complexification of a real semisimple Lie algebra , then complex conjugation on is an involution on . This is the Cartan involution on g {\displaystyle {\mathfrak {g}}} if and only if g 0 {\displaystyle {\mathfrak {g}}_{0}} is the Lie algebra of a compact Lie group.
In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate g ¯ {\displaystyle {\overline {\mathfrak {g}}}} is a complex Lie algebra with the same underlying real vector space but with i = − 1 {\displaystyle i={\sqrt {-1}}} acting as − i ...
Notational abuse to be found below includes e X for the exponential map exp given an argument, writing g for the element (g, e H) in a direct product G × H (e H is the identity in H), and analogously for Lie algebra direct sums (where also g + h and (g, h) are used interchangeably). Likewise for semidirect products and semidirect sums.