[13] If G and H are both simply connected and have isomorphic Lie algebras, the above result allows one to show that G and H are isomorphic. exp By the first isomorphism theorem, g

[19] If G is connected, it fits into the exact sequence: where

Thus, given two elements near the identity {\displaystyle e^{X}e^{Y}} Lie

n

It is important to emphasize that the equivalence of the preceding conditions holds only under the assumption that G has finite center. Thus (as the differential of a constant function). of

U

3 $\begingroup$ I would like to know a complete description of the indecomposable representations of the two dimensional non-abelian Lie algebra over the complex numbers. →

: e

g

= .

Z , where. G )

( {\displaystyle \pi } The universal covering space G ~ is also a connected abelian Lie group of dimension n which is, of course, simply connected; in particular, the Lie algebra of G ~ is of dimension n and abelian. )

In other words, the group is an Abelian-by-discrete group. ⁡ g e

)

)

{\displaystyle GL_{n}(\mathbb {R} )}

l

j

′

⁡ is also compact.

g

(

It is called the adjoint group of ( d g

) {\displaystyle \phi } Then, For a subset A of

between the universal enveloping algebra of ( t exp

e

e

{\displaystyle \bigcup _{n>0}U^{n}} ⁡ exp =

x

)

is in fact a Hopf algebra.

x

⁡ G g

is called the adjoint representation of

/

Let G be a connected Lie group.

the previous §), G is abelian if and only if its Lie algebra is abelian. T

~ x

This requirement cannot be omitted; see also, harvnb error: no target: CITEREFBourbaki (, harvnb error: no target: CITEREFHall2015Section_4.7 (, Notes for Math 261A Lie groups and Lie algebras, https://en.wikipedia.org/w/index.php?title=Lie_group–Lie_algebra_correspondence&oldid=941145261, Creative Commons Attribution-ShareAlike License, There exists an invariant inner product on, This page was last edited on 16 February 2020, at 21:26. ⁡

⁡

R

{\displaystyle f(\operatorname {exp} (X))=\operatorname {exp} (df(X))}

is then an automorphism of the Lie algebra However, we can certainly conclude that the connected component containing the identity is Abelian. ⁡ . Let It is important to emphasize that this argument is only local, since the exponential map is only invertible in a small neighborhood of the identity in G and since the Baker–Campbell–Hausdorff formula only holds if X and Y are small. X

)

For example, the Lie algebras of SO(3) and SU(2) are isomorphic,[11] but there is no corresponding homomorphism of SO(3) into SU(2).

e (

G T

{\displaystyle \Gamma }

⁡

Since

{\displaystyle \operatorname {Lie} (G)} and the set of one-parameter subgroups of G.[16], One approach to proving the second part of the Lie group-Lie algebra correspondence (the homomorphisms theorem) is to use the Baker–Campbell–Hausdorff formula, as in Section 5.7 of Hall's book.

Y G

∞

L l {\displaystyle \displaystyle \exp(X)=e^{X}=\sum _{0}^{\infty }{X^{j}/j!}} G

{\displaystyle \operatorname {exp} :{\mathfrak {g}}\to G}

Y

is a Lie subalgebra.

= H (The differential

G = ρ give the canonical identification: If

{\displaystyle l_{g}} R

Then the differentials f

[7] In general, if U is a neighborhood of the identity element in a connected topological group G, then

) d

are Lie subgroups of a Lie group, then

{\displaystyle {\mathfrak {g}}\subset {\mathfrak {gl}}_{n}(\mathbb {R} )=\operatorname {Lie} (GL_{n}(\mathbb {R} ))}

{\displaystyle {\mathfrak {g}}}

ad ,

H

G

(

z {\displaystyle \operatorname {Lie} (H\cap H')=\operatorname {Lie} (H)\cap \operatorname {Lie} (H').}. (

)

π g Lie ; the differential

)

0

⁡ {\displaystyle {\mathfrak {g}}}

n ↦

G A

.

exp

R

) ( Lie algebra of an Abelian Lie group is Abelian, If the Lie algebra is Abelian then the group is nearly so, View other analogues in Lie algebras of group properties, https://groupprops.subwiki.org/w/index.php?title=Abelian_Lie_algebra&oldid=12276, Analogues in Lie algebras of group properties.

is determined by the group law on G. By Lie's third theorem, there exists a subgroup

c Lie Let G be the subgroup of

∑

, which in particular implies that the Lie bracket of l

ad A

L g l

= →

g

⁡

One can show )

/

)

g G {\displaystyle \operatorname {Lie} (G)} and

.

whose Lie algebra is

g

⁡ {\displaystyle {\mathfrak {g}}} Γ

) = {\displaystyle {\mathfrak {gl}}_{n}} e

f

defines a local homeomorphism from a neighborhood of the zero vector to the neighborhood of the identity element. {\displaystyle f:G\to H} Suppose G is a closed subgroup of GL(n;C), and thus a Lie group, by the closed subgroups theorem.

= G This follows from the fact that for a Lie group, the Lie bracket of two tangent vectors and is the limit of a certain difference expression of conjugates.

{\displaystyle \operatorname {Lie} (G)}

Ad

X

G In this article, manifolds (in particular Lie groups) are assumed to be second countable; in particular, they have at most countably many connected components.

G ( =

) →

( Lie [18] (This observation is related to the distinction between integer spin and half-integer spin in quantum mechanics.) x

⁡

More explicitly, there is a natural action of the Lie group on its Lie algebra, called the adjoint representation, and the commutator can be defined as the differential at of .

=

c ) Thus, for example, if G is compact with finite center, the universal cover ⁡ )

{\displaystyle G=G_{1}\times \cdots \times G_{r}} ( [22] The kernel of it is a discrete group (since the dimension is zero) called the integer lattice of G and is denoted by

) X

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