Lie's theorems
/ lee /
The grand bargain of the whole subject is a dictionary: hard, curved, global Lie groups on one side, and easy, linear, finite-dimensional Lie algebras on the other. Lie's theorems are the rules of this dictionary — they say how much of a Lie group you can read off from its Lie algebra, and which way the translation is faithful. The slogan is that, up to covering and connectedness, a Lie group is its Lie algebra.
Three theorems, often called Lie's three theorems, set up the correspondence (modern statements, partly due to Cartan). (i) Each Lie group G has a Lie algebra g = T_e G, and a Lie group homomorphism F: G -> H induces a Lie algebra homomorphism dF_e: g -> h; for CONNECTED G this assignment is functorial and faithful in the sense that F is determined by dF_e. (ii) Conversely, every Lie algebra homomorphism phi: g -> h integrates to a Lie group homomorphism F: G -> H provided G is connected AND simply connected. (iii) Lie's third theorem (the existence/Cartan theorem): every finite-dimensional real Lie algebra is the Lie algebra of some Lie group — in fact of a unique connected simply connected one. Putting these together: the category of finite-dimensional real Lie algebras is equivalent to the category of connected, simply connected Lie groups. Any connected Lie group with algebra g is then a quotient of that universal simply connected group G~ by a discrete central subgroup.
These theorems are why you may compute in the linear world of g and transport the answer to G, why su(2) ~ so(3) yet SU(2) is the simply connected cover of SO(3) (quotient by the center {+-I}), and why representations of a simply connected group correspond exactly to representations of its Lie algebra. The honesty caveats are precisely the words 'connected' and 'simply connected.' Without simple connectedness, a Lie algebra map need not integrate (only locally), and without connectedness the Lie algebra is blind to extra components. The correspondence is between ALGEBRAS and SIMPLY CONNECTED groups; for everything else you must track the fundamental group and the covering.
The isomorphism of Lie algebras su(2) ~ so(3) does NOT come from an isomorphism of groups: SU(2) is simply connected (it is S^3), while SO(3) is not (pi_1(SO(3)) = Z/2). Lie's theorems say SU(2) is THE simply connected group for this algebra, and SO(3) = SU(2)/{+-I} is the quotient by the center — the famous spin double cover.
One algebra su(2)~so(3), two groups: SU(2) (simply connected) covers SO(3) two-to-one.
The equivalence is between Lie algebras and CONNECTED, SIMPLY CONNECTED groups. Drop simple connectedness and an algebra map integrates only locally; drop connectedness and the algebra cannot see extra components.