Editing Lie group (section)

=== Lie group versus Lie algebra isomorphisms ===
Isomorphic Lie groups necessarily have isomorphic Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras.

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 finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra.<ref>{{harvnb|Hall|2015}} Theorem 5.20</ref>

On the other hand, Lie groups with isomorphic Lie algebras need not be isomorphic. Furthermore, this result remains true even if we assume the groups are connected. To put it differently, the ''global'' structure of a Lie group is not determined by its Lie algebra; for example, if ''Z'' is any discrete subgroup of the center of ''G'' then ''G'' and ''G''/''Z'' have the same Lie algebra (see the [[table of Lie groups]] for examples). An example of importance in physics are the groups [[Special_unitary_group#The_group_SU(2)|SU(2)]] and [[Rotation group SO(3)|SO(3)]]. These two groups have isomorphic Lie algebras,<ref>{{harvnb|Hall|2015}} Example 3.27</ref> but the groups themselves are not isomorphic, because SU(2) is simply connected but SO(3) is not.<ref>{{harvnb|Hall|2015}} Section 1.3.4</ref>

On the other hand, if we require that the Lie group be [[simply connected]], then the global structure is determined by its Lie algebra: two simply connected Lie groups with isomorphic Lie algebras are isomorphic.<ref>{{harvnb|Hall|2015}} Corollary 5.7</ref> (See the next subsection for more information about simply connected Lie groups.) In light of Lie's third theorem, we may therefore say that there is a one-to-one correspondence between isomorphism classes of finite-dimensional real Lie algebras and isomorphism classes of simply connected Lie groups.