1.1. Hilbert’s fifth problem 9 Remark 1.1.4. In some literature, Lie groups are required to be connected (and occasionally, are even required to be simply connected), but we will not adopt this convention here. One can also define infinite-dimensional Lie groups, but in this text all Lie groups are understood to be finite dimen- sional. Example 1.1.5. Every group can be viewed as a Lie group if given the discrete topology (and the discrete smooth structure). (Note that we are not requiring Lie groups to be connected.) Example 1.1.6. Finite-dimensional vector spaces such as Rd are (additive) Lie groups, as are sublattices such as Zd or quotients such as Rd/Zd. How- ever, nonclosed subgroups such as Qd are not manifolds (at least with the topology induced from Rd) and are thus not Lie groups similarly, quotients such as Rd/Qd are not Lie groups either (they are not even Hausdorff). Also, infinite-dimensional topological vector spaces (such as RN with the product topology) will not be Lie groups. Example 1.1.7. The general linear group GLn(C) of invertible n × n com- plex matrices is a Lie group. A theorem of Cartan (Theorem 3.0.14) asserts that any closed subgroup of a Lie group is a smooth submanifold of that Lie group and is in particular also a Lie group. In particular, closed lin- ear groups (i.e., closed subgroups of a general linear group) are Lie groups examples include the real general linear group GLn(R), the unitary group Un(C), the special unitary group SUn(C), the orthogonal group On(R), the special orthogonal group SOn(R), and the Heisenberg group ⎛ ⎝0 1 R R 1 R⎠ 0 0 1 ⎞ of unipotent upper triangular 3 × 3 real matrices. Many Lie groups are isomorphic to closed linear groups for instance, the additive group R can be identified with the closed linear group 1 R 0 1 , the circle R/Z can be identified with SO2(R) (or U1(C)), and so forth. However, not all Lie groups are isomorphic to closed linear groups. A some- what trivial example is that of a discrete group with cardinality larger than the continuum, which is simply too large to fit inside any linear group. A less pathological example is provided by the Weil-Heisenberg group (1.3) G := ⎛ ⎝0 1 R R/Z 1 R 0 0 1 ⎞ ⎠ := ⎛ ⎝0 1 R R 1 R⎠ 0 0 1 ⎞ / ⎛ ⎝0 1 0 Z 1 0⎠ 0 0 1 ⎞

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