By Newman M. F. (Ed)
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Extra info for Topics in Algebra: Proceedings, 18th Summer Research Institute of the Australian Mathematical Society
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Here ω(x, y) is the skew-symmetric bilinear form ni=1 xi yi+n − yi xi+n (which, up to a change of basis, is the unique nondegenerate skew-symmetric bilinear form on K2n ). Equivalently, one can write ω(x, y) = (Jx, y), where ( , ) is the standard symmetric bilinear form on K2n and J = 0 In −In . 5) Note that there is some ambiguity with the notation for symplectic group: the group we denoted Sp(n, K) would be written in some books as Sp(2n, K). • U(n) (note that this is a real Lie group, even though its elements are matrices with complex entries) • SU(n) • Group of unitary quaternionic transformations Sp(n) = Sp(n, C) ∩ SU(2n).
Fundamental theorems of Lie theory Let us summarize the results we have so far about the relation between Lie groups and Lie algebras. 2); we will write g = Lie(G). Every morphism of Lie groups ϕ : G1 → G2 defines a morphism of Lie algebras ϕ∗ : g1 → g2 . For connected G1 , the map Hom(G1 , G2 ) → Hom(g1 , g2 ) ϕ → ϕ∗ is injective. ) (2) As a special case of the previous, every Lie subgroup H ⊂ G defines a Lie subalgebra h ⊂ g. (3) The group law in a connected Lie group G can be recovered from the commutator in g; however, we do not yet know whether we can also recover the topology of G from g.
What is the exponential map? If ξ ∈ Vect(M ) is a vector field, then exp(tξ ) should be a one-parameter family of diffeomorphisms whose derivative is vector field ξ . So this is the solution of the differential equation d t ϕ (m)|t=0 = ξ(m). dt In other words, ϕ t is the time t flow of the vector field ξ . We will denote it by exp(tξ ) = t ξ. 7) 34 Lie groups and Lie algebras This may not be defined globally, but for the moment, let us ignore this problem. What is the commutator [ξ , η]? 3), we need to consider tξ sη t−ξ s−η .