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11. Invariant vector fields and Lie algebras. Let G be a (real) Lie group. A vector field ξ on G is called left invariant, if µ∗a ξ = ξ for all a ∈ G, where µ∗a ξ = T (µa−1 ) ◦ ξ ◦ µa as in section 3. 11) we have µ∗a [ξ, η] = [µ∗a ξ, µ∗a η], the space XL (G) of all left invariant vector fields on G is closed under the Lie bracket, so it is a sub Lie algebra of X(G). ξ(e). Thus the Lie algebra XL (G) of left invariant vector fields is linearly isomorphic to Te G, and on Te G the Lie bracket on XL (G) induces a Lie algebra structure, whose bracket is again denoted by [ , ].

Michor, 38 Chapter II. 6 −1 function theorem. a−1 = (µa ◦ ν)(x) we may conclude that ν is everywhere smooth. Xa . 4. Example. The general linear group GL(n, R) is the group of all invertible real n × n-matrices. It is an open subset of L(Rn , Rn ), given by det = 0 and a Lie group. Similarly GL(n, C), the group of invertible complex n × n-matrices, is a Lie group; also GL(n, H), the group of all invertible quaternionic n × n-matrices, is a Lie group, since it is open in the real Banach algebra LH (Hn , Hn ) as a glance at the von Neumann series shows; but the quaternionic determinant is a more subtle instrument here.

25. Theorem (local structure of singular foliations). Let E be an integrable (singular) distribution of a manifold M . Then for each x ∈ M there exists a chart (U, u) with u(U ) = {y ∈ Rm : |y i | < ε for all i} for some ε > 0, and a countable subset A ⊂ Rm−n , such that for the leaf L through x we have u(U ∩ L) = {y ∈ u(U ) : (y n+1 , . . , y m ) ∈ A}. Each leaf is an initial submanifold. If furthermore the distribution E has locally constant rank, this property holds for each leaf meeting U with the same n.

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### A brill - noether theory for k-gonal nodal curves by Ballico E.

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