Sumários
6 Novembro 2025, 10:30
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Marko Stosic
Simply connected topological spaces (manifolds, Lie groups)
Fundamental group.
Coverings. Existence of a universal covering of a Lie group.
Example of a double cover of the Lie group SO(3) by the simply connected Lie group SU(2).
30 Outubro 2025, 10:30
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Marko Stosic
Exponential map on the tangent space of a Lie group.
Exponential map for matricial Lie algebra and matricial Lie groups.
Explicit computation of classical Lie algebra, via exponential of matrices.
Third fundamental theorem for Lie groups and Lie algebras.
23 Outubro 2025, 10:30
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Marko Stosic
Local diffeomorphisms on a manifold corresponding to a vector field.
One-parameter subgroup of a Lie group.
Exponential map on a Lie algebra.
16 Outubro 2025, 10:30
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Marko Stosic
More examples of (abstract) Lie algebras.
Algebra of endomorphisms of a given vector space: both as associative algebra, and as a Lie algebra (with respect to the commutator)
Lie algebra of derivations on an algebra (as a subalgebra of the Lie algebra of endomorphisms).
Derivations on the algebra of smooth real valued function on a manifold (C^{\infty}(M)).
Vector fields on a manifold corresponds to derivations on C^{\infty}(M).
Commutator of vector fields corresponds to the commutator on Lie algebra of derivations on C^{\infty}(M).
Lie algebra corresponding to Lie group: Lie algebra of left-invariant vector fields, with respect to the commutator.
9 Outubro 2025, 10:30
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Marko Stosic
Vector fields on a Lie group.
Left-invariant vector fields on a Lie group.
Isomorphism with the tangent space at the identity element.
Lie algebra associated to a Lie group.
Abstract (purely algebraic) definition of a general Lie algebra.
Lie bracket (commutator).
First examples of Lie algebras.
Lie algebra of square matrices.
Lie subalgebras.
Example of the Lie algebra so(3).