Lie groups and their Lie algebras - Lec 13 - Frederic Schuller
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as a clear source on Lie groups, left-invariant vector fields and pushforwards, both when recommending it and when asking about specific steps in the lecture.
Recommended by 2 people across 3 different months, 2020 to 2024.
What it is
Lecture 13 of Frederic Schuller's 'Lectures on the Geometric Anatomy of Theoretical Physics' course, covering Lie groups and their Lie algebras. It builds on earlier lectures about manifolds, vector fields and pushforwards, and introduces left-invariant vector fields and the exponential map. Presented as a university-style blackboard lecture running about 1h43m.
What people said
Recommends md2perpe · 2020-06 · Mathematics Stack Exchange · on “What is the connection of Gell-Mann matrices with SU(3)?”
If you want to learn some about Lie groups and Lie algebras (and other mathematics used in physics) I can recommend the lessons by Frederic Schuller on Youtube: [this video]
Recommends Antoni Parellada · 2022-10 · Mathematics Stack Exchange · on “What is the definition of a homogenous vector field?”
The question came about listening to online lectures ( here [this video] ) on the exponential map that allows projecting back to a manifold a pushed forward point via a linear operation on the tangent space. OK, something like that - way over my head! At some point the idea of left-invariant vector fields is brought up as essential to the construct. There would be some group operation on the Lie group that would somehow be associated to a pushforward of the vector field. Again, probably more wrong than right, but to give some idea to the expert reader of the material I'm referring to. "We are not interested in any vector field on a Lie group. We are only interested in a special class of vector fi
Recommends Antoni Parellada · 2024-06 · Mathematics Stack Exchange · on “Interpreting one of the several expressions of the pushfoward of a vector field on a Lie group (Prof Schuller lecture)”
Given a vector field $X$ and a left-translation function $l_g$ by an element of the Lie group $g \in G$ defined as $l_g(h)=gh,$ and an arbitrary function $f$ $$X_h(f \circ l_g) = X(f) (g h)$$ seems to indicate that the $X_h$ (which is a single vector at the point $h$ ) applied to the composition $f \circ l_g$ is the same as the entire vector field $X$ acting on the function $f$ and evaluating at the point $gh$ . Why is the LHS of the equation correct? I am missing some expression of the pushforward - an $*$ somewhere: $X_h$ is the vector at the point $h$ of the vector field $X$ . $l_g$ just multiples $g$ times $h$ as per the group operation, resulting in the point $gh$ , i.e. $f \circ l_g$ .
Cites Mirod the best · 2021-08 · Mathematics Stack Exchange · on “Pull-back between covector fields”
[this video] I was following this lecture and at this point in the lecture the professor starts discussing why it is easier to construct pull-backs between covector fields rather than to construct push-forwards between vector fields. He explains that a smooth function $h:M\rightarrow N$ (with $M$ and $N$ smooth manifolds) is enough to construct a pull-back from $\Gamma(T^*N)$ to $\Gamma(T^*M)$ (by $\Gamma(T^*M)$ I mean the set of covector fields on $M$ ) since, because the domain of $h$ is the entirety of $M$ , the image of the pull-back will be the entirety of $\Gamma(T^*M)$ . But then he goes on to explain that one of the reasons for which defining a push-forward is not as simple is because the