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Results tagged with lie-algebras
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user 290
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
10
votes
Accepted
Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatornam...
This is an elaboration on what is in the other answers. First, general categorical arguments can be used to prove the following. Let $K$ be a field and $A$ a $K$-algebra. The profinite completion $\wi …
- 118k
10
votes
Counting degrees of freedom in Lie algebra structure constants (aka why are there any nontri...
It is simply not true that if you impose $k$ polynomial constraints on $n$ variables then the result has dimension $n-k$, even if the constraints "look independent," and this is itself an example. Whe …
- 118k
7
votes
How to define cohomology of algebraic structures?
There is a tremendous amount of abstract formalism answering this question in various levels of generality depending on what you want to do. I'll pick one in the middle: the machinery of derived funct …
- 118k
7
votes
What is the Molien series of the SO(2)-invariant ring on the plane (sometimes written C[X]^{...
Write $z = x + iy, \bar{z} = x - iy$ as usual, where $x, y \in \mathbb{C}[x, y]$ are regarded as complex-valued polynomial functions on the plane. The action of $SO(2)$ diagonalizes as
$$z \mapsto e^{ …
- 118k
8
votes
Accepted
Lie powers of a graded vector space and Klyachko's theorem
Let me work over $\mathbb{C}$ for simplicity. We have
$$L(V) \cong \bigoplus_{n\ge0} V^{\otimes n} \otimes_{S_n} \text{Lie}(n),$$
where $\text{Lie}(n)$ is the $n^{th}$ space of the Lie operad, with sp …
- 118k
9
votes
Accepted
Peter–Weyl decomposition for compact Lie groups with isomorphic Lie algebras
It's just not true that having isomorphic Lie algebras implies a bijection between the irreducibles (presumably you mean a bijection compatible with the isomorphism between the Lie algebras). For exam …
- 118k
12
votes
Accepted
Bilinear forms in compact/semisimple Lie group theory
(Edit: I rewrote this answer. In the first draft I tried to take some shortcuts and found that they didn't work.)
Let $G$ be a compact Lie group acting faithfully on a f.d. vector space $V$ over $\mat …
- 118k
4
votes
How can one show $G/T$ is a coadjoint orbit for a compact Lie group $G$ and $T$ its maximal ...
To be very explicit let's take a look at the case $G = U(n), T = U(1)^n$. As Allen says, by finding a suitable invariant form we can look at adjoint orbits rather than coadjoint orbits. Here $\mathfra …
- 118k
3
votes
Commutator 2-forms on Lie groups
It seems to me more natural to dispose with the arbitrary choice of $f \in \mathfrak{g}^{\ast}$ and consider the corresponding $\mathfrak{g}$-valued $2$-form. If I'm not mistaken, this form is more or …
- 118k
17
votes
Accepted
A particular Lie algebra $L_{n}$ and (various) lie groups whose Lie algebra is isomorphic to...
Your Lie algebra consists of $X$ such that $Xv = 0$ where $v$ is the all-ones vector. So the corresponding Lie group in $GL_n(\mathbb{R})$ consists of $g \in GL_n(\mathbb{R})$ such that $gv = v$.
Th …
- 118k
6
votes
Representation Theory of Lie Groups: Reference Request
Neither statement requires compactness as a hypothesis. The key result to both, and the only place where any work is needed, is the following:
If $G$ is a connected Lie group and $H$ is a Lie group, …
- 118k
6
votes
Generators of invariant polynomials of semisimple Lie algebra
Here are some topological considerations that privilege a choice of generators up to scale. First, Chern-Weil theory gives an isomorphism
$$S(\mathfrak{g}^{\ast})^{\mathfrak{g}} \cong H^{\bullet}(BG, …
- 118k
17
votes
What is significant about the half-sum of positive roots?
Well, no one's explicitly talked about the relevance of spin structures to this story yet, so here's a sketch of the story as I understand it. For references see, for example, the nLab. I'll be blithe …
- 118k
37
votes
Why do Lie algebras pop up, from a categorical point of view?
The category of Lie algebras is equivalent to a certain category of cocommutative Hopf algebras, with the equivalence given by sending a Lie algebra $\mathfrak{g}$ to its universal enveloping algebra …
- 118k
6
votes
Getting the story of Dynkin and Satake diagrams straight
2 is false. The smallest counterexample is $\mathfrak{sl}_2(\mathbb{R})$. A necessary and sufficient condition for a semisimple real Lie algebra to be the Lie algebra of a compact Lie group is that th …
- 118k