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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.
3
votes
Exceptional Lie algebras
I am certainly no expert, but one answer to your second question is that one can think of the classical Lie groups / algebras as certain constructions on $\mathbb{R}, \mathbb{C}$, and $\mathbb{H}$ whi …
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9
votes
2
answers
617
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Does any identity holding in all finite-dimensional Lie algebras hold in all Lie algebras?
Equivalently, is the free Lie algebra on finitely many generators over a fixed field $k$ (say of characteristic not equal to $2$) residually finite-dimensional in the sense that any nonzero element re …
- 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
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
21
votes
Accepted
Basis-free definition of Casimir element?
The Casimir element is dual to the Killing form. (I think. I am somewhat uncertain about this because nobody has ever said this to me, even though it seems like the right thing to say, and frankly I …
- 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
7
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2
answers
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Reference request: equivalence of formal group laws and Lie algebras in characteristic zero
Let $k$ be a field of characteristic zero. Wikipedia states that the natural functor from finite-dimensional formal group laws over $k$ to finite-dimensional Lie algebras over $k$ is an equivalence of …
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6
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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
4
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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 …
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15
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3
answers
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What's the origin of the naming convention for the standard basis of $\mathfrak{sl}_2(\mathb...
$\mathfrak{sl}_2(\mathbb{C})$ is usually given a basis $H, X, Y$ satisfying $[H, X] = 2X, [H, Y] = -2Y, [X, Y] = H$. What is the origin of the use of the letter $H$? (It certainly doesn't stand for …
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26
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3
answers
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How are these two ways of thinking about the cross product related?
I was always bothered by the definition of the cross product given in e.g. a calculus course because it's never made clear how one would go about defining the cross product in a coordinate-free manner …
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7
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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 …
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6
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What is the categorical significance of the trivial $\mathfrak{g}$-module in the category of...
The $0$-dimensional Lie algebra $0$ is the terminal object in the category of Lie algebras; that is, every Lie algebra admits a unique morphism $\mathfrak{g} \to 0$. This morphism gives rise to a rest …
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5
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Decomposition into irreducibles of symmetric powers of irreps.
For what it's worth, here's a small general observation for representations of compact groups $G$. As Humphreys mentioned the idea is to consider the entire symmetric algebra $S(V)$ of a representatio …
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