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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.
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
26
votes
3
answers
4k
views
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 …
- 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
18
votes
2
answers
4k
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What is the physical meaning of a Lie algebra symmetry?
The physical meaning of a Lie group symmetry is clear: for example, if you have a quantum system whose states have values in some Hilbert space $H$, then a Lie group symmetry of the system means that …
- 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
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
15
votes
3
answers
1k
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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 …
- 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 …
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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
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
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
9
votes
2
answers
617
views
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
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
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^{ …
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