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Results tagged with lie-algebras
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user 13972
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.
81
votes
Beautiful descriptions of exceptional groups
It is not always clear what one means by 'the simplest description' of one of the exceptional Lie groups. In the examples you've given above, you quote descriptions of these groups as automorphisms o …
- 108k
40
votes
Accepted
Isometry group of a homogeneous space
Here is an algorithm to compute the Lie algebra of the group of isometries of a homogeneous space $G/H$ endowed with a $G$-invariant (pseudo-)Riemannian metric $g$. It is phrased in terms of essentia …
- 108k
37
votes
Accepted
decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$
The $E_6$ and $E_7$ decompositions you list are explained in Cartan's 1894 thesis (see pages 89–92 for these formulae). For $E_8$, Cartan instead gives a decomposition (a $\mathbb{Z}_3$-grading) of t …
- 108k
25
votes
Accepted
What is the homomorphism between the third exterior and third symmetric power of the adjoint...
There is a natural homomorphism that I am pretty sure is not zero in general that can be described most easily using the dual language of the Lie algebra $\frak g$:
Recall that there is a differen …
- 108k
23
votes
First Explicit Irreducible Representations
If all you really want to know are the irreducible representations of ${\frak{so}}(7)$ and ${\frak{so}}(8)$ up to dimension $30$, I can save you the trouble of looking up these tables:
For ${\frak{so …
- 108k
19
votes
Accepted
Geodesics on $SU(4)$
In the OP's particular case, the situation is somehwat simpler than the general case that José discusses. That's because the family of left-invariant metrics on $\mathrm{SU}(4)$ that the OP wants to …
- 108k
16
votes
Accepted
The surjectivity of the exponential map for the isometry group
No. Consider, for example, $M=\mathrm{SL}(3,\mathbb{R})/\mathrm{SO(3)}$ endowed with its $\mathrm{SL}(3,\mathbb{R})$-invariant Riemannian metric $g$ (which is unique up to a positive constant multipl …
- 108k
16
votes
Accepted
Intuition for the Cartan connection and "rolling without slipping" in Cartan geometry
You don't need to be 'handwavy' at all, and, in the standard modern way to understand this, one does not need one to choose local sections and talk about infinitesimal motions. Here's the standard ap …
- 108k
15
votes
Accepted
Maurer-Cartan structure equation derivation
What you are asking for is an introduction to the theory of $G$-structures, for which the (Maurer-Cartan) structure equations are a basic tool.
There are many sources for this material, starting, o …
- 108k
15
votes
Accepted
Why, conceptually, does the torus normalizer in $G_2$ split?
Here's a description that doesn't use octonions; instead, it uses the definition of $\mathrm{G}_2$ as the stabilizer of a $3$-form on $\mathbb{R}^7$. For simplicity, I'll do this for the split-form, …
- 108k
14
votes
Matrix representation for $F_4$
While there have been many people who have done this, the first person to do so was Élie Cartan, who wrote down a basis for the matrix algebra ${\frak{f}}_4\subset{\frak{so}}(26)$ in his 1894 thesis S …
- 108k
12
votes
Accepted
A Manifold for which $\chi^{\infty}(M)$ is rich
The answers to your questions are 'no' and 'yes'.
In the first place, there is no finite dimensional manifold whose Lie algebra of vector fields contains all of the Lie algebras ${\frak{sl}}(n,\mathb …
- 108k
10
votes
Accepted
Geodesic in space of circulant matrices
The answer is as follows: When $U$ is a positive-definite, symmetric, circulant matrix in $\mathrm{GL}(n,\mathbb{R})$, then there is a symmetric circulant matrix $u$ such that $U = e^u$ and the curve …
- 108k
10
votes
Does $G/H$ (quotient of a real semisimple Lie group by a Cartan subgroup) have a natural sym...
Actually, the issue is that there is usually more than one 'natural' symplectic structure on $G/T$, where $T$ is a Cartan subgroup. The space of $G$-invariant closed $2$-forms on $G/T$ has dimension …
- 108k
10
votes
Accepted
SU(6) -> SU(3) branching rule
You appear to have made a mistake in your calculation of the branching rules. The answer given in the wiki is correct, but it seems that you are using the 'wrong' subgroup of $\mathrm{SU}(6)$. Perha …
- 108k