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Results tagged with lie-algebras
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user 32332
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.
2
votes
(Co-) Homology of free nilpotent Lie Algebras
There are several results on the homology of free-nilpotent Lie algebras with trivial coefficients, in particular for free two-step nilpotent Lie algebras.
Stefan Sigg has determined the homology in t …
- 12.1k
0
votes
Understanding lie bracket of simple Lie algebra $W(2)$
This Lie algebra also arises in the classification of $3$-dimensional simple Lie algebras as follows:
Theorem: let $L$ be a simple, $3$-dimensional Lie algebra over an algebraically closed field of c …
- 12.1k
4
votes
Lie algebras with trivial second cohomology group
Even if we consider more generally cohomologically rigid Lie algebras $L$, which satisfy $H^i(L,L)$ for all $i\ge 0$, a classification is not possible.
Roger Carles proved in 1985 in his paper "Sur ce …
- 12.1k
1
vote
Accepted
Jacobi identity for circular permutations
The condition $c_N=0$ does impose a restriction on a Lie algebra $L$ in general (with the exception of cases where, say, $L$ is nilpotent of class $c$, so that all terms of $c_N$ are trivially zero fo …
- 12.1k
3
votes
contraction identity and killing form
The usual generalisation of the contraction identity for the cross product is the Binet–Cauchy identity involving the dot and the wedge product (a cross-product exists only in dimension $3$ and $7$, b …
- 12.1k
4
votes
On radicals of a lie algebra
The two definitions do never agree for a non-trivial nilpotent Lie algebra, as Ben has remarked. Indeed, if $\mathfrak{g}$ is nilpotent then $\mathfrak{s}=[\mathfrak{g},\mathfrak{r}]=[\mathfrak{g},\ma …
- 12.1k
4
votes
System of weights for nilpotent Lie algebras
1.) No, the weight systems are not classified in general. However, the weight systems for complex nilpotent Lie algebras of dimension $n\le 7$ have been computed by Roger Carles, in "Weight systems f …
- 12.1k
10
votes
Which nilpotent Lie algebras appear as nilradicals of parabolic subalgabras?
The subclass of nilpotent Lie algebras formed by
arbitrary ideals of parabolic subalgebras consisting of nilpotent elements in reductive Lie algebras has been classifed in the article
Yu.B. Khakimdzh …
- 12.1k
1
vote
Accepted
why no Lie algebra degenerate to a rigid algebra? Why the closure of a rigid algebra forms t...
For the second question suppose that $L$ is a geometrically rigid Lie algebra,
i.e., the orbit $O(L)$ is open. The algebra $L$, and hence $O(L)$ is contained
in some irreducible component $\mathcal{ …
- 12.1k
3
votes
What is known on finite dimensional nilpotent Lie algebras with maximal index ?
It is known, that the index of a Lie algebra is a semi-invariant for degenerations
(by Ooms and Elashvili), i.e., if $L_1$ degenerates to $L_2$, then
$ind(L_1)\le ind(L_2)$. This is very useful.
For …
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1
vote
Generic Stabilizers in a Nilpotent Lie Algebra
In general, there need not exist a surjective morphism $L\rightarrow C_L(\xi)=L_{\xi}$.
Take a Heisenberg Lie algebra $L$ of dimension $2m+1\ge 5$, with basis $x_1,\ldots x_m,
y_1,\ldots ,y_m,z$ and b …
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9
votes
Rigid nilpotent Lie algebras
Vergne's conjecture is still open. It says that there is no complex $n$-dimensional nilpotent Lie algebra which is rigid in the variety $\mathcal{L}_n(\mathbb{C})$ of all $n$-dimensional complex Lie a …
- 12.1k
2
votes
Stabilizers and Quotients of a Nilpotent Lie Algebra
A well-known decomposition theorem for nilpotent Lie algebras is the weight space
decomposition, studied by R. Carles, L. J. Santharoubane and others (in the nilpotent case).
Here $L$ is a finite-dime …
- 12.1k
1
vote
Witt Lie algebras
Since it not exactly clear, which Lie algebra is meant, let me note that the Lie algebra $W(1;\underline{2})$ is $4$-dimensional over $\Bbb F_2$, see the article by Helmut Strade Lie algebras of small …
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2
votes
Can we write $sl(4,\mathbb{C})$ as the vector space sum of two copies of $sl(3,\mathbb{C})$?
A. L. Onishchik has determined such semisimple decompositions (not necessarily direct sums) in his article Decompositions of reductive Lie groups. For example, it is true that for simple Lie algebras
…
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