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Results tagged with lie-algebras
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user 98863
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
Symmetry of Casimirs of Lie algebras
Please expand your definition of "Casimir". Is a $g$-invariant element in $g^{\otimes n}$?
If $g$ is a simple algebra and $\kappa$ its Killing form, then the map
$$g\times g\times g\to k$$
$$(x,y,z) …
- 1,028
-1
votes
First adjoint cohomology space of simple Lie algebras
I don't known if I interprete correctly your question. First, if $g$ is simple, then always $H^1(g,g)=0$. ALSO, $H^1(L,L)=0$ do not implies that $L$ is a central extensión of a simple algebra.For exam …
- 1,028
1
vote
infinite fold tensor product of universal enveloping algebra
Assuming $\infty$= the cardinality of some set $I$, then consider the Lie algebra $\mathfrak a^{(I)}$= the direct sum of $I$-copies of $\mathfrak a$, with bracket coordinatewise. Then take its univers …
- 1,028
0
votes
What is the cubic Casimir element of $\mathfrak{sl}_3$?
I think $\sum_{i,j,k}X_{ij}X_{jk}X_{ki}$ should work for $\mathfrak{gl}(3)$. Now if you want for $\mathfrak{sl}(3)$, maybe you can change $X_{ii}$ by $X_{ii}-(1/3)\mathrm{Id}$.
The formula above is li …
- 1,028
2
votes
Lie algebra of a compact Lie group and derivations of the Hopf algebra of representative fun...
Some years ago we did something that -I think- answers the algebraic version of your question [FS].
If you have a Hopf algebra $H$, then the counit gives you a map $\varepsilon_* :\mathrm{Hom}(H,H)\t …
- 1,028
3
votes
Classifications of Lie bialgebras
Sorry for the self publicity, but for the especific example of $gl_n(k)$, you can view it as $gl_n(k)\cong sl_n(k)\times k$ and we did some work for trivial central extensions that allow you to produc …
- 1,028