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Results tagged with lie-algebras
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user 1306
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.
10
votes
Accepted
Reference for an old result of P. M. Cohn
This is in P. M. Cohn, "On the Embedding of Rings in Skew Fields", Proceedings of the London Mathematical Society, Volume s3-11, Issue 1 (1961), Pages 511-530. I do not think that the zero characteris …
- 16.9k
4
votes
Accepted
Second cohomology group of the contact Lie algebra $K_3$
Yes, it is true. In fact, it is true that $H^i(K_{2n+1},F)=0$ for $0<i\le 2n$. This can be deduced from the theorem of Feigin sketched in
Feigin, B.L. Cohomology of contact Lie algebras.
(Russian) C. …
- 16.9k
3
votes
The dimension of the Grassmannian cohomology ring $H^*(\mathrm{Gr}_{n,d})$ and the fundament...
I accidentally (looking for something else) came across another paper where a very elegant explanation is given:
Dan Laksov, Anders Thorup: Schubert Calculus on Grassmannians and Exterior Powers
India …
- 16.9k
10
votes
Accepted
The dimension of the Grassmannian cohomology ring $H^*(\mathrm{Gr}_{n,d})$ and the fundament...
This is very standard. For a compact complex variety admitting a cell decomposition, the (co)homology is the free Abelian group generated by the cells (over $\mathbb{C}$ there is no room for the diffe …
- 16.9k
12
votes
Accepted
Counting degrees of freedom in Lie algebra structure constants (aka why are there any nontri...
Linear independence does not really say much.
This algebraic variety is discussed in some detail in an old paper of Kirillov and Neretin: The variety $A_n$ of $n$-dimensional Lie algebra structures.
T …
- 16.9k
10
votes
Accepted
Breaking up the free Lie algebra into GL irreps
The Whitehouse module referred to in one of the other answers is not necessary, since it is related to the cyclic operad Lie, that is to the representation of $S_{n+1}$ in $Lie(n)$.
The decomposition …
- 4,713
1
vote
Is there a name for a noncommutative generalization of Poisson algebra?
This seems to first have been considered by Dirac under the name "quantum Poisson bracket" - an easy accessible reference is Fock's "Fundamentals of Quantum Mechanics", discussion around formula (2.10 …
- 16.9k
6
votes
Accepted
CE(g) for g infinite dimensional
A definition that always works and does agree with that one in the finite-dimensional case is the following: put
$$
C^k(\mathfrak{g})=({\Lambda}^k\mathfrak{g})^*=\operatorname{Hom}({\Lambda}^k\mathfra …
- 35.4k
2
votes
Twisted affine Lie algebras, Lie bracket and normalized standard invariant form
I think that there is just a little mess between things that are denoted $K$, $K'$ in the book, as well as $d$, $d'$. For that, let us examine these formulas carefully.
Using the first formula for the …
- 16.9k
1
vote
A positive formula for the dimensions of homogeneous components of free Lie algebras
Now, to the matter of "positive formulas".
Classical result of Kraskiewicz and Weyman (preprint W. Kraskiewicz, J. Weyman. Algebra of Invariants and the Action of a Coxeter Element. Math. Inst. Copern …
- 16.9k
2
votes
A positive formula for the dimensions of homogeneous components of free Lie algebras
For your question about Lie triple systems:
In my article "Veronese powers of operads and pure homotopy algebras" joint with Martin Markl and Elisabeth Remm, Eur. J. Math. 6 (2020), 829-863 (https://l …
- 16.9k
1
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Applications of the PBW theorem on enveloping algebras
One immediate corollary is that you get to know the "sizes" of induced representations. If $\mathfrak{h}\subset\mathfrak{g}$ is a Lie subalgebra, and $M$ is an $\mathfrak{h}$-module, then the underlyi …
- 16.9k
10
votes
Accepted
Poincaré duality for (co)homology of Lie algebras?
First, let me expand on the reply of Dietrich Burde: I got hold of the paper of Hazewinkel, and can now be more precise about what is and what is not there (last time I saw it was some years ago).
…
- 63.9k
4
votes
On the isomorphism problem of enveloping algebras
FWIW, a ten year old article states: "We stress that, in spite of all this, the
characteristic zero case of the isomorphism problem remains entirely open."
(https://link.springer.com/article/10.1007/ …
- 16.9k
4
votes
Cohomology of Infinite Dimensional Lie Algebra
A very useful source to learn about cohomology of infinite-dimensional Lie algebras is the book by D.B.Fuks "Cohomology of Infinite-Dimensional Lie Algebras" (shocking, I know). This source discusses …
- 16.9k