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2 votes

Non-example to PBW theorem

Let's answer the updated question. Let $A$ be an associative algebra generated by elements $x_1,\ldots,x_n$ with relations of the form $$ [x_i,x_j] = \sum_{ijk} c_{ijk}x_k.$$ Let $\mathfrak a$ denote ...
Grant B.'s user avatar
  • 526
6 votes

Solvability of derivation Lie algebras of local finite-dimensional commutative algebras

The answer is NO. Consider for instance the commutative algebra $A$ having the elements $a_0,a_1,\ldots,a_n$ as a $k$-basis such that $a_0$ is the unity of $A$ and $a_ia_j=0$ for all $i,j=1,\ldots,n$. ...
Salvatore Siciliano's user avatar
3 votes

Relation between enveloping algebras and algebras of differential operators

As you guessed, the kernel will be infinite dimensional (at least for $\dim X \geq 1$). To see this, you can take the order filtration on $\mathcal D(X)$ and the PBW filtration on $U(\mathcal V_X)$. ...
Phil Tosteson's user avatar
2 votes

Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?

This is not a complete answer, but it strongly suggests where to look. Since $\binom{m+2}{3}_q$ is the $q$-character of $\mathrm{Sym}^3\mathrm{Sym}^{m-1}(E)$, working over the complex numbers we have $...
Mark Wildon's user avatar
  • 10.7k
4 votes

What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?

Any direct sum of a $n$-step nilpotent Lie algebra and a nonzero abelian Lie algebra satisfies the required property. Acknowledgement: After posting this answer, I realized that this class of examples ...
Salvatore Siciliano's user avatar
1 vote

Wedderburn–Artin like theorem for infinite dimensional Lie algebras?

I had the opportunity to chat with an expert in nonassociative algebras, and I report here what he told me. The short answer is: as yet there is none Wedderburn-Artin theory for Lie algebras. The ...
jg1896's user avatar
  • 2,515
3 votes

Finite dimensional irreducible representations of $\frak{sl}_m$ with non-trivial zero weight spaces

At the request of the question-asker I am converting my comments to an answer. In general, for any semisimple Lie algebra $\mathfrak{g}$, is $\mu$ is a (dominant, integral) weight of $\mathfrak{g}$ ...
Sam Hopkins's user avatar
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