Questions tagged [pbw-theorems]

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15 votes
2 answers
2k views

Clifford PBW theorem for quadratic form

$\DeclareMathOperator\Cl{Cl}$Update Feb 3 '12: now with a question 2 which is much more elementary (and should be well-known!). Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-module, and $g:...
2 votes
1 answer
235 views

Non-example to PBW theorem

I am interested in a (simple) example of an associative algebra $A$ with 1 generated by $x_1, \ldots, x_n$ which is quadratic (i.e. all relations between $x_1, \ldots, x_n$ have degree $\leqslant 2$) ...
0 votes
1 answer
198 views

Adjoint action on the universal enveloping algebra and the PBW theorem

Let $\frak{g}$ be a semisimple Lie algebra and $U(\frak{g})$ its universal enveloping algebra. The adjoint action of $\frak{g}$ on itself extends to an action of $\frak{g}$ on $U(\frak{g})$. How does ...
9 votes
1 answer
426 views

$U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$

While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is ...
54 votes
9 answers
9k views

Nice proofs of the Poincaré–Birkhoff–Witt theorem

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $k$, with an ordered basis $x_1 < x_2 < ... < x_n$. We define the universal enveloping algebra $U(\mathfrak{g})$ of $\...
3 votes
0 answers
88 views

Reference of general version of the PBW theorem and its consequences

Let $A$ be a commutative ring with identity and $L$ be a Lie algebra which is also a free module over $A$. I have seen the following statements: The universal enveloping algebra $U(L)$ is isomorphic (...
8 votes
0 answers
154 views

Alternate proof of easiest special case of PBW theorem

Let $L$ be a Lie algebra over a field $k$ of characteristic $0$ (I'm happy for that field to be $\mathbb{C}$) and let $U(L)$ be its universal enveloping algebra. One of the standard consequences of ...
4 votes
2 answers
458 views

Applications of the PBW theorem on enveloping algebras

What are some nice corollaries or applications of the Poincaré Birkhoff Witt theorem? There's this immediate corollary that a Lie algebra sits inside the universal enveloping algebra so in particular, ...
1 vote
0 answers
55 views

Questions of the paper "PBW-pairs of varieties of linear algebras"

I am reading this paper "PBW-pairs of varieties of linear algebras", the link is here:https://www.tandfonline.com/doi/abs/10.1080/00927872.2012.720867. At page 672, there is a definition of PBW-pair. ...
17 votes
1 answer
516 views

Splitting the injection that you get from the Poincaré-Birkhoff-Witt theorem

Let $\mathfrak g$ be a Lie algebra over a field of characteristic zero, with universal enveloping algebra $U\mathfrak g$. By the Poincaré-Birkhoff-Witt theorem one knows that $i:\mathfrak g \to U\...
4 votes
0 answers
618 views

The existence of zero-divisors in the universal enveloping algebra of an infinite-dimensional Lie algebra

The intuition for this problem comes from $\S$17 Exercise 1 Humphreys' Introduction to Lie Algebras and Representation Theory which essentially asks us to use PBW in order to prove that if a Lie ...
2 votes
1 answer
495 views

What's an example of a commutative algebra over $\mathbb Q$ that fails to satisfy this version of the "PBW theorem"

In a recent question, I recalled the notion of differential operator, polyderivation, and principal symbol for a commutative algebra $A$ over some fixed commutative ring $k$. (I will not repeat those ...
9 votes
1 answer
672 views

For which algebras does \{Differential Operators\} satisfy a PBW-like theorem?

Let $k$ be a commutative ring, $A$ a commutative $k$-algebra, and for some other part of why I'm asking this question I only care about the case when $k \supseteq \mathbb Q$. Recall the following ...