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### PBW for Lie Superalgebras [migrated]

The PBW Theorem
In the literature there are many sources discussing the PBW-basis for Lie superalgebras, see for example M. Scheunert - The Theory of Lie Superalgebras theorem 1 and corollary in ...

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### $U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ 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 ...

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### 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 ...

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### 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 ...

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### Clifford PBW theorem for quadratic form

Update: 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 $g:L\to k$ be a quadratic form, i. e., a ...