## Koszul duality

Given a finite-dimensional $k$-vector space $V$ (I am happy taking $k = \mathbb{C}$ anywhere in the following if it makes a difference) and a subspace $R \subseteq V \otimes V$, we can form the *quadratic algebra*
$$A = A(V,R) = T(V)/ \langle R \rangle,$$
where $\langle R \rangle$ is the 2-sided ideal in the tensor algebra generated by $R$.

We can then form the quadratic algebra $A^! = A(V^*, R^\perp)$, where
$$ R^\perp = \{ \phi \in V^* \otimes V^* \mid \phi(R) = 0 \}, $$
and we have identified $V^* \otimes V^*$ with $(V \otimes V)^*$. This algebra $A^!$ is also quadratic by construction, and is known as the *Koszul dual* of $A$. It's pretty clear that $(A^!)^! \simeq A$.

One example of this is given by the symmetric and exterior algebras of a vector space and its dual, i.e. for a finite-dimensional vector space $V$, we have $$ S(V)^! \simeq \Lambda(V^*), \quad \Lambda(V)^! \simeq S(V^*). $$

## Clifford and Weyl algebras

Now suppose that $V$ is even-dimensional, say $\mathrm{dim}_\mathbb{C}(V) = 2n$, and let $h: V \otimes V \to k$ be a nondegenerate symmetric bilinear form on $V$. The *Clifford algebra* is the algebra
$$ \mathrm{Cl}(V,h) = T(V)/\langle x - h (x) \mid x \in S^2(V) \rangle, $$
and this can be viewed as a deformation of the exterior algebra in the sense that the Clifford algebra is naturally filtered and the associated graded is $\Lambda(V)$. If $h$ is nondegenerate, then (over $\mathbb{C}$, at least) we can show that $\mathrm{Cl}(V,h) \simeq M_{2^n}(\mathbb{C})$.

If we take instead a nondegenerate alternating (i.e. symplectic) form $g:V \otimes V \to k$, then we can form the *Weyl algebra*
$$ A_n = A_n(V,g) = T(V)/\langle x - g(x) \mid x \in \Lambda^2(V) \rangle. $$
This too has a natural filtration from the tensor algebra, and the associated graded is $S(V)$.

These two deformations share some features in common. For instance, the Weyl algebra is isomorphic to the algebra of polynomial differential operators on $\mathbb{C}[x_1, \dots, x_n]$, and one can think of the Clifford algebra as being a $\mathbb{Z}/2$-graded analogue of that via creation and annihilation operators on $\Lambda(V)$. Both algebras are simple.

## Main question

Is there any sort of non-quadratic Koszul duality that relates the Clifford and Weyl algebras?