Here is another proposal. The key will be to enlarge the category of quadratic vector spaces fairly substantially. Here are three hints leading towards the proposal:
- Clifford algebras can and should be thought of as $\mathbb{Z}_2$-graded; for example, Clifford algebras over $\mathbb{R}$ give every element of the $\mathbb{Z}_2$-graded Brauer group / Brauer-Wall group of $\mathbb{R}$.
- Thinking of Clifford algebras as deformations of exterior algebras, the analogous deformations of symmetric algebras are the Weyl algebras. It is possible to combine the construction of exterior algebras and symmetric algebras into a single construction, namely the construction of the symmetric algebra on a super vector space.
- Weyl algebras are almost universal enveloping algebras of certain Lie algebras. More precisely, let $(V, \omega)$ be a symplectic vector space. From this data we can construct a Lie algebra $V \oplus \mathbb{R}$ such that $\mathbb{R}$ is central and $[v, w] = \omega(v, w) \in \mathbb{R}$ where $v, w \in V$. Then the Weyl algebra constructed from $(V, \omega)$ is the quotient of the universal enveloping algebra $U(V \oplus \mathbb{R})$ by the extra relation that $1 \in \mathbb{R}$ acts as the identity.
So let's try this:
There is a forgetful functor from super algebras to a certain category of super Lie algebras with extra structure whose left adjoint restricts to 1) the symmetric algebra functor, 2) the exterior algebra functor, 3) the Weyl algebra functor, 4) the Clifford algebra functor, and 5) the universal enveloping algebra functor on suitable subcategories.
This left adjoint generalizes every functor discussed above. Some details:
The first category $\text{SAlg}$, the category of super algebras, is the category of monoid objects in super vector spaces. As a category it can be thought of as the category of $\mathbb{Z}_2$-graded algebras, but as a symmetric monoidal category it has a nontrivial braiding given by the Koszul sign rule as usual. In particular, a commutative super algebra is commutative in the super sense, not the usual sense.
The second category begins from the category $\text{SLieAlg}$ of super Lie algebras, which is the category of Lie algebra objects in super vector spaces; here it's quite important to distinguish this category from the category of $\mathbb{Z}_2$-graded Lie algebras because the Koszul sign rule changes some signs in the Lie algebra axioms. In particular, the skew-symmetry axiom becomes
$$[x, y] = - (-1)^{|x| |y|} [y, x]$$
for homogeneous elements $x, y$. Hence if either $x$ or $y$ is even then this is skew-symmetry in the usual sense, but if $x$ and $y$ are both odd then we actually have symmetry. This is crucial.
The category we're actually interested is not quite this category; instead it is the category of super Lie algebras $\mathfrak{g}$ equipped with a morphism $\mathbb{R}[0] \to \mathfrak{g}$, where $\mathbb{R}[0]$ denotes the abelian Lie algebra $\mathbb{R}$ in degree $0$, with central image. This might be called the category of "centrally pointed super Lie algebras," maybe.
The forgetful functor from $\text{SAlg}$ to the above category sends a super algebra $A$ to the underlying super vector space of $A$ equipped with the super Lie bracket
$$[x, y] = xy - (-1)^{|x| |y|} yx$$
on homogeneous elements, with the map $\mathbb{R}[0] \to A$ being given by the unit element of $A$.
The left adjoint to this forgetful functor sends $\mathbb{R}[0] \to \mathfrak{g}$ to the quotient of the universal enveloping (super) algebra $U(\mathfrak{g})$ by the extra relation that $1 \in \mathbb{R}[0]$ acts as the identity. I'll also call this functor $U$. Here are the five promised subcategories (they are not full subcategories):
- Given a vector space $V$, send it to the abelian super Lie algebra $V[0] \oplus \mathbb{R}[0]$. Then $U(V[0] \oplus \mathbb{R}[0])$ is the symmetric algebra on $V$.
- Given a vector space $V$, send it to $V[1] \oplus \mathbb{R}[0]$. Then $U(V[1] \oplus \mathbb{R}[0])$ is the exterior algebra on $V$.
- Given a symplectic vector space $(V, \omega)$, send it to $V[0] \oplus \mathbb{R}[0]$ with bracket given by $\omega$ as previously discussed. Then $U(V[0] \oplus \mathbb{R}[0])$ is the Weyl algebra of $(V, \omega)$.
- Given a quadratic vector space $(V, Q)$, let $B(v, w) = Q(v + w) - Q(v) - Q(w)$ be twice the symmetric bilinear form determined by $Q$ and send it to $V[1] \oplus \mathbb{R}[0]$ with bracket given by $B$, as previously discussed for the Weyl algebra. Then $U(V[1] \oplus \mathbb{R}[0])$ is the Clifford algebra of $(V, Q)$.
- Given a Lie algebra $\mathfrak{g}$, send it to $\mathfrak{g}[0] \oplus \mathbb{R}[0]$. Then $U(\mathfrak{g}[0] \oplus \mathbb{R}[0])$ is the usual universal enveloping algebra of $\mathfrak{g}$.