- 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.
The category we're actually interested is not quite this category; instead it is the category of central extensions
$$0 \to \mathbb{R}[0] \to \mathfrak{g} \to \mathfrak{h} \to 0$$
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 central extension
$$0 \to \mathbb{R}[0] \to A \to A' \to 0$$
where the underlying super vector space of $A$ is equipped with the super Lie bracket
on homogeneous elements and where, with the inclusionmap $\mathbb{R}[0] \to A$ isbeing given by the unit element of $A$.
The left adjoint to this forgetful functor sends a central extension $\mathbb{R}[0] \to \mathfrak{g} \to \mathfrak{h}$$\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 trivial central extensionabelian 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 the trivial central extension $V[1] \oplus \mathbb{R}[0]$. 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 the trivial central extension $\mathfrak{g}[0] \oplus \mathbb{R}[0]$. Then $U(\mathfrak{g}[0] \oplus \mathbb{R}[0])$ is the usual universal enveloping algebra of $\mathfrak{g}$.