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super algebra

Idea

In the general sense, superalgebra is the study of (higher) algebra over the base topos on superpoints.

More specifically, an associative superalgebra is an associative algebra in the context of superalgebra. As in the ordinary case, this is often just called a superalgebra , too.

In the following we first discuss

• Associatvive superalgebra

as monoids in the braided monoidal category of super vector spaces. Then we pass to the more abstract picture of

• Algebra in the topos over superpoints

and consider systematically algebra in the sheaf topos over the site of superpoints and show how this reproduces and generalizes the previous notions.

See (Sachse) and the references at super ∞-groupoid for some history of the topos-theoretic perspective on superalgebra.

Associative superalgebras

Definition

An ordinary associative algebra (a vector space with a linear and associative and unital product operation) is a monoid in the monoidal category Vect of vector spaces.

Throughout, fix a field $k$ of characteristic 0.

Definition

Write SVect for the symmetric monoidal category of super vector spaces over $k$. This is the category of ${\mathbb{Z}}_2$-graded vector spaces equipped with the unique non-trivial symmetric braided monoidal structure.

Objects are vector spaces with a direct sum decomposition

$$V=V_{\mathrm{even}}\oplus V_{\mathrm{odd}}$$V = V_{even} \oplus V_{odd}

and the tensor product is given in terms of that on vector spaces by

$$V\otimes W=(V_{\mathrm{even}}\otimes W_{\mathrm{even}}\oplus V_{\mathrm{odd}}\otimes W_{\mathrm{odd}})\oplus (V_{\mathrm{even}}\otimes W_{\mathrm{odd}}\oplus V_{\mathrm{odd}}\otimes W_{\mathrm{even}})$$V \otimes W = (V_{even} \otimes W_{even} \oplus V_{odd}\otimes W_{odd}) \oplus (V_{even} \otimes W_{odd} \oplus V_{odd} \otimes W_{even})

but equipped with the non-trivial braiding morphism

$$b_{V,W}:V\otimes W\to W\otimes V$$b_{V, W} : V \otimes W \to W \otimes V

that is the usual braiding isomorphism of Vect on $V_{\mathrm{even}}\otimes W_{\mathrm{even}}$ and on $V_{\mathrm{even}}\otimes W_{\mathrm{odd}}\oplus V_{\mathrm{odd}}\otimes W_{\mathrm{even}}$ but is $(-1)$ times this on $V_{\mathrm{odd}}\otimes W_{\mathrm{odd}}$.

Definition

A super algebra over $K$ is a monoid in the symmetric monoidal category SVect of super vector spaces.

A (graded)-commutative algebra over $K$ is a monoid in the symmetric monoidal category SVect of super vector spaces.

This means that a commutative superalgebra is a super vector space

$$A=A_{\mathrm{even}}\oplus A_{\mathrm{odd}}$$A = A_{even} \oplus A_{odd}

equipped with a morphism of super vector spaces

$$(-)\cdot (-):A\otimes A\to A$$(-)\cdot (-) : A \otimes A \to A

that is associative and commutative in the usual sense. Spcifically for the commutativity this means that with $a,b\in A_{\mathrm{odd}}$ we have

$$a\cdot b=-b\cdot a.$$a \cdot b = - b \cdot a \,.

Whereas if either of $a$ or $b$ is in $A_{\mathrm{even}}$ we have

$$a\cdot b=b\cdot a.$$a \cdot b = b \cdot a \,.

Examples

Grassmann algebra

• For $V$ a vector space or graded vector space the Grassmann algebra ${\wedge }^{•}V$ is a super algebra. This are the free superalgebras.

Clifford algebra

An class of examples of non-(graded)-commutative superalgebra are Clifford algebra.

In fact, let $V$ be a vector space equipped with symmetric inner product $⟨-,-range$. Write ${\wedge }^{•}V$ be the Grassmann algebra on $V$. The inner product makes this a super Poisson algebra. The Clifford algebra $\mathrm{Cl}(V,⟨rangl)$ is the deformation quantization of this.

Algebra in the topos over superpoints

We now consider higher algebra in the (∞,1)-topos over super points.

The topos

Definition

Write $\mathrm{SuperPoint}$ for the site of superpoints. Write

$$\mathrm{SuperSet}:=\mathrm{Sh}(\mathrm{SuperPoint})$$SuperSet := Sh(SuperPoint)

for the sheaf topos (a presheaf topos) over this site. Write

$$\mathrm{Super}\mathrm{\infty }\mathrm{Grpd}:={\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{SuperPoint})$$Super \infty Grpd := Sh_{(\infty,1)}(SuperPoint)

for the (∞,1)-sheaf (∞,1)-topos over this site: the (∞,1)-topos of super ∞-groupoids.

The line object $𝕂$

The canonical line object in $\mathrm{Sh}(\mathrm{SuperPoint})$ is

$${𝔸}^1=\mathrm{Spec}F(*)={ℝ}^{0\mid 1},$$\mathbb{A}^1 = Spec F(*) = \mathbb{R}^{0|1} \,,

the odd line.

As a presheaf ${ℝ}^{0\mid 1}:({\mathrm{Gr}}^{\mathrm{op}}{)}^{\mathrm{op}}\to \mathrm{Set}$ this assigns

$${𝔸}^1:\mathrm{Spec}{\wedge }^{•}V\mapsto ({\wedge }^{•}V{)}_{\mathrm{odd}}.$$\mathbb{A}^1 : Spec \wedge^\bullet V \mapsto (\wedge^\bullet V)_{odd} \,.

Definition

Consider analogously the presheaf $𝕂={𝕂}^1$ defined by

$${𝕂}^1:\mathrm{Spec}{\wedge }^{•}V\mapsto ({\wedge }^{•}V{)}_{\mathrm{even}}.$$\mathbb{K}^1 : Spec \wedge^\bullet V \mapsto (\wedge^\bullet V)_{even} \,.

This is canonically equipped with the structure of a (unital) commutative ring in $\mathrm{SuperSet}$.

In (Sachse) this appears around (21).

$𝕂$-Modules

Definition

Write ${\mathrm{Mod}}_{𝕂}(\mathrm{SuperSet})$ for the category of modules over $𝕂$ in $\mathrm{SuperSet}$.

Proposition

There is a functor

$$j:{\mathrm{SVect}}_k={\mathrm{Mod}}_k(\mathrm{Set})\hookrightarrow {\mathrm{Mod}}_{𝕂}(\mathrm{SuperSet})$$j : SVect_k = Mod_k(Set) \hookrightarrow Mod_{\mathbb{K}}(SuperSet)

from super vector spaces over $k$ to internal modules over $𝕂$ that sends $V\in {\mathrm{SVect}}_k$ to

$$j(V):\mathrm{Spec}\Lambda \mapsto (\Lambda {\otimes }_{k}V{)}_{\mathrm{even}}=({\Lambda }_{\mathrm{even}}{\otimes }_k{V}_{\mathrm{even}})\oplus ({\Lambda }_{\mathrm{odd}}{\otimes }_k{V}_{\mathrm{odd}}).$$j(V) : Spec \Lambda \mapsto (\Lambda \otimes_k V)_{even} = (\Lambda_{even} \otimes_k V_{even}) \oplus (\Lambda_{odd} \otimes_k V_{odd}) \,.

This is a full and faithful functor.

This appears as (Sachse, corollary 3.2).

Proof

The proof is a variation on the Yoneda lemma.

This means that ordinary super vector spaces are embedded as a full subcategory in $𝕂$-modules in the topos over super point.

Associative and Lie Superalgebras

Proposition

The functor $j$ from prop 1 induces a full and faithful functor

$${\mathrm{SAlg}}_k(\mathrm{Set})\hookrightarrow {\mathrm{Alg}}_{𝕂}(\mathrm{SuperSet})$$SAlg_k(Set) \hookrightarrow Alg_{\mathbb{K}}(SuperSet)

of superalgebras over $k$ as in def. 2 and internal associative algebras over $𝕂$ in $\mathrm{SuperSet}$.

Similarly we have a faithful embedding

$${\mathrm{SLieAlg}}_k(\mathrm{Set})\hookrightarrow {\mathrm{LieAlg}}_{𝕂}(\mathrm{SuperSet})$$SLieAlg_k(Set) \hookrightarrow LieAlg_{\mathbb{K}}(SuperSet)

of ordinary super Lie algebras over $k$ into the internal Lie algebras over $𝕂$.

This appears as (Sachse, corollary 3.3).

References

Basics of pedestrian superalgebra are reviewed in section 2 and the topos-theoretic reformulation is discussed in section 3 of

• Christoph Sachse, A Categorical Formulation of Superalgebra and Supergeometry (arXiv:0802.4067)