nLab
super algebra

Idea

Basic idea

In the general sense, superalgebra is the study of (higher) algebra

• internal to the symmetric monoidal category of ${\mathbb{Z}}_2$-graded vector spaces (super vector spaces);

equivalently

• 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 superalgebras

as monoids in the symmetric monoidal category of super vector spaces. Then we pass to the perspective 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.

Abstract idea

Superalgebra is universal in the following sense. The crucial super-grading rule (the “Koszul sign rule”)

$$a\otimes b=(-1{)}^{\mathrm{deg}(a)\mathrm{deg}(b)}b\otimes a$$a \otimes b = (-1)^{deg(a) deg(b)} b \otimes a

in the symmetric monoidal category of $\mathbb{Z}$-graded vector spaces is induced from the subcategory which is the abelian 2-group of metric graded lines. This in turn is the free abelian 2-group (groupal symmetric monoidal category) on a single generator. (This point of view is amplified for instance in the first part of (Kapranov 13)). Generally then super-grading and hence super-algebra arises from the 2-truncation (3-coskeleton) of the free abelian ∞-group on a single generator, which is the sphere spectrum $𝕊$. So the ${\mathbb{Z}}_2$-grading of superalgebra comes from the stable homotopy groups of spheres ${\pi }_n(𝕊)$ in degree 1 and 2:

$n=$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$\cdots$
${\pi }_n(𝕊)=$	$\mathbb{Z}$	${\mathbb{Z}}_2$	${\mathbb{Z}}_2$	${\mathbb{Z}}_{24}$	$0$	$0$	${\mathbb{Z}}_2$	${\mathbb{Z}}_{240}$	$\cdots$
meaning:	degree	boson/fermion super-degree	spin	string	$-$	$-$	?	?	$\cdots$
free object on single generator:	abelian group	abelian 2-group	abelian 3-group	abelian 4-group			abelian 7-group	abelian 8-group	abelian ∞-group

Remark

That ${\mathbb{Z}}_{24}$ in degree 3 here coresponds to string structure/string geometry is witnessed by many occurences of this number in this context. Maybe the most direct one concerning supergeometry is the $24$ factor in the periodicity of “2-Clifford algebras” in their incarnation as objects in the 3-category of fermionic conformal nets.

Remark

Extrapolating from the pattern, the question mark entries might be filled by Fivebrane 6-group structure corresponding to super 5-branes (NS-5-brane/M5-brane.

Associative superalgebras

Definition

Superalgebras

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

A (graded)-commutative (associative) 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 \,.

Related notions

Definition

The center of a superalgebra $A$ is the sub-superalgebra $Z(A)\hookrightarrow A$ spanned by all those elements $z\in A$ of homogeneous degree which graded-commute with all other homogeneois elements $a$.

Definition

For $A$ a superalgebra, its opposite $A^{\mathrm{op}}$ is the superalgebra with the same underlying super vector space as $A$, and with multiplication defined on homogeneous elements by

$$a_1{\cdot }_{A^{\mathrm{op}}}{a}_2≔(-1{)}^{\mid a_1\mid \mid a_2\mid }{a}_2{\cdot }_A{a}_1.$$a_1 \cdot_{A^{op}} a_2 \coloneqq (-1)^{{\vert a_1\vert}{\vert a_2\vert}} a_2 \cdot_{A} a_1 \,.

Definition

A superalgebra $A$ is called central simple if

1. its center, def. 3 is the ground field;

2. its only 2-sided graded ideals are $0$ and $A$ itself.

Definition

Write $2\mathrm{sVect}\simeq \mathrm{sAlg}$ for the 2-category equivalent to the one whose objects are superalgebra, 1-morphisms are bimodules and 2-morphisms are intertwiners. This is naturally a monoidal 2-category.

Remark

By the discussion at 2-vector space this is equivalently the 2-category of super 2-vector spaces. Equivalence in $2\mathrm{sVect}\simeq \mathrm{sAlg}$ is also called Morita equivalence of super-algebras.

Definition

A superalgebra is an Azumaya algebra if it is an invertible object in the monoidal 2-category $s2\mathrm{Vect}\simeq \mathrm{sAlg}$, def. 6.

Remark

The group of equivalence classes of Azumaya super algebras is called the super Brauer group, see there for more details.

Examples

Endomorphisms algebras, matrix algebras

Definition

For $V\in \mathrm{SVect}$ a super vector space, its endomorphism ring is canonically a super-algebra. Superalgebras isomorphic to ones of this form, are also called matrix super algebras.

Proposition

A matrix superalgebra, def. 9 is central simple, def. 5.

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.

Example

There is a superalgebra over the complex numbers of the form

$$A=ℂ\oplus ℂ⟨u⟩,$$A = \mathbb{C} \oplus \mathbb{C}\langle u\rangle \,,

where the single odd generator satisfies $u\cdot u=1$.

Properties

General

Proposition

A superalgebra is isomorphic to a matrix algebra, def. 9 precisely if it is equivalent in $2\mathrm{sVect}\simeq \mathrm{Alg}$, def. 6, (Morita equivalent) to the ground field super algebra.

Picard 3-group, Brauer group

We discuss the Picard 3-group of $2\mathrm{sVect}\simeq \mathrm{sAlg}$, def. 6, hence the corresponding Brauer group.

Theorem

A superalgebra is invertible/Azumaya, def. 8, precisely if it is finite dimensional and central simple, def. 5.

This is due to (Wall).

Theorem

The Brauer group of superalgebras over the complex numbers is the cyclic group of order 2. That over the real numbers is cyclic of order 8:

$$\mathrm{sBr}(ℂ)\simeq {\mathbb{Z}}_2$$sBr(\mathbb{C}) \simeq \mathbb{Z}_2

$$\mathrm{sBr}(ℝ)\simeq {\mathbb{Z}}_8.$$sBr(\mathbb{R}) \simeq \mathbb{Z}_8 \,.

The non-trivial element in $\mathrm{sBr}(ℝ)$ is that presented by the superalgebra $ℂ\oplus ℂu$ of example 1, with $u\cdot u=1$.

This is due to (Wall).

The following generalizes this to the higher homotopy groups.

Proposition

The homotopy groups of the braided 3-group ${\mathrm{sAlg}}^{\times }$ of Azumaya superalgebra are

	${\mathrm{sAlg}}_{ℂ}^{\times }$	${\mathrm{sAlg}}_{ℝ}^{\times }$
${\pi }_2$	${ℂ}^{\times }$	${ℝ}^{\times }$
${\pi }_1$	${\mathbb{Z}}_2$	${\mathbb{Z}}_2$
${\pi }_0$	${\mathbb{Z}}_2$	${\mathbb{Z}}_8$

where the groups of units ${ℂ}^{\times }$ and ${ℝ}^{\times }$ are regarded as discrete groups.

This appears in (Freed, (1.38)).

Algebra in the topos over superpoints

We now consider higher algebra in the (∞,1)-topos over super points: the cohesive (∞,1)-topos $H=$ Super∞Grpd.

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 $ℝ$

Definition

Write

$$j:\mathrm{SuperSmthMfd}\hookrightarrow \mathrm{Sh}(\mathrm{SuperPoint})$$j \colon SuperSmthMfd \hookrightarrow Sh(SuperPoint)

for the restricted Yoneda embedding of supermanifolds given by the canonical inclusion $\mathrm{SuperPoint}\hookrightarrow \mathrm{SuperSmoothManifold}$.

Definition

Write

$$ℝ:=j(ℝ)\in \mathrm{Sh}(\mathrm{SuperPoint})$$\mathbb{R} := j(\mathbb{R}) \in Sh(SuperPoint)

for the presheaf represented by the real line, regarded as a supermanifold. Equipped with its canonical internal ring structure this is

$$ℝ\in \mathrm{Ring}(\mathrm{Sh}(\mathrm{SuperPoint})).$$\mathbb{R} \in Ring(Sh(SuperPoint)) \,.

Remark

By the discussion at supermanifold (in the section As locally ringed spaces - Properties) $ℝ$ sends the formal dual of a Grassmann algebra to its even subalgebra

$$ℝ:\mathrm{Spec}{\wedge }^{•}V\mapsto ({\wedge }^{•}V{)}_{\mathrm{even}}.$$\mathbb{R} : 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).

$ℝ$-Modules

Definition

Write ${\mathrm{Mod}}_{ℝ}(\mathrm{SuperSet})$ for the category of modules over $ℝ$ of def. 12 in $\mathrm{SuperSet}$.

Proposition

The restriction of the embedding of def. 11 to supermanifolds which are super vector spaces is a functor

$$j:{\mathrm{SVect}}_{ℝ}={\mathrm{Mod}}_{ℝ}(\mathrm{Set})\hookrightarrow {\mathrm{Mod}}_{𝕂}(\mathrm{SuperSet})$$j : SVect_{\mathbb{R}} = Mod_{\mathbb{R}}(Set) \hookrightarrow Mod_{\mathbb{K}}(SuperSet)

from real super vector spaces to internal modules over $ℝ$ that sends $V\in {\mathrm{SVect}}_{ℝ}$ to

$$j(V):\mathrm{Spec}\Lambda \mapsto (\Lambda {\otimes }_{ℝ}V{)}_{\mathrm{even}}=({\Lambda }_{\mathrm{even}}{\otimes }_{ℝ}{V}_{\mathrm{even}})\oplus ({\Lambda }_{\mathrm{odd}}{\otimes }_{ℝ}{V}_{\mathrm{odd}}).$$j(V) : Spec \Lambda \mapsto (\Lambda \otimes_\mathbb{R} V)_{even} = (\Lambda_{even} \otimes_\mathbb{R} V_{even}) \oplus (\Lambda_{odd} \otimes_\mathbb{R} 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 points.

Associative and Lie Superalgebras

Proposition

The functor $j$ from prop 4 induces a full and faithful functor

$${\mathrm{SAlg}}_{ℝ}(\mathrm{Set})\hookrightarrow {\mathrm{Alg}}_{ℝ}(\mathrm{SuperSet})$$SAlg_{\mathbb{R}}(Set) \hookrightarrow Alg_{\mathbb{R}}(SuperSet)

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

Similarly we have a faithful embedding

$${\mathrm{SLieAlg}}_{ℝ}(\mathrm{Set})\hookrightarrow {\mathrm{LieAlg}}_{ℝ}(\mathrm{SuperSet})$$SLieAlg_{\mathbb{R}}(Set) \hookrightarrow LieAlg_{\mathbb{R}}(SuperSet)

of ordinary super Lie algebras over $ℝ$ into the internal Lie algebras over $ℝ$.

This appears as (Sachse, corollary 3.3).

Properties

(…)

Related concepts

• smooth superalgebra

• supermanifold

References

Basics of 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)

Brauer groups of superalgebras are discussed in

• Daniel Freed, Lectures on twisted K-theory and orientifolds (pdf)

• C. T. C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213 (1963/1964), 187-199.

• Pierre Deligne, Notes on spinors in Quantum Fields and Strings

• Peter Donovan, Max Karoubi, Graded Brauer groups and K-theory with local coefficients, Publications Math. IHES 38 (1970), 5-25 (pdf)

See also at super line 2-bundle for more on this.

Discussion of superalgebra as induced from free groupal symmetric monoidal categories (abelian 2-groups) and hence ultimately from the sphere spectrum is in

• Mikhail Kapranov, Categorification of supersymmetry and stable homotopy groups of spheres, April 2013 (abstract, video)