super algebra



Basic idea

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


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

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

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

We discuss the general abstract raison d’ être of super algebra. Readers looking for just the plain definition should probably skip to below on first reading.

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

ab=(1) deg(a)deg(b)ba 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), whose second part is about super 2-algebra). 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\mathbb{S}. So the 2\mathbb{Z}_2-grading of superalgebra comes from the stable homotopy groups of spheres π n(𝕊)\pi_n(\mathbb{S}) in degree 1 and 2:

n=n = 0011223344556677\cdots
π n(𝕊)=\pi_n(\mathbb{S}) = \mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}0 0 00 2\mathbb{Z}_2 240\mathbb{Z}_{240}\cdots
meaning:degreeboson/fermion super-degreespinstring--??\cdots
free object on single generator:abelian groupabelian 2-groupabelian 3-groupabelian 4-groupabelian 7-groupabelian 8-groupabelian ∞-group

This suggests (as hinted at in (Kapranov 13)) that in full generality higher supergeometry is to be thought of as 𝕊\mathbb{S}-graded geometry, hence dually as higher algebra with ∞-group of units augmented over the sphere spectrum.

But notice that this is canonically so for every E-∞ ring, see at ∞-group of units – Augmented definition. This would mean: In higher geometry/higher algebra supergeometry/superalgebra is intrinsic, canonically given.

Using this together with Sati’s Geometric and topological structures related to M-branes and the image of the J-homomorphism

Whitehead tower of orthogonal grouporientationspinstringfivebraneninebrane
homotopy groups of stable orthogonal groupπ n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2
stable homotopy groups of spheresπ n(𝕊)\pi_n(\mathbb{S})\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}00 2\mathbb{Z}_2 240\mathbb{Z}_{240} 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 6\mathbb{Z}_6 504\mathbb{Z}_{504}0 3\mathbb{Z}_3 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2
image of J-homomorphismim(π n(J))im(\pi_n(J))0 2\mathbb{Z}_20 24\mathbb{Z}_{24}000 240\mathbb{Z}_{240} 2\mathbb{Z}_2 2\mathbb{Z}_20 504\mathbb{Z}_{504}000 480\mathbb{Z}_{480} 2\mathbb{Z}_2

we can derive the terminology in the above table as indicated now.

The following uses the notions of motivic quantization as indicated there, to be expanded.

Associative 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 kk of characteristic 0.


Write SVect for the symmetric monoidal category of super vector spaces over kk. This is the category of 2\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 evenV odd V = V_{even} \oplus V_{odd}

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

VW=(V evenW evenV oddW odd)(V evenW oddV oddW 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:VWWV b_{V, W} : V \otimes W \to W \otimes V

that is the usual braiding isomorphism of Vect on V evenW evenV_{even} \otimes W_{even} and on V evenW oddV oddW evenV_{even} \otimes W_{odd} \oplus V_{odd} \otimes W_{even} but is (1)(-1) times this on V oddW oddV_{odd}\otimes W_{odd}.


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

A (graded)-commutative (associative) algebra over KK 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 evenA odd A = A_{even} \oplus A_{odd}

equipped with a morphism of super vector spaces

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

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

ab=ba. a \cdot b = - b \cdot a \,.

Whereas if either of aa or bb is in A evenA_{even} we have

ab=ba. a \cdot b = b \cdot a \,.

Related notions


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


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

a 1 A opa 2(1) |a 1||a 2|a 2 Aa 1. a_1 \cdot_{A^{op}} a_2 \coloneqq (-1)^{{\vert a_1\vert}{\vert a_2\vert}} a_2 \cdot_{A} a_1 \,.

A superalgebra AA is called central simple if

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

  2. its only 2-sided graded ideals are 00 and AA itself.


Write 2sVectsAlg2sVect \simeq 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.


By the discussion at 2-vector space this is equivalently the 2-category of super 2-vector spaces. Equivalence in 2sVectsAlg2sVect \simeq sAlg is also called Morita equivalence of super-algebras.


A superalgebra is an Azumaya algebra if it is an invertible object in the monoidal 2-category s2VectsAlgs2Vect \simeq sAlg, def. 6.


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


Endomorphisms algebras, matrix algebras


For VSVectV \in 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.


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

Grassmann algebra

Clifford algebra

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

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


There is a superalgebra over the complex numbers of the form

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

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


Relation to ordinary commutative algebras


Given some ground field kk, write

ι:CAlg kSCAlg k \iota \colon CAlg_k \hookrightarrow SCAlg_k

for the full subcategory of ordinary commutative algebras over kk into commutative super-algebras (as those having trivial odd part).


The inclusion ι\iota of def. 10 has

  1. a right adjoint () even(-)_{even} given by restricting a superalgebra to its even part;

  2. a left adjoint ()/(() odd)(-)/((-)_{odd}) given by forming the “body”, the quotient by the ideal generated by the odd part (by the “soul”).

This is immediate, but conceptually important, it is made explicit for instance in (Carchedi-Roytenberg 12, example 3.18).


Prop. 2 gives an adjoint triple of the form

CAlg k() even()/(() odd)SCAlg k CAlg_k \stackrel{\overset{(-)/((-)_{odd})}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{(-)_{even}}{\longleftarrow}}} SCAlg_k

and hence an adjoint cylinder, which induces a pair of adjoint modalities (fermionic modality \dashv bosonic modality). See at super smooth infinity-groupoid for more on this.

Relation to matrix algebras


A superalgebra is isomorphic to a matrix algebra, def. 9, precisely if it is equivalent in 2sVectAlg2 sVect \simeq Alg, def. 6, (Morita equivalent) to the ground field super algebra.

Picard 3-group, Brauer group

We discuss the Picard 3-group of 2sVectsAlg2sVect \simeq sAlg, def. 6, hence the corresponding Brauer group. See also at super line 2-bundle.


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

This is due to (Wall).


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:

sBr() 2 sBr(\mathbb{C}) \simeq \mathbb{Z}_2
sBr() 8. sBr(\mathbb{R}) \simeq \mathbb{Z}_8 \,.

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

This is due to (Wall).

The following generalizes this to the higher homotopy groups.


The homotopy groups of the braided 3-group sAlg ×sAlg^\times of Azumaya superalgebra are

sAlg ×sAlg^\times_{\mathbb{C}}sAlg ×sAlg^\times_{\mathbb{R}}
π 2\pi_2 ×\mathbb{C}^\times ×\mathbb{R}^\times
π 1\pi_1 2\mathbb{Z}_2 2\mathbb{Z}_2
π 0\pi_0 2\mathbb{Z}_2 8\mathbb{Z}_8

where the groups of units ×\mathbb{C}^\times and ×\mathbb{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=\mathbf{H} = Super∞Grpd.

The topos


Write SuperPointSuperPoint for the site of superpoints. Write

SuperSet:=Sh(SuperPoint) SuperSet := Sh(SuperPoint)

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

SuperGrpd:=Sh (,1)(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 \mathbb{R}



j:SuperSmthMfdSh(SuperPoint) j \colon SuperSmthMfd \hookrightarrow Sh(SuperPoint)

for the restricted Yoneda embedding of supermanifolds given by the canonical inclusion SuperPointSuperSmoothManifoldSuperPoint \hookrightarrow SuperSmoothManifold.



:=j()Sh(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

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

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

:Spec V( V) 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 SuperSetSuperSet.

In (Sachse) this appears around (21).



Write Mod (SuperSet)Mod_{\mathbb{R}}(SuperSet) for the category of modules over \mathbb{R} of def. 13 in SuperSetSuperSet.


The restriction of the embedding of def. 12 to supermanifolds which are super vector spaces is a functor

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

from real super vector spaces to internal modules over \mathbb{R} that sends VSVect V \in SVect_{\mathbb{R}} to

j(V):SpecΛ(Λ V) even=(Λ even V even)(Λ odd V 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).


The proof is a variation on the Yoneda lemma.

This means that ordinary super vector spaces are embedded as a full subcategory in 𝕂\mathbb{K}-modules in the topos over super points.

Associative and Lie Superalgebras


The functor jj from prop 5 induces a full and faithful functor

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

of superalgebras over \mathbb{R} as in def. 2 and internal associative algebras over \mathbb{R} in SuperSetSuperSet.

Similarly we have a faithful embedding

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

of ordinary super Lie algebras over \mathbb{R} into the internal Lie algebras over \mathbb{R}.

This appears as (Sachse, corollary 3.3).





The concept of Grassmann algebra is due to

Discussion of superalgebra as algebra in certain symmetric monoidal tensor categories is in

Lecture notes include

The observation that the study of super-structures in mathematics is usefully regarded as taking place over the base topos on the site of super points has been made around 1984 in

and in

  • V. Molotkov., Infinite-dimensional 2 k\mathbb{Z}_2^k-supermanifolds , ICTP preprints, IC/84/183, 1984.

A summary/review is in the appendix of

  • Anatoly Konechny and Albert Schwarz,

    On (kl|q)(k \oplus l|q)-dimensional supermanifolds in Supersymmetry and Quantum Field Theory (D. Volkov memorial volume) Springer-Verlag, 1998 , Lecture Notes in Physics, 509 , J. Wess and V. Akulov (editors)(arXiv:hep-th/9706003)

    Theory of (kl|q)(k \oplus l|q)-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486

  • Albert Schwarz, I. Shapiro, Supergeometry and Arithmetic Geometry (arXiv:hep-th/0605119)

For more along these lines see also the references at supermanifold and at super infinity-groupoid.

Discussion in terms of smooth algebras (synthetic differential supergeometry) is in

Brauer groups and Picard 2-groupoid

Brauer groups of superalgebras are discussed in

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

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

Revised on March 4, 2015 17:06:11 by Urs Schreiber (