nLab super algebra

from Grassmann 1844, p. 61 and 84

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, a supercommutative superalgebra is an commutative algebra in the context of superalgebra.

See at geometry of physics – superalgebra for more on this.

In the following we first discuss

• Associative 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

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.

One way to understand the relevance of supercommutative superalgebra is Deligne's theorem on tensor categories, which states that well-behaved tensor categories over the complex numbers are equivalent to categories of representations of supergroups. From this perspective the crucial sign rule is related to the symmetric braiding in tensor categories. This in turn may itself be understood from a more general perspective as follows.

Superalgebra is universal in the following sense. The crucial super-grading rule (the “Koszul sign rule”, Grassmann 1844, §37, §55)

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

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 in the first part of (Kapranov 13), whose second part is about super 2-algebra, more details in Kapranov 15). 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 $\mathbb{S}$. So the $\mathbb{Z}_2$-grading of superalgebra comes from the stable homotopy groups of spheres $\pi_n(\mathbb{S})$ in degree 1 and 2:

$n =$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$\cdots$
$\pi_n(\mathbb{S}) =$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_{24}$	$0$	$0$	$\mathbb{Z}_2$	$\mathbb{Z}_{240}$	$\cdots$
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

This suggests (as indicated in (Kapranov 13, Kapranov 15)) that in full generality higher supergeometry is to be thought of as $\mathbb{S}$-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

	$n$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Whitehead tower of orthogonal group		orientation	spin group		string group				fivebrane group				ninebrane group
higher versions		special orthogonal group	spin group		string 2-group				fivebrane 6-group				ninebrane 10-group
homotopy groups of stable orthogonal group	$\pi_n(O)$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$
stable homotopy groups of spheres	$\pi_n(\mathbb{S})$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_{24}$	0	0	$\mathbb{Z}_2$	$\mathbb{Z}_{240}$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_6$	$\mathbb{Z}_{504}$	0	$\mathbb{Z}_3$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_{480} \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism	$im(\pi_n(J))$	0	$\mathbb{Z}_2$	0	$\mathbb{Z}_{24}$	0	0	0	$\mathbb{Z}_{240}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}_{504}$	0	0	0	$\mathbb{Z}_{480}$	$\mathbb{Z}_2$

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.

• $d = 1$ sigma-model

  The local coefficients for quantizing the (spinning) particle on the boundary of the string ending on a D-brane (by K-theoretic geometric quantization by push-forward/D-brane charge) are

  $$\array{ B BU(1) &\simeq& B K(\mathbb{Z},2) &\to& B gl_1(KU) &\to& KU \text{-}Mod }$$

  for $KU$ the complex K-theory spectrum E-∞ ring, and hence the characteristic twists are in degree 2 of the group of units, hence of the graded ∞-group of units

  $$gl^{gr}_1(KU) \to \mathbb{S}$$

  hence are graded by the second homotopy group

  $$\pi_2(\mathbb{S}) \simeq \mathbb{Z}_2$$

  of the sphere spectrum.

• $d = 2$ sigma-model

  The local coefficients for quantizing the string (on the boundary of the M2-brane ending on an M9-brane) are

  $$\array{ B B^2 U(1) &\simeq& B K(\mathbb{Z},3) &\to& B gl_1(tmf) &\to& tmf \text{-}Mod }$$

  for the tmf E-∞ ring, and hence the characteristic twists are in degree 3 of the group of units, hence of the graded group of units

  $$gl^{gr}_1(tmf) \to \mathbb{S}$$

  hence are graded by the third homotopy group

  $$\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24}$$

  of the sphere spectrum.

• $d = 5$ sigma-model

  The local coefficients for quantizing the Yang monopole (on the boundary of the M5-brane ending on an M9-brane) are

  $$\array{ B B^5 U(1) &\simeq& B K(\mathbb{Z},6) &\to& B gl_1(K(5)) &\to& K(5) \text{-}Mod } \,,$$

  and hence the characteristic twists are in degree 6 of the group of units, hence of the graded group of units

  $$gl^{gr}_1(K(5)) \to \mathbb{S}$$

  hence are graded by the sixth homotopy group

  $$\pi_6(\mathbb{S}) \simeq \mathbb{Z}_{2}$$

  of the sphere spectrum.

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

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

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

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

but equipped with the non-trivial braiding morphism

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

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

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 commutative monoid in the symmetric monoidal category SVect of super vector spaces.

This means that a commutative superalgebra is a super vector space

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

equipped with a morphism of super vector spaces

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

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

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

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

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

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

$$a_1 \cdot_{A^{op}} a_2 \coloneqq (-1)^{{\vert a_1\vert}{\vert a_2\vert}} a_2 \cdot_{A} a_1 \,.$$

Definition

A superalgebra $A$ is called central simple if

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

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

Definition

Write $2sVect \simeq sAlg$ 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.

Remark

By the discussion at 2-vector space this is equivalently the 2-category of super 2-vector spaces. Equivalence in $2sVect \simeq sAlg$ 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 $s2Vect \simeq sAlg$, def. .

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 $V \in SVect$ 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. is central simple, def. .

Grassmann algebra

• For $V$ a vector space or graded vector space the Grassmann algebra $\wedge^\bullet V$ 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 $\langle -,- \rangle$. Write $\wedge^\bullet V$ be the Grassmann algebra on $V$. The inner product makes this a super Poisson algebra. The Clifford algebra $Cl(V, \langle -,- \rangle)$ is the deformation quantization of this.

Example

There is a superalgebra over the complex numbers of the form

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

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

Properties

Relation to ordinary commutative algebras

Definition

Given some ground field $k$, write

$$\iota \colon CAlg_k \hookrightarrow SCAlg_k$$

for the full subcategory of ordinary commutative algebras over $k$ into supercommutative superalgebras (as those having trivial odd part).

Proposition

The inclusion $\iota$ of def. has

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

2. a left adjoint $(-)/((-)_{odd})$ 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).

Remark

Prop. gives an adjoint triple of the form

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

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

Proposition

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

Picard 3-group, Brauer group

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

Theorem

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

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:

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

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

The non-trivial element in $sBr(\mathbb{R})$ is that presented by the superalgebra $\mathbb{C} \oplus \mathbb{C} u$ of example , 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 $sAlg^\times$ of Azumaya superalgebra are

	$sAlg^\times_{\mathbb{C}}$	$sAlg^\times_{\mathbb{R}}$
$\pi_2$	$\mathbb{C}^\times$	$\mathbb{R}^\times$
$\pi_1$	$\mathbb{Z}_2$	$\mathbb{Z}_2$
$\pi_0$	$\mathbb{Z}_2$	$\mathbb{Z}_8$

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 $\mathbf{H} =$ Super∞Grpd.

The topos

Definition

Write $SuperPoint$ for the site of superpoints. Write

$$SuperSet := Sh(SuperPoint)$$

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

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

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

The line object $\mathbb{R}$

Definition

Write

$$j \colon SuperSmthMfd \hookrightarrow Sh(SuperPoint)$$

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

Definition

Write

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

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

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

Remark

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

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

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

In (Sachse) this appears around (21).

$\mathbb{R}$-Modules

Definition

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

Proposition

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

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

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

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

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 $\mathbb{K}$-modules in the topos over super points.

Associative and Lie Superalgebras

Proposition

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

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

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

Similarly we have a faithful embedding

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

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

This appears as (Sachse, corollary 3.3).

Properties

(…)

Related concepts

• super module, super vector space

• supercommutative algebra, differential graded-commutative superalgebra

  model structure on differential graded-commutative superalgebras

• smooth superalgebra

• supermanifold

• super 2-algebra

References

General

The concept of Grassmann algebra and the super-sign rule is due to

• Hermann Grassmann, Ausdehnungslehre (1844):

Early modern account:

• Felix A. Berezin (edited by Alexandre A. Kirillov): Introduction to Superanalysis, Mathematical Physics and Applied Mathematics 9, Springer (1987) [doi:10.1007/978-94-017-1963-6]

Review of basic superalgebra:

• Yuri Manin, Chapter 3 in: Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften 289, Springer (1988) [doi:10.1007/978-3-662-07386-5]

• Dennis Westra, Superrings and supergroups, 2009 (pdf)

• Pierre Deligne, John Morgan, Ch 1 in: Notes on Supersymmetry (following Joseph Bernstein), in: Quantum Fields and Strings, A course for mathematicians, 1, Amer. Math. Soc. Providence (1999) 41-97 [ISBN:978-0-8218-2014-8, web version, pdf]

• Ivan Mirković, Sec 1 in: Notes on Super Math, in Quantum Field Theory Seminar, lecture notes (2004) [pdf, pdf]

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

• Pierre Deligne, Catégorie Tensorielle (pdf)

(see also at Deligne's theorem on tensor categories).

Lecture notes include

• Daniel Freed, Five lectures on supersymmetry

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

• Albert Schwarz, On the definition of superspace, Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 37–42, (russian original pdf)

• Alexander Voronov, Maps of supermanifolds , Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 43–48

and in

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

A summary/review is in the appendix of

• Anatoly Konechny and Albert Schwarz,

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

  Theory of $(k \oplus l|q)$-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

• David Carchedi, Dmitry Roytenberg, On theories of superalgebras of differentiable functions, Theory and Applications of Categories, Vol. 28, 2013, No. 30, pp 1022-1098. (arxiv:1211.6134, TAC)

Brauer groups and Picard 2-groupoid

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, talk at Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013) (April 2013) [pdf, video:YT]

  Abstract:. The “minimal sign skeleton” necessary to formulate the Koszul sign rule is a certain Picard category, a symmetric monoidal category with all objects and morphisms invertible. It can be seen as the free Picard category generated by one object and corresponds, by Grothendieck‘s dictionary, to the truncation of the spherical spectrum $S$ in degrees 0 and 1, so that $\{\pm 1\}$ appears as the first stable homotopy group of spheres $\pi_{n+1}(S^n)$. This suggest a “higher” or categorified versions of super-mathematics which utilize deeper structure of $S$. The first concept on this path is that of a supersymmetric monoidal category which is categorified version of the concept of a supercommutative algebra.

  (cf. super 2-algebra and spectral super-scheme)

• Mikhail Kapranov, Supergeometry in mathematics and physics, in Gabriel Catren, Mathieu Anel, (eds.) New Spaces for Mathematics and Physics (arXiv:1512.07042)

• Mikhail Kapranov, Super-geometry, talk at New Spaces for Mathematics & Physics, IHP Paris (Oct-Sept 2015) [video recording]