and
In the general sense, superalgebra is the study of (higher) algebra
equivalently
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.
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”)
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 $\mathbb{S}$. 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 hinted at in (Kapranov 13)) 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 | string | fivebrane | ninebrane | |||||||||||||
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
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
hence are graded by the second homotopy group
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
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
hence are graded by the third homotopy group
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
and hence the characteristic twists are in degree 6 of the group of units, hence of the graded group of units
hence are graded by the sixth homotopy group
of the sphere spectrum.
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.
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
and the tensor product is given in terms of that on vector spaces by
but equipped with the non-trivial braiding morphism
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 $V_{odd}\otimes W_{odd}$.
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
equipped with a morphism of super vector spaces
that is associative and commutative in the usual sense. Spcifically for the commutativity this means that with $a,b \in A_{odd}$ we have
Whereas if either of $a$ or $b$ is in $A_{even}$ we have
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 $a$.
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 superalgebra $A$ is called central simple if
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.
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.
A superalgebra is an Azumaya algebra if it is an invertible object in the monoidal 2-category $s2Vect \simeq sAlg$, def. 6.
The group of equivalence classes of Azumaya super algebras is called the super Brauer group, see there for more details.
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.
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 $\langle -,- \range$. Write $\wedge^\bullet V$ be the Grassmann algebra on $V$. The inner product makes this a super Poisson algebra. The Clifford algebra $Cl(V, \langle \rangl)$ is the deformation quantization of this.
There is a superalgebra over the complex numbers of the form
where the single odd generator satisfies $u \cdot u = 1$.
Given some ground field $k$, write
for the full subcategory of ordinary commutative algebras over $k$ into commutative super-algebras (as those having trivial odd part).
The inclusion $\iota$ of def. 10 has
a right adjoint $(-)_{even}$ given by restricting a superalgebra to its even part;
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).
Prop. 2 gives an adjoint triple of the form
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.
A superalgebra is isomorphic to a matrix algebra, def. 9, precisely if it is equivalent in $2 sVect \simeq Alg$, def. 6, (Morita equivalent) to the ground field super algebra.
We discuss the Picard 3-group of $2sVect \simeq sAlg$, 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:
The non-trivial element in $sBr(\mathbb{R})$ is that presented by the superalgebra $\mathbb{C} \oplus \mathbb{C} u$ of example 1, with $u \cdot u = 1$.
This is due to (Wall).
The following generalizes this to the higher homotopy groups.
The homotopy groups of the braided 3-group $sAlg^\times$ 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)).
We now consider higher algebra in the (∞,1)-topos over super points: the cohesive (∞,1)-topos $\mathbf{H} =$ Super∞Grpd.
Write $SuperPoint$ for the site of superpoints. Write
for the sheaf topos (a presheaf topos) over this site. Write
for the (∞,1)-sheaf (∞,1)-topos over this site: the (∞,1)-topos of super ∞-groupoids.
Write
for the restricted Yoneda embedding of supermanifolds given by the canonical inclusion $SuperPoint \hookrightarrow SuperSmoothManifold$.
Write
for the presheaf represented by the real line, regarded as a supermanifold. Equipped with its canonical internal ring structure this is
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
This is canonically equipped with the structure of a (unital) commutative ring in $SuperSet$.
In (Sachse) this appears around (21).
Write $Mod_{\mathbb{R}}(SuperSet)$ for the category of modules over $\mathbb{R}$ of def. 13 in $SuperSet$.
The restriction of the embedding of def. 12 to supermanifolds which are super vector spaces is a functor
from real super vector spaces to internal modules over $\mathbb{R}$ that sends $V \in SVect_{\mathbb{R}}$ to
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 $\mathbb{K}$-modules in the topos over super points.
The functor $j$ from prop 5 induces a full and faithful functor
of superalgebras over $\mathbb{R}$ as in def. 2 and internal associative algebras over $\mathbb{R}$ in $SuperSet$.
Similarly we have a faithful embedding
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
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
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 (D. 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
Brauer groups of superalgebras are discussed in
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