Super-Algebra and Super-Geometry
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.
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)
in the symmetric monoidal category of -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 . So the -grading of superalgebra comes from the stable homotopy groups of spheres in degree 1 and 2:
This suggests (as hinted at in (Kapranov 13)) that in full generality higher supergeometry is to be thought of as -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 group||orientation||spin||string||fivebrane||ninebrane|
|homotopy groups of stable orthogonal group||0||0||0||0||0||0||0||0|
|stable homotopy groups of spheres||0||0||0|
|image of J-homomorphism||0||0||0||0||0||0||0||0||0|
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.
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 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.
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.
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 of characteristic 0.
Write SVect for the symmetric monoidal category of super vector spaces over . This is the category of -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 and on but is times this on .
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. Specifically for the commutativity this means that with we have
Whereas if either of or is in we have
The center of a superalgebra is the sub-superalgebra spanned by all those elements of homogeneous degree which graded-commute with all other homogeneois elements .
For a superalgebra, its opposite is the superalgebra with the same underlying super vector space as , and with multiplication defined on homogeneous elements by
A superalgebra is called central simple if
its center, def. 3 is the ground field;
its only 2-sided graded ideals are and itself.
Endomorphisms algebras, matrix algebras
For 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.
An class of examples of non-(graded)-commutative superalgebra are Clifford algebra.
In fact, let be a vector space equipped with symmetric inner product . Write be the Grassmann algebra on . The inner product makes this a super Poisson algebra. The Clifford algebra is the deformation quantization of this.
There is a superalgebra over the complex numbers of the form
where the single odd generator satisfies .
Relation to ordinary commutative algebras
The inclusion of def. 10 has
a right adjoint given by restricting a superalgebra to its even part;
a left adjoint 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).
Relation to matrix algebras
A superalgebra is isomorphic to a matrix algebra, def. 9, precisely if it is equivalent in , def. 6, (Morita equivalent) to the ground field super algebra.
Picard 3-group, Brauer group
We discuss the Picard 3-group of , 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 is that presented by the superalgebra of example 1, with .
This is due to (Wall).
The following generalizes this to the higher homotopy groups.
The homotopy groups of the braided 3-group of Azumaya superalgebra are
where the groups of units and 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 Super∞Grpd.
The line object
for the restricted Yoneda embedding of supermanifolds given by the canonical inclusion .
for the presheaf represented by the real line, regarded as a supermanifold. Equipped with its canonical internal ring structure this is
In (Sachse) this appears around (21).
Write for the category of modules over of def. 13 in .
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 that sends to
This is a full and faithful functor.
This appears as (Sachse, corollary 3.2).
This means that ordinary super vector spaces are embedded as a full subcategory in -modules in the topos over super points.
Associative and Lie Superalgebras
The functor from prop 5 induces a full and faithful functor
of superalgebras over as in def. 2 and internal associative algebras over in .
Similarly we have a faithful embedding
of ordinary super Lie algebras over into the internal Lie algebras over .
This appears as (Sachse, corollary 3.3).
The concept of Grassmann algebra and the super-sign rule is due to
Review of basic superalgebra includes
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
- V. Molotkov., Infinite-dimensional -supermanifolds , ICTP preprints, IC/84/183, 1984.
A summary/review is in the appendix of
Anatoly Konechny and Albert Schwarz,
On -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 -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