A generalization of the notion of tensor product of
vector spaces, in which
is associative only up to isomorphism and the transposition
isomorphism
need not square to the identity.
The full definition is: a
category
of objects
,
,
,
etc., equipped with a functor
,
and a collection of functorial isomorphisms
(called the
associator)
between any three objects and
(called the
braiding)
between any two objects.
"Functorial"
means that these isomorphisms commute with any morphisms between
objects. Thus,
if the morphism is applied before or after the braiding. Similarly
for functoriality in the other arguments of
and in the arguments of
.
The precise definition is that
and
are natural equivalences, where
is the functor sending
to
,
etc., and
is the functor sending
to
.
In addition, these functors are required to be
coherent.
The
coherence for
is the
pentagon identity
Figure: b120420a
It says that the two ways to reverse the bracketings as shown
coincide.
MacLane's coherence theorem
says that then all other
routes between two bracketed tensor products also coincide. In
effect, this means that one may generalize constructions in linear
algebra exactly as if
were strictly associative, dropping
brackets. Afterwards one may add brackets, for example with
brackets accumulating to the left, and then insert applications of
as needed for the desired compositions to make sense; all
different ways to do this will yield the same net result.
One also requires a unit object
and an associated collection of functorial isomorphisms
and
obeying a
triangle coherence identity
This structure
is called a
monoidal category
(see also
Triple).
The coherence conditions for the additional structure
of a
braided category are the so-called
hexagon identities
Here and below, the bracketings and
needed to make sense of these identities are omitted. (When they are inserted,
each
identity corresponds to a diagram with six arrows.) One can show
that compatibility of
with
is then automatic.
Notice that although
generalizes the concept of transposition of vector spaces, one does
not
demand that
.
If this does hold for all
,
,
then a
symmetric monoidal category
or
tensor category
is obtained.
In this case the two hexagon identities are equivalent
and ensure that all ways to go
(
a permutation) by composing
(and
)
yield the same result. In particular, there is an action of the
symmetric group
in
for any object
.
For a general braided category there is a similar result in terms
of braids. To explain this, the following notational
device is used: instead of writing the morphisms
and
in the usual way as arrows, one writes them as
braids and assumes that they point downwards:
Figure: b120420b
One writes another morphism
,
say, as a node on a strand connecting
to
and assumed to be pointing downwards.
Functoriality says that morphisms can be pulled through braid
crossings as
Figure: b120420c
Similarly for morphisms on the other strand. When the tensor
product of several morphisms is applied, one writes their strands
separately side by side, connecting the relevant objects in the
tensor product. On the other hand, one is free to group some of
the objects in the tensor product together as a single object and
represent morphisms to and from it by a single strand. This
notation is consistent precisely because of the hexagon
conditions, which become
Figure: b120420d
The doubled strands on the left-hand side could be replaced by
single strands for the composite morphisms.
The coherence theorem for braided categories then asserts that
different routes between tensor product expressions by repeated
applications of
,
and their inverses compose to the same
morphism if the corresponding braids are the same. In particular,
for any object
there is an action of the pure
braid group
on
.
The former can be presented as
where
is represented by
acting in the
th
copies of
.
The representation usually has a kernel and
modulo this kernel is the
Hecke algebra
associated to an object in a braided category.
Finally, an object
in a braided category is
rigid
if there are another object
and morphisms
and
such that
(suppressing
,
,
).
Here,
is called a
left dual
of
and is unique up to isomorphism (there is a similar notion of
right dual).
is called
rigid
if every object has such duals.
Using diagrammatic notation one writes
Figure: b120420e
and the above axioms become
Figure: b120420f
Note that in a rigid braided category, morphisms
look like knots (cf. also
Knot theory;
Braid theory).
Every knot presented on paper is in
fact the closure of some braid, so with a little more structure
(notably, both left and right duals) one can arrange that every
oriented knot can be read from top to bottom as a morphism
.
Fixing an object
,
can be read as a left or right
and
can be read as a left or right
(in accordance with orientation). One can read braid crossings as
or
.
The result is not quite a knot invariant
but can usually be adjusted to become one.
Examples.
Some standard examples of braided categories are provided by the
following constructions over a ground field
.
Its invertible elements are denoted
.
1)
Fix
,
an
th
root of unity. The category of
anyspaces
consists of
-graded
vector spaces
(where the elements of
have degree
).
Morphisms are degree-preserving linear mappings. One takes the
associator trivial and the braiding
for
and
of degree
,
,
respectively.
Clearly,
is the usual category of vector spaces, while
is the category of linear
superspaces
or
-graded
vector spaces. The category
of
-graded
spaces is braided similarly for any
.
A similar construction works for grading by any Abelian group
equipped with a bicharacter
(a function multiplicative in each argument) and braiding
on elements of degree
.
2)
If
is a group and
a group
-cocycle,
then the category of
-graded
spaces is monoidal with associator
on elements of degree
.
If
is Abelian and equipped with a quasi-bicharacter with respect to
,
then the category is braided.
For example, the
octonion algebra
lives naturally in such a category with
and
a certain coboundary.
3)
The category of finite-dimensional representations of
the
quantum enveloping algebras
(cf. also
Quantum groups;
Universal enveloping algebra)
associated to a semi-simple
Lie algebra
.
Also, certain classes of
infinite-dimensional representations. Associated to the
standard
representation of
is the standard Hecke algebra defined by the additional relation
.
The knot invariant associated to this same representation is the
Jones knot polynomial.
In an algebraic formulation, if
,
,
is a
quasi-triangular Hopf algebra,
then its category
of representations is braided with
,
where one first acts with the quasi-triangular structure
and then applies the usual transposition mapping
.
4)
The category of
co-modules
under the matrix quantum groups (cf.
Quantum groups)
.
For example, the quantum plane generated by
modulo the relations
is covariant under
and hence lives as an algebra in its braided category of
co-modules. Likewise, the category of co-modules under a
quantum matrix bi-algebra
associated to an invertible solution
of the
Yang–Baxter equation
is braided.
In a general algebraic formulation, if
,
,
is a
dual (or co-)
quasi-triangular Hopf algebra,
then its category of co-modules is braided by
where
denotes the result of the co-action
.
5)
If
is any
Hopf algebra
with invertible antipode, then its category
of
crossed modules
(also called
Drinfel'd–Radford–Yetter modules)
is
braided. Here,
objects are vector spaces
which are both modules and co-modules under
,
the two being compatible in the sense
for
and
,
where
denotes the action and
denotes the co-product. The braiding is
For example, if
is any Lie algebra and
is its enveloping algebra (cf. also
Universal enveloping algebra),
then
is a crossed module by the adjoint action and the co-product of
.
The braiding induced on it is
6)
Let
be any monoidal category. Then its
representation-theoretic
dual category
(also called the
double
or
centre
)
is a braided category. Objects are pairs
where
is an object of
and
is a collection of functorial isomorphisms representing the
of
in the sense
(
,
,
suppressed in this notation). The braiding is
.
These constructions are in roughly increasing order of generality.
Thus, the categorical dual or double construction applied to the
category of
-modules
yields the category of crossed modules
.
These, in turn, are an elementary reformulation (and
thereby a slight generalization to infinite-dimensional
)
of the category of
-modules,
where
is the quantum double
quasi-triangular Hopf algebra associated to any
finite-dimensional
Hopf algebra
.
Meanwhile, a bicharacter on an Abelian group
extends by linearity to a dual-quasi-triangular structure on
the
corresponding group algebra.