Braided category
braided monoidal category, quasi-tensor category
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
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(called the associator) between any three objects and
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(called the braiding) between any two objects. "Functorial" means that these isomorphisms commute with any morphisms between objects. Thus,
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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
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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
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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
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( 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
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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
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(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
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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
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on elements of degree .
2) If is a group and
a group
-cocycle, then the category of
-graded spaces is monoidal with associator
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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
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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
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for and
, where
denotes the action and
denotes the co-product. The braiding is
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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
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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
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(,
,
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.
References
[a1] | S. MacLane, "Categories for the working mathematician" , GTM , 5 , Springer (1974) |
[a2] | A. Joyal, R. Street, "Braided monoidal categories" Math. Reports Macquarie Univ. , 86008 (1986) |
[a3] | S. Majid, "Foundations of quantum group theory" , Cambridge Univ. Press (1995) |
Braided category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Braided_category&oldid=18411