In mathematics, a fusion category is a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
that is
rigid,
semisimple
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
,
-linear,
monoidal and has only finitely many isomorphism classes of
simple objects, such that the monoidal unit is simple. If the
ground field In mathematics, a ground field is a field ''K'' fixed at the beginning of the discussion.
Use
It is used in various areas of algebra:
In linear algebra
In linear algebra, the concept of a vector space may be developed over any field.
In algeb ...
is
algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
, then the latter is equivalent to
by
Schur's lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations
of a group ' ...
.
Examples
*
Representation Category of a finite group
Reconstruction
Under
Tannaka–Krein duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topologic ...
, every fusion category arises as the representations of a
weak Hopf algebra.
References
Category theory
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