tensor category
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| tensor category | |
|---|---|
| Name | Tensor category |
| Field | Category theory, Representation theory |
| Introduced | 1960s–1980s |
| Notable | Jean-Pierre Serre; Alexandre Grothendieck; Alexander Grothendieck; Graeme Segal; Vladimir Drinfeld; Michael Atiyah |
tensor category
A tensor category is an algebraic structure studied in Category theory, arising in contexts such as Representation theory, Quantum field theory, Low-dimensional topology, and Operator algebras. It formalizes a notion of objects that can be tensored together, equipped with associativity and unit constraints, and appears in the work of figures like Vladimir Drinfeld, Michio Jimbo, Graeme Segal, and Michael Atiyah. Tensor categories connect to constructions in Algebraic geometry, Number theory, Statistical mechanics, and Quantum groups.
Definition and basic examples
A tensor category is a monoidal category with additional finiteness or linearity conditions studied by authors such as Jean-Pierre Serre and Pierre Deligne. Basic examples include categories of finite-dimensional representations of a group like Lie groups such as SU(2), SL(2,C), algebraic groups like GL_n over fields considered by Alexander Grothendieck, and quantum analogs from Quantum group theory developed by Vladimir Drinfeld and Michio Jimbo. Other examples arise from Hopf algebras studied by Sweedler and Nichols, categories of modules over a Vertex operator algebra linked to work by Richard Borcherds and Victor Kac, and categories of bimodules in the theory of von Neumann algebras investigated by Vaughan Jones.
Monoidal structure and coherence
The monoidal structure involves a tensor product functor, unit object, and natural associators studied in coherence theorems by Saunders Mac Lane and developments in higher category theory by John Baez and Jacob Lurie. Coherence constraints are governed by Mac Lane's coherence theorem and further refined in contexts handled by Max Kelly and Ross Street. Examples of coherence issues occur in braided constructions introduced by Andruskiewitsch and Dmitri Nikshych and in categorical constructions used by Tom Leinster and Emily Riehl.
Braided, symmetric, and ribbon tensor categories
Braided tensor categories, formalized after work of Drinfeld and V. G. Kac, admit a braiding natural isomorphism; symmetric tensor categories appear in classical settings like categories of representations of Symmetric groups or of Algebraic groups considered by Pierre Deligne. Ribbon categories, which admit twist morphisms, are crucial in constructions related to invariants of links and 3-manifolds developed by Edward Witten, Vaughan Jones, and Nicolai Reshetikhin. The classification of braided and modular categories connects to work by Gregory Moore, Nikita Nekrasov, Anton Kapustin, and Paul Seidel.
Module categories and representations
Module categories over a tensor category generalize the notion of representations and were studied in the context of subfactor theory by Vaughan Jones and in fusion theory by Pavel Etingof and Shlomo Gelaki. The theory interacts with induction and restriction functors studied by George Mackey and with categorical actions considered by Henning Krause and Ben Webster. Module categories provide bridges to the representation theory of algebras like Hecke algebras, Temperley–Lieb algebras, and quantum algebras linked to Jimbo and Drinfeld.
Fusion categories and semisimplicity
Fusion categories are semisimple tensor categories with finitely many simple objects, studied extensively by Pavel Etingof, Victor Ostrik, and Shlomo Gelaki. They generalize representation categories of finite groups such as S_3 and finite-dimensional Hopf algebras like those investigated by Hans-Jürgen Schneider and George Lusztig. Semisimplicity relates to theorems of Wedderburn and to categorical dimensions introduced in the work of Kazhdan and Lusztig. Classification efforts link to techniques from Modular representation theory and invariants from Conformal field theory developed by Belavin, Polyakov, and Zamolodchikov.
Duality, pivotal structures, and traces
Duality and pivotal structures provide categorical analogs of dual vector spaces and are central in the study of spherical and pivotal categories used by Michael Atiyah and in TQFT constructions by Edward Witten and Graeme Segal. Traces and dimensions in pivotal categories underpin quantum invariants introduced by Reshetikhin and Turaev and link to index theory from Atiyah–Singer frameworks. Notions of left and right duals play roles in coherence conditions explored by Mac Lane and in monoidal bicategory theory developed by Street and Bénabou.
Applications in topology and mathematical physics
Tensor categories provide algebraic underpinnings for topological quantum field theories (TQFTs) as formulated by Atiyah and implemented by constructions of Witten and Reshetikhin–Turaev. They produce link and 3-manifold invariants used in low-dimensional topology studied by William Thurston and Greg Kuperberg, and model anyon statistics in condensed matter contexts investigated by Alexei Kitaev and Michael Freedman. Connections to conformal field theory and string theory involve works by Belavin, Polyakov, Zamolodchikov, Edward Witten, and Cumrun Vafa, and influence developments in quantum computation studied by Peter Shor and Frederic T. Chong.