mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Tannakian category is a particular kind of monoidal category ''C'', equipped with some extra structure relative to a given field ''K''. The role of such categories ''C'' is to approximate, in some sense, the category of linear representations of an algebraic group ''G'' defined over ''K''. A number of major applications of the theory have been made, or might be made in pursuit of some of the central conjectures of contemporary

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

and number theory. The name is taken from Tadao Tannaka and Tannaka–Krein duality, a theory about compact groups ''G'' and their representation theory. The theory was developed first in the school of Alexander Grothendieck. It was later reconsidered by Pierre Deligne, and some simplifications made. The pattern of the theory is that of Grothendieck's Galois theory, which is a theory about finite permutation representations of groups ''G'' which are

profinite group In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...

s. The gist of the theory, which is rather elaborate in detail in the exposition of Saavedra Rivano, is that the fiber functor Φ of the Galois theory is replaced by a tensor functor ''T'' from ''C'' to ''K''-Vect. The group of natural transformations of Φ to itself, which turns out to be a profinite group in the Galois theory, is replaced by the group (''a priori'' only a monoid) of natural transformations of ''T'' into itself, that respect the tensor structure. This is by nature not an algebraic group, but an inverse limit of algebraic groups ( pro-algebraic group).

Formal definition

A neutral Tannakian category is a rigid

abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...

tensor category, such that there exists a ''K''-tensor functor to the category of finite dimensional ''K''-vector spaces that is

exact Exact may refer to: * Exaction, a concept in real property law * ''Ex'Act'', 2016 studio album by Exo * Schooner Exact, the ship which carried the founders of Seattle Companies * Exact (company), a Dutch software company * Exact Change, an Ameri ...

and faithful.

Applications

The construction is used in cases where a Hodge structure or

l-adic representation In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...

is to be considered in the light of group representation theory. For example, the Mumford–Tate group and

motivic Galois group In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham coho ...

are potentially to be recovered from one cohomology group or Galois module, by means of a mediating Tannakian category it generates. Those areas of application are closely connected to the theory of motives. Another place in which Tannakian categories have been used is in connection with the

Grothendieck–Katz p-curvature conjecture In mathematics, the Grothendieck–Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogous to the result in the Chebotarev density the ...

; in other words, in bounding monodromy groups. The

Geometric Satake equivalence In mathematics, the Satake isomorphism, introduced by , identifies the Hecke algebra of a reductive group over a local field with a ring of invariants of the Weyl group. The geometric Satake equivalence is a geometric version of the Satake isomorph ...

establishes an equivalence between representations of the Langlands dual group

^L G

of a reductive group ''G'' and certain equivariant

perverse sheaves The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was int ...

on the

affine Grassmannian In mathematics, the affine Grassmannian of an algebraic group ''G'' over a field ''k'' is an ind-scheme—a colimit of finite-dimensional schemes—which can be thought of as a flag variety for the loop group ''G''(''k''((''t''))) and which desc ...

associated to ''G''. This equivalence provides a non-combinatorial construction of the Langlands dual group. It is proved by showing that the mentioned category of perverse sheaves is a Tannakian category and identifying its Tannaka dual group with

^L G

Extensions

has established partial Tannaka duality results in the situation where the category is ''R''-linear, where ''R'' is no longer a field (as in classical Tannakian duality), but certain valuation rings. showed a Tannaka duality result if ''R'' is a Dedekind ring. has initiated the study of Tannaka duality in the context of infinity-categories.

References

* * * * * *

Formal definition

Applications

Extensions

References

Further reading