現代数学の系譜工学物理雑談古典ガロア理論も読む79

[過去ﾛｸﾞ] 現代数学の系譜工学物理雑談古典ガロア理論も読む79 (1002ﾚｽ)
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201: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/29(金) 00:19:47.46 ID:KnsCfpdu

>>200
>which is a fortiori an equivalence of tannakian categories (Ginzburg 2000).

淡中先生（＾＾
https://en.wikipedia.org/wiki/Tannakian_formalism
Tannakian formalism

In 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 and number theory.

The name is taken from 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 groups.

Contents
1 Formal definition
2 Applications
3 Extensions

つづく

http://rio2016.5ch.net/test/read.cgi/math/1573769803/201

202: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/29(金) 00:20:19.92 ID:KnsCfpdu

>>201
つづき

Applications

The Geometric Satake equivalence establishes an equivalence between representations of the Langlands dual group  {}^{L}G}  of a reductive group G and certain equivariant perverse sheaves on the affine Grassmannian 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
Wedhorn (2004) 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. Duong & Hai (2017) showed a Tannaka duality result if R is a Dedekind ring.

Iwanari (2014) has initiated the study of Tannaka duality in the context of infinity-categories.

http://rio2016.5ch.net/test/read.cgi/math/1573769803/202

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