On the Hall algebra of coherent sheaves on P^1 over F_1

Szczesny, Matt

doi:10.48550/arxiv.1009.3570

Cited by 2 publications

(2 citation statements)

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“…For instance, he considers (Dynkin) quivers [Szc10b] (where hall(Q−rep F 1 ) is related (but not necessarily isomorphic!) to the positive part of the universal enveloping algebra of the corresponding simple Lie algebra) and coherent sheaves on P 1 [Szc10a]. ♦…”

Section: The Canonical Isomorphism Hall(a)mentioning

confidence: 99%

Hall monoidal categories and categorical modules

Walde

2016

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We construct so called Hall monoidal categories (and Hall modules thereover) and exhibit them as a categorification of classical Hall and Hecke algebras (and certain modules thereover). The input of the (functorial!) construction are simplicial groupoids satisfying the 2-Segal conditions (as introduced by Dyckerhoff and Kapranov [DKa]), the main examples come from Waldhausen's S-construction. To treat the case of modules, we introduce a relative version of the 2-Segal conditions.Furthermore, we generalize a classical result about the representation theory of symmetric groups to the case of wreath product groups: We construct a monoidal equivalence between the category of complex G ≀ S n -representations (for a fixed finite group G and varying n ∈ N) and the category of "Gequivariant" polynomial functors; we use this equivalence to prove a version of Schur-Weyl duality for wreath products.

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Section: The Canonical Isomorphism Hall(a)mentioning

confidence: 99%

Hall monoidal categories and categorical modules

Walde

2016

Preprint

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“…Section 2.2.8) to derive the following characterization. A detailed proof can be found in [32,Thm. 4].…”

Section: 43mentioning

confidence: 99%

Sheaves and $K$-theory for $\mathbb{F}_1$-schemes

Chu,

Lorscheid,

Santhanam

2010

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This paper is devoted to the open problem in F 1 -geometry of developing K-theory for F 1 -schemes. We provide all necessary facts from the theory of monoid actions on pointed sets and we introduce sheaves for M 0 -schemes and F 1 -schemes in the sense of Connes and Consani. A wide range of results hopefully lies the background for further developments of the algebraic geometry over F 1 . Special attention is paid to two aspects particular to F 1 -geometry, namely, normal morphisms and locally projective sheaves, which occur when we adopt Quillen's Q-construction to a definition of G-theory and K-theory for F 1 -schemes. A comparison with Waldhausen's S • -construction yields the ring structure of K-theory. In particular, we generalize Deitmar's K-theory of monoids and show that K * (Spec F 1 ) realizes the stable homotopy of the spheres as a ring spectrum.

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On the Hall algebra of coherent sheaves on P^1 over F_1

Cited by 2 publications

References 12 publications

Hall monoidal categories and categorical modules

Hall monoidal categories and categorical modules

Sheaves and $K$-theory for $\mathbb{F}_1$-schemes

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