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'Trace Functor and Categorified Quantum Sln' Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2017 Trace functor and categorified quantum sln Guliyev, Zaur Abstract: M. Khovanov and A. Lauda [35] define a 2-category U such that the split Grothendieck group K0(U) is isomorphic to an integral version of the universal enveloping algebra U(sl_n), n 2. The trace Tr is a decategorification functor that is an alternative to the usual decategorification givenby K0. We compute the trace TrU using the diagrammatic presentation of U. We find a graded algebra homomor- phism between TrU and the current algebra U(sl_n[t]), which is the universal enveloping algebra of the Lie algebra sl_n C[t]. More recently, this homomorphism is proven to be an isomorphism in [5]. Thus, TrU has a richer structure than K0(U) = U(sl_n). A 2-representation is a categorification of the notion of a representation. In particular, a 2-representation of U is a 2-functor from U to a linear, additive 2-category. Our next result provides a framework on how to get a current algebra module from any 2-representation of U. We are interested in the 2-representation, defined by Khovanov-Lauda using bimodules over cohomology rings of flag varieties. This 2-representation induces an action of the current algebra U(sl_n[t]) on the cohomology rings. We explicitly compute the action of U(sl_n[t]) generators using the trace functor. It turns out that the obtained current algebra module is related to another family of U(sl_n[t])-modules, called local Weyl modules. Using known results about the cohomology rings, we are able to provide a new proof of the character formula for the local Weyl modules. Finally, we apply the trace functor to the Bao-Shan-Wang-Webster [2] categorification U J of the coideal sub- J algebra of quantized sl2n + 1. We compute the trace of the 2-category U in this thesis. We define a new algebra, called the current coideal algebra, and we show that TrU J is homomorphic to the cur- rent coideal algebra. M. Khovanov and A. Lauda [35] definieren eine 2-Kategorie U, so dass die geteilte Grothendieck-Gruppe K0(U) zu einer integralen Version der universellen einhüllenden Algebra U(sl_n), n 2 isomorph ist. Die Spur Tr ist eine Alternative zu dem De-kategorisierungsfunktor, normalerweise gegeben durch K0. Wir berechnen TrU mithilfe der diagrammatischen Präsentation von U. Wir finden einen graduierten Algebra-Homomorphismus zwischen TrU und der Stromalgebra U(sln[t]), welche die universelle einh üllende Algebra der Lie-Algebra U(sl_n C[t]) ist. Kürzlich wurde in [5] bewiesen, dass dieser Homomorphismus sogar einen Isomorphismus darstellt. Somit hat TrU eine reichere Struktur als K0(U) = U(sl_n). Eine 2-Darstellung ist eine Kategorisierung des Begriffs der Darstellung. Insbesondere ist eine 2-Darstellung von U ein 2-Funktor aus U in eine lineare, additive 2-Kategorie. Unser nächstes Ergebnis stellt einen Rahmen dar, um aus einer beliebigen 2-Darstellung ein Stromalgebra-Modul zu erhalten. Wir interessieren uns für die 2-Darstellung, definierte von Khovanov-Lauda durch Bimod- uln über Kohomologien der Fahnenmannigfaltigkeiten. Diese 2-Darstellung induziert eine Wirking der Stromalgebra U(sl_n[t])) auf die Kohomologieringe. Wir berechnen die explizite Stromalgebrawirkung durch den Spurfunktor. Es wird festgestellt, dass der erhaltene Stromalgebramodul sich auf eine andere Familie von U(sl_n[t])-Moduln bezieht, bekannt als lokale Weyl-Moduln. Mithilfe bekannter Ergebnisse geben wir einen neuen Beweis der Charakterformeln von lokalen Weyl-Moduln. Schliesslich wenden wir den Spurfunktor auf die Bao-Shan-Wang-Webster Kategorisierung [2] U J der koidealen Unteralgebra der J quantisierten Lie-Algebra sl2n + 1 an. Wir berechnen die Spur der 2-Kategorie U in dieser Arbeit. Wir definieren eine neue Algebra, als koideale Stromalgebra genannt, und wir zeigen,U dassTr J homomorph zu der koidealen Stromalgebra ist. Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-170241 Dissertation Published Version Originally published at: Guliyev, Zaur. Trace functor and categorified quantum sln. 2017, University of Zurich, Faculty of Science. 2 Trace Functor and Categorified Quantum sln Dissertation zur zur Erlangung der naturwissenschaftlichen Doktorw¨urde (Dr. sc. nat.) vorgelegt der Mathematisch-naturwissenschaftlichen Fakult¨at der Universit¨atZ¨urich von Zaur Guliyev aus Aserbaidschan Promotionskommission Prof. Dr. Anna Beliakova (Vorsitz) Prof. Dr. Valentin F´eray Prof. Dr. Andrew Kresch Z¨urich, 2017 1 Abstract M. Khovanov and A. Lauda [35] define a 2-category U such that the split Grothendieck group K0(U) is isomorphic to an integral version of the universal enveloping algebra U(sln), n ≥ 2. The trace Tr is a decategorification functor that is an alternative to the usual decate- gorification given by K0. We compute the trace Tr U using the diagrammatic presentation of U. We find a graded algebra homomorphism between Tr U and the current algebra U(sln[t]), which is the universal enveloping algebra of the Lie algebra sln ⊗ C[t]. More recently, this homomorphism is proven to be an isomorphism in [5]. Thus, Tr U has a richer structure than K0(U)= U(sln). A 2-representation is a categorification of the notion of a representation. In particular, a 2-representation of U is a 2-functor from U to a linear, additive 2-category. Our next result provides a framework on how to get a current algebra module from any 2-representation of U. We are interested in the 2-representation, defined by Khovanov-Lauda using bimodules over cohomology rings of flag varieties. This 2-representation induces an action of the current algebra U(sln[t]) on the cohomology rings. We explicitly compute the action of U(sln[t]) generators using the trace functor. It turns out that the obtained current algebra module is related to another family of U(sln[t])-modules, called local Weyl modules. Using known results about the cohomology rings, we are able to provide a new proof of the character formula for the local Weyl modules. Finally, we apply the trace functor to the Bao-Shan-Wang-Webster [2] categorification | U of the coideal subalgebra of quantized sl2n+1. We compute the trace of the 2-category U | in this thesis. We define a new algebra, called the current coideal algebra, and we show that Tr U | is homomorphic to the current coideal algebra. 2 Zusammenfassung M. Khovanov and A. Lauda [35] definieren eine 2-Kategorie U, so dass die geteilte Grothendieck- Gruppe K0(U) zu einer integralen Version der universellen einh¨ullendenAlgebra U(sln), n ≥ 2 isomorph ist. Die Spur Tr ist eine Alternative zu dem De-kategorisierungsfunktor, normalerweise gegeben durch K0. Wir berechnen Tr U mithilfe der diagrammatischen Pr¨asentation von U. Wir finden einen graduierten Algebra-Homomorphismus zwischen Tr U und der Stromalgebra U(sln[t]), welche die universelle einh¨ullende Algebra der Lie- Algebra sln ⊗ C[t] ist. K¨urzlich wurde in [5] bewiesen, dass dieser Homomorphismus sogar einen Isomorphismus darstellt. Somit hat Tr U eine reichere Struktur als K0(U)= U(sln). Eine 2-Darstellung ist eine Kategorisierung des Begriffs der Darstellung. Insbesondere ist eine 2-Darstellung von U ein 2-Funktor aus U in eine lineare, additive 2-Kategorie. Unser n¨achstes Ergebnis stellt einen Rahmen dar, um aus einer beliebigen 2-Darstellung ein Stromalgebra-Modul zu erhalten. Wir interessieren uns f¨urdie 2-Darstellung, definierte von Khovanov-Lauda durch Bimoduln ¨uber Kohomologien der Fahnenmannigfaltigkeiten. Diese 2-Darstellung induziert eine Wirking der Stromalgebra U(sln[t]) auf die Kohomolo- gieringe. Wir berechnen die explizite Stromalgebrawirkung durch den Spurfunktor. Es wird festgestellt, dass der erhaltene Stromalgebramodul sich auf eine andere Familie von U(sln[t])-Moduln bezieht, bekannt als lokale Weyl-Moduln. Mithilfe bekannter Ergebnisse geben wir einen neuen Beweis der Charakterformeln von lokalen Weyl-Moduln. Schliesslich wenden wir den Spurfunktor auf die Bao-Shan-Wang-Webster Kategorisierung | [2] U der koidealen Unteralgebra der quantisierten Lie-Algebra sl2n+1 an. Wir berechnen die Spur der 2-Kategorie U | in dieser Arbeit. Wir definieren eine neue Algebra, als koideale Stromalgebra genannt, und wir zeigen, dass Tr U | homomorph zu der koidealen Stromal- gebra ist. 3 Contents Introduction . .6 I Trace of categorified quantum groups 18 1 Trace functor . 18 1.1 Trace of a ring . 18 1.2 Trace of a linear category . 21 1.3 Idempotent completions and split Grothendieck group . 22 1.4 Pivotal categories . 24 1.5 2-categories and trace . 25 1.6 Kar, K0 and Tr for 2-categories . 26 1.7 Horizontal trace of a 2-category . 29 2 The categorified quantum sln ......................... 31 2.1 Cartan datum for sln ......................... 31 2.2 The quantum group Uq(sln)...................... 32 2.3 The 2-category UQ(sln)......................... 33 2.4 The current algebra and the trace of U ................ 40 2.5 Proof of the homomorphism theorem . 44 2.6 Horizontal trace of U .......................... 52 II Trace and 2-representations 54 3 Local Weyl modules of current algebra . 54 3.1 Local Weyl modules . 54 3.2 Duals of local Weyl modules . 56 4 2-representations . 58 4.1 Integrable 2-representations . 58 4.2 Cyclotomic KLR algebras . 59 4.3 Web algebra and foamation 2-representation . 63 5 The 2-category PolN .............................. 67 5.1 Symmetric polynomials . 67 5.2 The 2-functor U! PolN ....................... 71 4 5.3 Z(PolN ) as a current algebra module . 76 λ 5.4 The coinvariant algebra Cν ...................... 79 III Trace of categorified quantum symmetric pairs 88 6 Categorification of the coideal algebra . 88 6.1 The coideal algebra U| ........................ 88 6.2 Coideal 2-category U | .......................... 90 7 Trace of U | ..................................
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