Webs of type Q

Gordon C Brown,Jonathan R Kujawa

doi:10.5802/alco.191

Abstract

Howe dualities lead to diagrammatic categories which describe the representations of Lie-type objects as a monoidal category (that is, via generators and relations). Applying this philosophy to the type Q Howe duality of Cheng–Wang and Sergeev, we introduce diagrammatic web supercategories of type Q via generators and relations and show they describe the full subcategory of supermodules for the Lie superalgebra of type Q given by the tensor products of supersymmetric tensor powers of the natural supermodule.

Highlights

For g = sl(n) a diagrammatic presentation for the monoidal category generated by the fundamental representations was conjectured for n = 4 by Kim [23] and for general n by Morrison [25]
In 2014, Cautis–Kamnitzer–Morrison gave a complete combinatorial description of the monoidal category generated by the fundamental representations of U (sl(n)) [11]; that is, the full subcategory of all modules of the form Λk1 (Vn)⊗· · ·⊗Λkt (Vn), where Vn is the natural module for sl(n)
In Definition 4.1 we introduce the supercategory of upward oriented webs of type Q, q-Web↑

Summary

Introduction

Kuperberg completed the rank 2 case by giving a diagrammatic presentation for the monoidal subcategory generated by the fundamental representations [24]. In 2014, Cautis–Kamnitzer–Morrison gave a complete combinatorial description of the monoidal category generated by the fundamental representations of U (sl(n)) [11]; that is, the full subcategory of all modules of the form Λk1 (Vn)⊗· · ·⊗Λkt (Vn), where Vn is the natural module for sl(n). Perhaps of greater significance is the method of proof used therein They show that a diagrammatic description of the category follows from a skew Howe duality between gl(m) and sl(n). Centralizing actions are understood to naturally lead to presentations of monoidal categories. This philosophy has since been applied in a number of settings.

Monoidal supercategories

Lie superalgebras of type Q

Upward Webs

Oriented Webs

Main Theorems