Braided and coboundary monoidal categories

In this expository paper, we discuss and compare the notions of braided and coboundary monoidal categories. Coboundary monoidal categories are analogues of braided monoidal categories in which the role of the braid group is replaced by the cactus group. We focus on the categories of representations of quantum groups and crystals and explain how while the former is a braided monoidal category, this structure does not pass to the crystal limit. However, the categories of representations of quantum groups of finite type also possess the structure of a coboundary category which does behave well in the crystal limit. We explain this construction and also a recent interpretation of the coboundary structure using quiver varieties. This geometric viewpoint allows one to show that the category of crystals is in fact a coboundary monoidal category for arbitrary symmetrizable Kac-Moody type.Comment: 24 pages; v2: minor typos corrected. To appear in the proceedings of the conference "Algebras, Representations and Applications" (Lie and Jordan Algebras, their Representations and Applications - III) in Honour of Prof. Ivan Shestakov's 60th Birthda

Enantioselective synthesis of (+)-petromyroxol, enabled by rhodium-catalyzed denitrogenation and rearrangement of a 1-sulfonyl-1,2,3-triazole

Petromyroxol is a non-racemic mixture of enantiomeric oxylipids isolated from water conditioned with larval sea lamprey. The (+)-antipode exhibits interesting biological properties but only 1 mg was isolated from >100000 L of water. Recently, transition metal-catalyzed denitrogenation of 1-sulfonyl-1,2,3-triazoles has emerged as a powerful strategy for the synthesis of value-added products, including efficient diastereocontrolled construction of tetrahydrofurans. This methodology enabled the rapid development of the first synthesis of (+)-petromyroxol in 9 steps and 20% overall yield from a readily accessible starting material

Quiver varieties and Demazure modules

Using subvarieties, which we call Demazure quiver varieties, of the quiver varieties of Nakajima, we give a geometric realization of Demazure modules of Kac-Moody algebras with symmetric Cartan data. We give a natural geometric characterization of the extremal weights of a representation and show that Lusztig's semicanonical basis is compatible with the filtration of a representation by Demazure modules. For the case of affine sl_2, we give a characterization of the Demazure quiver variety in terms of a nilpotency condition on quiver representations and an explicit combinatorial description of the Demazure crystal in terms of Young pyramids.Comment: 14 pages, 2 figures; v2: Minor corrections and reference added; v3: Proofs of Proposition 6.1 and Theorem 8.1 corrected. This version incorporates an Erratum to the published versio

NP coordination in underspecified scope representations

Accurately capturing the quantifier scope behaviour of coordinated NPs can be problematic for underspecification systems that define constraints over semantic constructors. We present an extension to a hole-semantics like language that allows a natural representation of coordinated NPs, and a translation from partial scope requirements into constraints on the constructors. We conclude that the efficient decision procedures developed for constraints on semantic constructors enable the possible meanings of sentences containing coordinated NPs to be fully underspecified

Rhodium(II)-catalyzed stereocontrolled synthesis of 2-tetrasubstituted saturated heterocycles from 1-Sulfonyl-1,2,3-triazoles

Rhodium(II) acetate catalyzes the denitrogenative transformation of 4-substituted 1-sulfonyl-1,2,3-triazoles with pendent allyl and propargyl ethers and thioethers to onium ylides that undergo [2,3]-sigmatropic rearrangement to give 2-tetrasubstituted heterocycles with high yield and diastereoselectivity