Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0705.2630

A geometric categorification of tensor products of $U_q(sl_2)$-modules

Hao Zheng
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give a purely geometric categorification of tensor products of finite-dimensional simple $U_q(sl_2)$-modules and $R$-matrices on them. The work is developed in the framework of category of perverse sheaves and the categorification theorems are understood as consequences of Deligne's theory of weights.

Explore related subjects

math.RTmath.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hao Zheng. 2007-06-09. A geometric categorification of tensor products of $U_q(sl_2)$-modules. https://arxiv.org/abs/0705.2630

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Stable Brauer--Thrall II$'$ conjecture for finite-dimensional Jacobian algebras

We prove that finite-dimensional Jacobian algebras associated with non-degenerate quivers with potentials satisfy the stable Brauer--Thrall II$'$ conjecture. In particular, this implies that the brick Brauer--Thrall II$'$ conjecture (also known as the $τ$-Brauer--Thrall II$'$ conjecture) holds for finite-dimensional Jacobian algebras.

math.RT↗
arXiv2026

Quasi-equivalence of relabeled positroid cluster structures

We prove the Fraser--Sherman-Bennett conjecture on the quasi-equivalence of positroid cluster structures arising from admissibly relabeled plabic graphs. We construct a tilting module from the boundary necklace and obtain a Frobenius categorification of the relabeled cluster structure. By identifying localized characters of rank-one modules with Plücker coordinates, we match the stable images of the mutable face modules with the summands of a reachable cluster-tilting object in the stable category. The induced derived equivalence yields a unimodular transformation of relative index lattices intertwining the full extended exchange matrices. These comparisons show that the identity map on the coordinate ring is a quasi-cluster isomorphism between the two cluster structures.

math.RT↗
arXiv2026

Multiplicity One and van Dijk's Conjecture for Connected Complex Symmetric Spaces

We prove regularity for symmetric pairs of connected complex reductive groups by combining invariant theory with the analysis of Schwartz distributions. For the classical residual families, a Gaussian lifting argument reduces the required equivariant vanishing to invariant theory in quadratic moment quotients: contraction and supertrace identities settle DIII and balanced CII, while Gram-matrix invariants and volume tensors handle BDI/Spin. In the exceptional EVII case, transverse slices and explicit radial differential operators establish the vanishing of the full sign-equivariant Schwartz-distribution space. Together with the reduction to irreducible factors and finite-component assembly, these results establish regularity for all symmetric pairs of connected complex reductive groups. We further establish multiplicity one for irreducible Casselman-Wallach representations whenever the complex symmetric space is connected, allowing disconnected ambient groups. In particular, van Dijk's unitary Gelfand conjecture follows in this setting. Finally, we construct an invariant tempered distribution supported on an EVII nilpotent orbit closure whose Fourier transform also has nilpotent support, showing that the EVII isotropy representation is not special.

math.RT↗