The Center of Small Quantum Groups I: The Principal Block in Type A

2016/04/30 by Anna Lachowska, You Qi · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Block (permutation group theory) #Center (category theory) #Combinatorics #Conjecture #Crystallography #Diagonal #Geometry #Group (periodic table) #Lie algebra #Mathematics #Physics #Pure mathematics #Quantum #Quantum group #Quantum mechanics #Root of unity #Space (punctuation) #Type (biology) #math.AG #math.QA #math.RT #msc:14L99 #msc:17B37 #msc:20G05

paper · pdf · doi:10.1093/imrn/rnx062

published as Int. Math. Res. Not. IMRN 2018, no. 20, 6349-6405 · 44 pages, V3 contains corrections from the referee and updated references, comments welcome, International Mathematics Research Notices 2017

openalex publication_date 2017/03/01 · arxiv created 2017/04/21 · arxiv updated 2021/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We develop an elementary algebraic method to compute the center of the principal block of a small quantum group associated with a complex semisimple Lie algebra at a root of unity. The cases of |\mathfraksl₃| and |\mathfraksl₄| are computed explicitly. This allows us to formulate the conjecture that, as a bigraded vector space, the center of a regular block of the small quantum |\mathfraksl_m| at a root of unity is isomorphic to Haiman’s diagonal coinvariant algebra for the symmetric group |S_m|⁠.

Citations

• Representations of Semisimple Lie Algebras in the BGG Category 𝒪
• Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras
• The small quantum group and the Springer resolution
• Invariants of 3-manifolds via link polynomials and quantum groups
• Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center
• The center of small quantum groups II: singular blocks
• An approach to categorification of some small quantum groups
• Algebraic Geometry

Cited by

• The center of small quantum groups II: singular blocks