2023/12/28 by Michael Lau, Olivier Mathieu, Lau, Michael +1
Mathematics · #17B10 (primary) #17B60 #17C05 #17C50 (secondary) #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2312.16766
openalex created_date 2023/12/02 · openalex publication_date 2023/12/28 · openalex updated_date 2026/07/28
We consider bounded weight modules for the universal central extension \mathfraksl2(J) of the Tits-Kantor-Koecher algebra of a unital Jordan algebra J. Universal objects called Weyl modules are introduced and studied, and a combinatorial dominance criterion is given for analogues of highest weights. Specializing J to the free Jordan algebra J(r) of rank r, the category Cfin of finite-dimensional ℤ-graded \mathfraksl2(J)-modules shares many properties with the representation theory of algebraic groups. Using a deep result of Zelmanov, we show that this subcategory admits Weyl modules. By analogy, we conjecture that Cfin is a highest weight category. The resulting homological properties would then imply cohomological vanishing results previously conjectured as a way of determining graded dimensions of free Jordan algebras.