2022/07/22 by Alexandru Chirvăsitu, Chirvasitu, Alexandru
Computer Science · Decision Sciences · Mathematics · #16D60 #16T05 #17B05 #17B10 #Advanced Topics in Algebra #Category Theory (math.CT) #Constraint Satisfaction and Optimization #FOS: Mathematics #Fuzzy and Soft Set Theory #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2207.11338
openalex publication_date 2022/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathfrakg be a Lie algebra over an algebraically closed field \Bbbk of characteristic zero. Define the universal grading group C(\mathfrakg) as having one generator gρ for each irreducible \mathfrakg-representation ρ, one relation gπ = gρ-1 whenever π is weakly contained in the dual representation ρ^* (i.e. the kernel of π in the enveloping algebra U(\mathfrakg) contains that of ρ^*), and one relation gρ = gρ'gρ" whenever ρ is weakly contained in ρ'⊗ρ". The main result is that attaching to an irreducible representation its central character gives an isomorphism between C(\mathfrakg) and the dual \mathfrakz^* of the center \mathfrakz≤ \mathfrakg when \mathfrakg is (a) finite-dimensional solvable; (b) finite-dimensional semisimple. The group C(\mathfrakg) is also trivial when the enveloping algebra U(\mathfrakg) has a faithful irreducible representation (which happens for instance for various infinite-dimensional algebras of interest, such as \mathfraksl(∞), \mathfrako(∞) and \mathfraksp(∞)). These are analogues of a result of Müger's for compact groups and a number of results by the author on locally compact groups, and provide further evidence for the pervasiveness of such center-reconstruction phenomena.