2012/05/31 by Andrew R. Linshaw · 20 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Combinatorics #Conjecture #Current algebra #Group theory #Invariant (physics) #Lie algebra #Mathematical physics #Mathematics #Nilpotent #Physics #Pure mathematics #Quotient #Reductive group #Subalgebra #Symplectic geometry #Vertex operator algebra #math.QA #math.RT
paper · pdf · doi:10.1007/s00220-015-2502-x
published in Communications in Mathematical Physics 345(2), 545-585 (Springer Science+Business Media) · Final version
arxiv created 2015/11/23 · openalex publication_date 2015/11/30 · arxiv updated 2020/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The Lie algebra D of regular differential operators on the circle has a universal central extension D. The invariant subalgebra D+ under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum D+-module with central charge c∈ℂ, and its irreducible quotient Vc, possess vertex algebra structures, and Vc has a nontrivial structure if and only if c∈ (1)/(2)ℤ. We show that for each integer n>0, Vn/2 and V-n are W-algebras of types W(2,4,…,2n) and W(2,4,…, 2n2+4n), respectively. These results are formal consequences of Weyl's first and second fundamental theorems of invariant theory for the orthogonal group O(n) and the symplectic group Sp(2n), respectively. Based on Sergeev's theorems on the invariant theory of Osp(1,2n) we conjecture that V-n + 1/2 is of type W(2,4,…, 4n2+8n+2), and we prove this for n=1. As an application, we show that invariant subalgebras of βγ-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.