2023/06/02 by Andrey Mudrov, Mudrov, Andrey, Vladimir Stukopin +1 · 2 citations
Mathematics · #17B10 #17B37 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2306.01592
openalex publication_date 2023/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be an associative algebra containing either classical or quantum universal enveloping algebra of a semi-simple complex Lie algebra \mathfrakg. We present a construction of the Mickelsson algebra Z(A,\mathfrakg) relative to the left ideal in A generated by positive root vectors. Our method employs a calculus on Hasse diagrams associated with classical or quantum \mathfrakg-modules. We give an explicit expression for a PBW basis in Z(A,\mathfrakg) in the case when A=U(\mathfraka) of a finite-dimensional Lie algebra \mathfraka⊃ \mathfrakg. For A=Uq(\mathfraka) and \mathfrakg the commutant of a Levi subalgebra in \mathfraka, we construct a PBW basis in terms of quantum Lax operators, upon extension of the ground ring of scalars to ℂ[[ℏ]].