Factorization of quasitriangular structures of smash biproduct bialgebras

2025/05/07 by Fujun Wang, Wang, Fujun
Mathematics · #16S40 #16T10 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2505.04188

openalex publication_date 2025/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the factorization and reconstruction of quasitriangular structures of smash biproduct bialgebras. Let A_τ×_σB be a smash biproduct bialgebra. Under condition that σ is right conormal, we prove that A_τ×_σB is quasitriangular if and only if there exists a set of normalized elements W∈ B⊗ B, X∈ A⊗ B, Y∈ B⊗ A and Z∈ A⊗ A satisfying a certain series of identities. In this case, the quasitriangular structure of A_τ×_σB is given as ∑ Z ¹_τ1τ2X¹_τ3X¹⊗ W¹Y¹⊗ Z² Y²_σ1σ2ε_B(1_Bτ1σ2 X²_σ1)⊗1_Bτ21_Bτ3X²W². Our result generalizes the similar results for Radford's biproduct Hopf algebras studied by L. Zhao and W. Zhao, for bicrossproduct Hopf algebras studied by Zhao, Wang and Jiao, and for the dual Hopf algebras of double cross product Hopf algebras studied by Jiao.

Citations

• Classification of Finite-Dimensional Triangular Hopf Algebras With The Chevalley Property
• On the Classification of Finite-Dimensional Triangular Hopf Algebras
• Oriented Quantum Algebras and Invariants of Knots and Links
• Quantum Invariants of Knots and 3-Manifolds

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