2024/03/11 by Alfons Van Daele, Van Daele, Alfons
Mathematics · #16T05 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2403.06863
openalex publication_date 2024/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a non-degenerate algebra over the complex numbers and Δ a homomorphism from A to the multiplier algebra M(A⊗ A). Consider the linear maps T1 and T2 from A⊗ A to M(A⊗ A) defined by T1(a⊗ b)=Δ(a)(1⊗ b) \qquadand T2(c⊗ a)=(c⊗ 1)Δ(a). The pair (A,Δ) is a multiplier Hopf algebra if these two maps have range in A⊗ A and are bijections from A⊗ A to itself. In our recent paper on the Larson-Sweedler theorem, single sided multiplier Hopf algebras emerge in a natural way. For this case, instead of requiring the above for the maps T1 and T2, we now have this property for the maps T1 and T4 or for T2 and T3 where T3(a⊗ b)=(1⊗ b)Δ(a) \qquadand T4(c⊗ a)=Δ(a)(c⊗ 1). As it turns out, also for these single sided multiplier Hopf algebras, the existence of a unique counit and antipode can be proven. In fact, rather surprisingly, using the properties of the antipode, one can actually show that for a single sided multiplier Hopf algebra all four canonical maps are bijections from A⊗ A to itself. In other words, (A,Δ) is automatically a regular multiplier Hopf algebra. We take the advantage of this approach to reconsider some of the known results for a regular multiplier Hopf algebra.