1994/01/01 by Alfons Van Daele · 8 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Algorithm #Annotation #Artificial intelligence #Computer science #Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.1090/s0002-9947-1994-1220906-5
openalex publication_date 1994/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21
In this paper we generalize the notion of Hopf algebra. We consider an algebra <italic>A</italic> , with or without identity, and a homomorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:annotation encoding="application/x-tex">Δ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from <italic>A</italic> to the multiplier algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis upper A circled-times upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo> ⊗ </mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M(A ⊗ A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A circled-times upper A"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo> ⊗ </mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A ⊗ A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We impose certain conditions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:annotation encoding="application/x-tex">Δ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (such as coassociativity). Then we call the pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A comma normal upper Delta right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(A,Δ )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a multiplier Hopf algebra. The motivating example is the case where <italic>A</italic> is the algebra of complex, finitely supported functions on a group <italic>G</italic> and where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Delta f right-parenthesis left-parenthesis s comma t right-parenthesis equals f left-parenthesis s t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(Δ f)(s,t) = f(st)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s comma t element-of upper G"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">s,t ∈ G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper A"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f ∈ A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove the existence of a counit and an antipode. If <italic>A</italic> has an identity, we have a usual Hopf algebra. We also consider the case where <italic>A</italic> is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="asterisk"> <mml:semantics> <mml:mo> ∗ </mml:mo> <mml:annotation encoding="application/x-tex">∗</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra. Then we show that (a large enough) subspace of the dual space can also be made into a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="asterisk"> <mml:semantics> <mml:mo> ∗ </mml:mo> <mml:annotation encoding="application/x-tex">∗</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra.