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• W2023401178 abstract "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<!-- Δ --></mml:mi> <mml:annotation encoding=application/x-tex>Delta</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 otimes 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 otimes 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<!-- Δ --></mml:mi> <mml:annotation encoding=application/x-tex>Delta</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<!-- Δ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(A,Delta )</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<!-- 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>(Delta 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 in 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 in 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>ast</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>ast</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra." @default.
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• W2023401178 date "1994-01-01" @default.
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• W2023401178 title "Multiplier Hopf algebras" @default.
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