An Algebraic Framework for Group Duality
An Algebraic Framework for Group Duality
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DOI：
10.1006/aima.1998.1775
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发表时间：
1998-12
影响因子：
1.7
通讯作者：
A. V. Daele
中科院分区：
文献类型：
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作者：
A. V. Daele
Hopf代数是一对（A， Δ），其中Hopf代数是一个具有恒等式的结合代数andΔa同态format_1满足一定条件。如果我们放弃A有单位元的假设，如果我们allowΔto在所谓的乘数代数（A⊗A）中有值，我们就得到了Hopf代数概念的自然扩展。我们称之为乘数Hopf代数。具有启发性的例子是群上有限支持的复函数的代数，其乘法定义为群内乘积的对偶。同样，对于这些乘子Hopf代数，线性泛函（在Hopf代数理论中称为积分）有一个自然的左右不变性概念。我们证明，如果存在这样的不变泛函，它们是唯一的（直到一个标量）和忠实的。对于具有不变泛函的正则乘子Hopf代数（a， Δ）（即具有可逆对映），我们以规范的方式构造对偶（Â， Δ）。它又是一个带不变泛函的正则乘子Hopf代数。还证明了（Â， Δ）的对偶与原乘子Hopf代数（A， Δ）是规范同构的。在这里可以概括抽象谐波分析的许多方面。我们可以定义傅里叶变换；可以证明Plancherel公式。因为任何有限维Hopf代数都是正则乘子Hopf代数并且具有不变泛函，所以对偶定理适用于所有有限维Hopf代数。然后它就符合了Hopf代数的对偶性。但是我们的乘子Hopf代数的范畴在某种程度上也包括离散（量子）群和紧（量子）群。我们的对偶包括离散量子群和紧量子群之间的对偶。特别地，它包括紧阿贝尔群和离散阿贝尔群之间的对偶性。我们理论的一个很好的特点是我们将这种对偶扩展到了非阿贝尔情况，但是在一个范畴内。在本文的最后一节中，我们介绍了紧型代数和离散型代数。我们也证明了它们是对偶的。我们用一个足够普遍的例子来说明我们理论的大多数不同特征。在这个范畴内也可以构造德林菲尔德的量子双元。这提供了更广泛的示例类别。因此，在这个设置中，我们得到的不仅仅是紧致和离散量子。
Abstract A Hopf algebra is a pair (A, Δ) whereAis an associative algebra with identity andΔa homomorphism formAtoA⊗Asatisfying certain conditions. If we drop the assumption thatAhas an identity and if we allowΔto have values in the so-called multiplier algebraM(A⊗A), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A, Δ) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (Â, Δ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (Â, Δ) is canonically isomorphic with the original multiplier Hopf algebra (A, Δ). It is possible to generalize many aspects of abstract harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting.