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Quantum Physics and LinguisticsA Compositional, Diagrammatic Discourse$
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Chris Heunen, Mehrnoosh Sadrzadeh, and Edward Grefenstette

Print publication date: 2013

Print ISBN-13: 9780199646296

Published to Oxford Scholarship Online: May 2013

DOI: 10.1093/acprof:oso/9780199646296.001.0001

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Some Graphical Aspects of Frobenius Algebras

Some Graphical Aspects of Frobenius Algebras

(p.22) CHAPTER 2 Some Graphical Aspects of Frobenius Algebras
Quantum Physics and Linguistics

Fauser Bertfried

Oxford University Press

Frobenius algebras surface at many places in mathematics and physics. Quite recently, using a convenient graphical notation, Frobenius algebras have been used to investigate foundational issues of quantum theory— references will be given below. Also, as shown elsewhere in this book, Frobenius algebras emerge in the semantic analysis of natural languages. The aim of this chapter is to present the basic results of Frobenius algebras and their relation to finite-dimensional Hopf algebras, with special emphasis on use of graphical notation. Frobenius structures, which are related to ring theory, need to be considered too. Frobenius structures encode such notions as semi-simplicity and separability of rings. We occasionally need to extend the graphical calculus to encode properties of underlying rings, but will not venture properly into 2-categorical notions. Moreover, no new results may be found in this chapter, and far from everything that is known about the subject is covered here. However, in passing we will give some pointers to the literature, which unfortunately is far too large to be considered completely.

Keywords:   frobenius algebras, graphical notation, hopf algebras, rings, quantum groups

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