The Trace Function and Hopf Algebras

Radford, David E.

doi:10.1006/jabr.1994.1033

Cited by 124 publications

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“…which is a special case of Radford's formula given in [Rad94]. Recall that H acts on H * from the right by f ↼ h, h ′ = f, hh ′ .…”

Section: Integral Theorymentioning

confidence: 98%

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The monoidal center and the character algebra

Shimizu

2017

Journal of Pure and Applied Algebra

128

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Abstract. For a pivotal finite tensor category C over an algebraically closed field k, we define the algebra CF(C) of class functions and the internal character ch(X) ∈ CF(C) for an object X ∈ C by using an adjunction between C and its monoidal center Z(C). We also develop the theory of integrals and the Fourier transform in a unimodular finite tensor category by using the same adjunction. Our main result is that the map ch : Gr k (C) → CF(C) given by taking the internal character is a well-defined injective homomorphism of kalgebras, where Gr k (C) is the scalar extension of the Grothendieck ring of C to k. Moreover, under the assumption that C is unimodular, the map ch is an isomorphism if and only if C is semisimple.As an application, we show that the algebra Gr k (C) is semisimple if C is a non-degenerate pivotal fusion category. If, moreover, Gr k (C) is commutative, then we define the character table of C based on the integral theory. It turns out that the character table is obtained from the S-matrix if C is a modular tensor category. Generalizing corresponding results in the finite group theory, we prove the orthogonality relations and the integrality of the character table.

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“…which is a special case of Radford's formula given in [Rad94]. Recall that H acts on H * from the right by f ↼ h, h ′ = f, hh ′ .…”

Section: Integral Theorymentioning

confidence: 98%

“…for all a, h ∈ H. The former condition means that λ is a right cointegral on H. By the result of Radford [Rad94], a non-zero right cointegral on H satisfies the latter condition if and only if H is unimodular. Now we suppose that H is unimodular.…”

Section: Integral Theorymentioning

confidence: 98%

The monoidal center and the character algebra

Shimizu

2017

Journal of Pure and Applied Algebra

128

View full text Add to dashboard Cite

show abstract

“…We close this section with the remark that ba = S 2 (a)b has been shown to hold for unimodular, finite dimensional ribbon Hopf algebras (see [19]). Using (61) in (54) we conclude that it actually holds for (the wider class of) quasitriangular FDHAs.…”

Section: Lemmamentioning

confidence: 85%

“…where, in the second line, we used (19). Two useful lemmas follow Lemma 4 For A a FDHA and σ ′ = 0, it holds…”

Section: Further Propertiesmentioning

confidence: 99%