Hopf-Frobenius Algebras and a Simpler Drinfeld Double
@article{Collins2019HopfFrobeniusAA,
title={Hopf-Frobenius Algebras and a Simpler Drinfeld Double},
author={Joseph Collins and Ross Duncan},
journal={Electronic Proceedings in Theoretical Computer Science},
year={2019}
}The ZX-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of $†$-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, Hopf-Frobenius algebras, starting from a single Hopf algebra which is not necessarily commutative or cocommutative. We provide the necessary and sufficient condition for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf…
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References
SHOWING 1-10 OF 40 REFERENCES
Interacting Frobenius Algebras are Hopf
- Mathematics2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2016
Theories featuring the interaction between a Frobenius algebra and a Hopf algebra are shown to form Hopf algebras, and the system of Bonchi et al as a subtheory in the prime power dimensional case is recovered.
Kitaev Lattice Models as a Hopf Algebra Gauge Theory
- Mathematics
- 2016
We prove that Kitaev’s lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to the combinatorial quantisation of Chern–Simons theory for the Drinfeld double D(H). This shows…
Frobenius Algebras and 2-D Topological Quantum Field Theories
- Mathematics
- 2004
This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The precise…
Quantum doubles in monoidal categories
- Mathematics
- 2000
Let H be a Hopf algebra in a rigid symmetric monoidal category C then the evaluation map τis a convolution-invertible skew pairing. In the previous paper, we constructed a Hopf algebra D(H)=H ⋈ r H…
Quantum field theory meets Hopf algebra
- Mathematics
- 2006
This paper provides a primer in quantum field theory (QFT) based on Hopf algebra and describes new Hopf algebraic constructions inspired by QFT concepts. The following QFT concepts are introduced:…
Infinite-dimensional Categorical Quantum Mechanics
- MathematicsQPL
- 2016
Non-standard analysis is used to define a category suitable for categorical quantum mechanics in arbitrary separable Hilbert spaces, and it is shown that standard bounded operators can be suitably embedded in it.