On the Unitary Representations of the Braid Group B 6

Malak M. Dally and Mohammad N. Abdulrahim
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Malak M. Dally: Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, P.O. Box 11-5020 Beirut, Lebanon
Mohammad N. Abdulrahim: Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, P.O. Box 11-5020 Beirut, Lebanon

Mathematics, 2019, vol. 7, issue 11, 1-7

Abstract: We consider a non-abelian leakage-free qudit system that consists of two qubits each composed of three anyons. For this system, we need to have a non-abelian four dimensional unitary representation of the braid group B 6 to obtain a totally leakage-free braiding. The obtained representation is denoted by ρ . We first prove that ρ is irreducible. Next, we find the points y ∈ C * at which the representation ρ is equivalent to the tensor product of a one dimensional representation χ ( y ) and μ ^ 6 ( ± i ) , an irreducible four dimensional representation of the braid group B 6 . The representation μ ^ 6 ( ± i ) was constructed by E. Formanek to classify the irreducible representations of the braid group B n of low degree. Finally, we prove that the representation χ ( y ) ⊗ μ ^ 6 ( ± i ) is a unitary relative to a hermitian positive definite matrix.

Keywords: braid group; unitarity; equivalence; irreducibility (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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