Next Article in Journal
Single-Valued Neutrosophic Power Shapley Choquet Average Operators and Their Applications to Multi-Criteria Decision-Making
Previous Article in Journal
Cournot Duopoly Games: Models and Investigations
Open AccessArticle

On the Unitary Representations of the Braid Group B6

Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, P.O. Box 11-5020 Beirut, Lebanon
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(11), 1080; https://doi.org/10.3390/math7111080
Received: 28 September 2019 / Revised: 6 November 2019 / Accepted: 7 November 2019 / Published: 9 November 2019
(This article belongs to the Section Mathematics and Computers Science)
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 braid group; unitarity; equivalence; irreducibility
MDPI and ACS Style

Dally, M.M.; Abdulrahim, M.N. On the Unitary Representations of the Braid Group B6. Mathematics 2019, 7, 1080.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop