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Abstract
This work is a development of braids, tensor categories and Yang–Baxter operators. According to Li [Li, F. (1998). Weak Hopf algebras and some new solution of quantum Yang-Baxter equation. J. Algebra 208:72–100; Li, F. (2000). Solutions of Yang-Baxter equation in endomorphism semigroup and quasi-(co)braided almost bialgebras. Comm. Algebra 28(5):2253–2270], it can be seen as a continuation of studying (not necessarily invertible) solutions of the (quantum) Yang–Baxter equation. We firstly introduce the right braid monoids and discuss their properties. Then, we define pre-tensor categories, pre-tensor functors and quasi-braided pre-tensor categories, and investiage their characterizations. Three examples are given from respectively a weak Hopf algebra, a crossed S-set of a Clifford monoid and the (strict) right braid category. Two universalities of the (strict) right braid category are gotten in order to characterize a category of general Yang–Baxter operators and a quasi-braided pre-tensor category. In a pre-tensor category we build a general centre of a pre-tensor category as a generalization of a centre and show that it is a quasi-braided pre-tensor category. At the end, a categorical interpretation of the quantum quasi-double of a weak Hapf algebra is obtained under a certain condition.
Acknowledgment
Project supported by the Natural Science Foundation of Zhejiang Province of China (No.102028) and the National Natural Science Foundation of China (No.19971074). The author finished the manuscript of this article in 1998 when he was working as a visiting scholar at Institute of Mathematics, Chinese Academy of Sciences. The author takes this opportunity to express his thanks to Professor Fu-an Li for his kind assistance and advice.
Notes
#Communicated by E. Zelmanov.