EXTENDING TABLE ALGEBRAS TO HOPF ALGEBRAS

Yıl 2019, , 13 - 28, 11.07.2019

Allen Herman Gurmail Singh

https://doi.org/10.24330/ieja.586882

Cited By: 1

Öz

Let $A$ be a table algebra with standard basis $\mathbf{B}$, multiplication $\mu$, unit map $\eta$, skew-linear involution $*$, and degree map $\delta$.  In this article we study the possible coalgebra structures $(A,\Delta, \delta)$ on $A$ for which $(A, \mu, \eta, \Delta, \delta)$ becomes a Hopf algebra with respect to some antipode.  We show that such Hopf algebra structures are not always available for noncommutative table algebras.  On the other hand, commutative table algebras will always have a Hopf algebra structure induced from an algebra-isomorphic group algebra.  To illustrate our approach, we derive Hopf algebra comultiplications on table algebras of dimension 2 and 3.

Anahtar Kelimeler

Table algebra, Hopf algebra, fusion ring, association scheme

Kaynakça

• H. I. Blau, Table algebras, European J. Combin., 30(6) (2009), 1426-1455.
• A. Hanaki, Character products of association schemes, J. Algebra, 283(2) (2005), 596-603.
• A. Herman, M. Muzychuk and B. Xu, The recognition problem for table algebras and reality-based algebras, J. Algebra, 479 (2017), 173-191.
• A. Masuoka, Semisimple Hopf algebras of dimension 6,8, Israel J. Math., 92 (1995), 361-373.
• A. Masuoka, The p^n theorem for semisimple Hopf algebras, Proc. Amer. Math. Soc., 124 (1996), 735-737.
• D. E. Radford, Hopf Algebras, Series on Knots and Everything, 49, World Scientic Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.
• Y. Zhu, Hopf algebras of prime dimension, Internat. Math. Res. Notices, 1 (1994), 53-59.