### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 004, 11 pages      arXiv:1510.04408      https://doi.org/10.3842/SIGMA.2016.004

### Generalized Clifford Algebras as Algebras in Suitable Symmetric Linear Gr-Categories

Tao Cheng ab, Hua-Lin Huang a and Yuping Yang a
a) School of Mathematics, Shandong University, Jinan 250100, China
b) School of Mathematical Science, Shandong Normal University, Jinan 250014, China

Received October 22, 2015, in final form January 06, 2016; Published online January 12, 2016

Abstract
By viewing Clifford algebras as algebras in some suitable symmetric Gr-categories, Albuquerque and Majid were able to give a new derivation of some well known results about Clifford algebras and to generalize them. Along the same line, Bulacu observed that Clifford algebras are weak Hopf algebras in the aforementioned categories and obtained other interesting properties. The aim of this paper is to study generalized Clifford algebras in a similar manner and extend the results of Albuquerque, Majid and Bulacu to the generalized setting. In particular, by taking full advantage of the gauge transformations in symmetric linear Gr-categories, we derive the decomposition theorem and provide categorical weak Hopf structures for generalized Clifford algebras in a conceptual and simpler manner.

Key words: generalized Clifford algebra; symmetric Gr-category; twisted group algebra.

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References

1. Albuquerque H., Majid S., Quasialgebra structure of the octonions, J. Algebra 220 (1999), 188-224, math.QA/9802116.
2. Albuquerque H., Majid S., Clifford algebras obtained by twisting of group algebras, J. Pure Appl. Algebra 171 (2002), 133-148, math.QA/0011040.
3. Bulacu D., The weak braided Hopf algebra structure of some Cayley-Dickson algebras, J. Algebra 322 (2009), 2404-2427.
4. Bulacu D., A Clifford algebra is a weak Hopf algebra in a suitable symmetric monoidal category, J. Algebra 332 (2011), 244-284.
5. Drinfeld V.G., Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419-1457.
6. Huang H.-L., Liu G., Ye Y., The braided monoidal structures on a class of linear Gr-categories, Algebr. Represent. Theory 17 (2014), 1249-1265, arXiv:1206.5402.
7. Huang H.-L., Liu G., Ye Y., On braided linear Gr-categories, arXiv:1310.1529.
8. Jagannathan R., On generalized Clifford algebras and their physical applications, in The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, Springer, New York, 2010, 465-489, arXiv:1005.4300.
9. Kassel C., Quantum groups, Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York, 1995.
10. Knus M.A., A generalisation of Clifford algebras, Math. Z. 110 (1969), 171-176.
11. Long F.W., Generalized Clifford algebras and dimodule algebras, J. London Math. Soc. 13 (1976), 438-442.
12. Majid S., Algebras and Hopf algebras in braided categories, in Advances in Hopf Algebras (Chicago, IL, 1992), Lecture Notes in Pure and Appl. Math., Vol. 158, Dekker, New York, 1994, 55-105, q-alg/9509023.
13. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
14. Milnor J., Introduction to algebraic $K$-theory, Annals of Mathematics Studies, Vol. 72, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1971.
15. Morier-Genoud S., Ovsienko V., A series of algebras generalizing the octonions and Hurwitz-Radon identity, Comm. Math. Phys. 306 (2011), 83-118, arXiv:1003.0429.
16. Morris A.O., On a generalized Clifford algebra, Quart. J. Math. 18 (1967), 7-12.
17. Morris A.O., On a generalized Clifford algebra. II, Quart. J. Math. 19 (1968), 289-299.
18. Thomas E., A generalization of Clifford algebras, Glasgow Math. J. 15 (1974), 74-78.