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Generalized Clifford Algebras as Algebras in Suitable Symmetric Linear Gr-Categories
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: generalized Clifford algebra; symmetric Gr-category; twisted group algebra. 
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Tao Cheng; Hua-Lin Huang; Yuping Yang. Generalized Clifford Algebras as Algebras in Suitable Symmetric Linear Gr-Categories. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a3/

[1] Albuquerque H., Majid S., “Quasialgebra structure of the octonions”, J. Algebra, 220 (1999), 188–224, arXiv: math.QA/9802116 | DOI | MR | Zbl

[2] Albuquerque H., Majid S., “Clifford algebras obtained by twisting of group algebras”, J. Pure Appl. Algebra, 171 (2002), 133–148, arXiv: math.QA/0011040 | DOI | MR | Zbl

[3] Bulacu D., “The weak braided Hopf algebra structure of some Cayley–Dickson algebras”, J. Algebra, 322 (2009), 2404–2427 | DOI | MR | Zbl

[4] Bulacu D., “A Clifford algebra is a weak Hopf algebra in a suitable symmetric monoidal category”, J. Algebra, 332 (2011), 244–284 | DOI | MR | Zbl

[5] Drinfeld V. G., “Quasi-Hopf algebras”, Leningrad Math. J., 1 (1990), 1419–1457 | MR | Zbl

[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 | DOI | MR | Zbl

[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”, The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, Springer, New York, 2010, 465–489, arXiv: 1005.4300 | DOI | MR | Zbl

[9] Kassel C., Quantum groups, Graduate Texts in Mathematics, 155, Springer-Verlag, New York, 1995 | DOI | MR | Zbl

[10] Knus M. A., “A generalisation of Clifford algebras”, Math. Z., 110 (1969), 171–176 | DOI | MR | Zbl

[11] Long F. W., “Generalized Clifford algebras and dimodule algebras”, J. London Math. Soc., 13 (1976), 438–442 | DOI | MR | Zbl

[12] Majid S., “Algebras and Hopf algebras in braided categories”, Advances in Hopf Algebras (Chicago, IL, 1992), Lecture Notes in Pure and Appl. Math., 158, Dekker, New York, 1994, 55–105, arXiv: q-alg/9509023 | MR | Zbl

[13] Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995 | DOI | MR | Zbl

[14] Milnor J., Introduction to algebraic $K$-theory, Annals of Mathematics Studies, 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971 | MR | Zbl

[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 | DOI | MR | Zbl

[16] Morris A. O., “On a generalized Clifford algebra”, Quart. J. Math., 18 (1967), 7–12 | DOI | MR | Zbl

[17] Morris A. O., “On a generalized Clifford algebra, II”, Quart. J. Math., 19 (1968), 289–299 | DOI | MR | Zbl

[18] Thomas E., “A generalization of Clifford algebras”, Glasgow Math. J., 15 (1974), 74–78 | DOI | MR | Zbl