Abstract
The main ingredient about generalized Clifford algebras (GCAs) in this paper is viewing them as algebras in certain symmetric linear categories of graded vector spaces. With this new view, we provide new derivations of some known results in a more direct manner. In particular, we prove the periodicity results on GCAs and propose a generalized Clifford process and show that any GCA can be obtained by repeating the process.
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Ablamowicz, R., Fauser, B.: Mathematics of CLIFFORD: a Maple package for Clifford and Grassmann algebras. Adv. Appl. Clifford Algebras 15(2), 157–181 (2005)
Ablamowicz, R., Fauser, B.: On the transposition anti-involution in real Clifford algebras I: the transposition map. Linear Multilinear Algebra 59(12), 1331–1358 (2011)
Ablamowicz, R., Fauser, B.: On the transposition anti-involution in real Clifford algebras II: stabilizer groups of primitive idempotents. Linear Multilinear Algebra 59(12), 1359–1381 (2011)
Ablamowicz, R., Fauser, B.: On the transposition anti-involution in real Clifford algebras III: the automorphism group of the transposition scalar product on spinor spaces. Linear Multilinear Algebra 60(6), 621–644 (2012)
Ablamowicz, R., Fauser, B.: Using periodicity theorems for computations in higher dimensional Clifford algebras. Adv. Appl. Clifford Algebrs 24(2), 569–587 (2014)
Ablamowicz, R., Fauser, B.: CLIFFORD: a Maple package for Clifford and Grassmann algebras (2016). http://math.tntech.edu/rafal/
Albuquerque, H., Majid, S.: Quasialgebra structure of the octonions. J. Algebra 220, 188–224 (1999)
Albuquerque, H., Majid, S.: Clifford algebras obtained by twisting of group algebras. J. Pure Appl. Algebra 171, 133–148 (2002)
Bulacu, D.: A Clifford algebra is a weak Hopf algebra in a suitable symmetric monoidal category. J. Algebra 332, 244–284 (2011)
Cheng, T., Huang, H.-L., Yang, Y.: Generalized Clifford algebras as algebras in suitable symmetric linear Gr-categories. SIGMA Symmetry Integr. Geom. Methods Appl. 12, 004, 11 (2016)
Cheng, T., Huang, H.-L., Yang, Y., Zhang, Y.-H.: The octonions form an Azumaya algebra in certain braided linear Gr-categories. Adv. Appl. Clifford Algebras 27, 1055–1064 (2017)
Hu, Y.-Q., Huang, H.-L., Zhang, C.: \({\mathbb{Z}}_2^n\)-graded quasialgebras and the Hurwitz problem on compositions of quadratic forms. Trans. Am. Math. Soc. 370(1), 241–263 (2018)
Koç, C.: Clifford algebras of \(d\)-forms. Turk. J. Math. 19(3), 301–311 (1995)
Koç, C.: C-lattices and decompositions of generalized Clifford algebras. Adv. Appl. Clifford Algebras 20, 313–320 (2010)
Morris, A.O.: On a generalized Clifford algebra. Q. J. Math. Oxf. Ser. 18(2), 7–18 (1967)
Morris, A.O.: On a generalized Clifford algebra(II). Q. J. Math. Oxf. Ser. 19(2), 287–299 (1968)
Thomas, E.: A generalization of Clifford algebras. Glasg. Math. J. 15, 74–78 (1974)
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The authors would like to express their sincere thanks to the referees for their kind suggestions which led to a great improvement of this manuscript.
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Communicated by Rafał Abłamowicz.
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This research was supported by NSFC 11701339 and 11701468.
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Cheng, T., Li, H. & Yang, Y. A New View of Generalized Clifford Algebras. Adv. Appl. Clifford Algebras 29, 42 (2019). https://doi.org/10.1007/s00006-019-0966-z
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DOI: https://doi.org/10.1007/s00006-019-0966-z