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Structure of a twisted group algebra for an algebra obtained by the Cayley-Dickson process. (English. Ukrainian original) Zbl 1523.16030

Ukr. Math. J. 74, No. 6, 859-870 (2022); translation from Ukr. Mat. Zh. 74, No. 6, 761-771 (2022).
Summary: Following some ideas proposed in [J. W. Bales, Missouri J. Math. Sci. 21, No. 2, 83–93 (2009; Zbl 1187.17002)], we present an algorithm for computing basis elements in an algebra obtained by the Cayley-Dickson process. As a consequence of this result, we prove that an algebra obtained with the help of the Cayley-Dickson process is a twisted group algebra for the group \(G = \mathbb{Z}_2^n\), \(n = 2^t\), \(t \in \mathbb{N} ,\) over a field \(K\) with \(\operatorname{char}(K \neq 2)\). We also present some properties and applications of the quaternion nonassociative algebras.

MSC:

16S35 Twisted and skew group rings, crossed products
16S34 Group rings

Citations:

Zbl 1187.17002
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Full Text: DOI

References:

[1] J. W. Bales, “A tree for computing the Cayley-Dickson twist,” Missouri J. Math. Sci., 21, No. 2, 83-93 (2009). · Zbl 1187.17002
[2] C. Flaut and V. Shpakivskyi, “Holomorphic functions in generalized Cayley-Dickson algebras,” Adv. Appl. Clifford Algebras, 25, No. 1, 95-112 (2015). · Zbl 1311.30031
[3] S. Pumplün, “How to obtain division algebras from a generalized Cayley-Dickson doubling process,” J. Algebra, 402, 406-434 (2014). · Zbl 1298.17006
[4] Reynolds, WF, Twisted group algebras over arbitrary fields, Illinois J. Math., 3, 91-103 (1971) · Zbl 0224.20002
[5] Schafer, RD, An Introduction to Nonassociative Algebras (1966), New York: Academic Press, New York · Zbl 0145.25601
[6] R. D. Schafer, “On the algebras formed by the Cayley-Dickson process,” Amer. J. Math., 76, 435-446 (1954). · Zbl 0059.02901
[7] Waterhouse, WC, Nonassociative quaternion algebras, Algebras, Groups, Geometries, 4, 365-378 (1987) · Zbl 0643.17001
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