Clifford algebras of binary homogeneous forms. (English) Zbl 1262.15028
Summary: We study the generalized Clifford algebras associated to homogeneous binary forms of prime degree \(p\), focusing on exponentiation forms of \(p\)-central spaces in a division algebra.
For a two-dimensional \(p\)-central space, we make the simplifying assumption that one basis element is a sum of two eigenvectors with respect to conjugation by the other. If the product of the eigenvalues is 1 then the Clifford algebra is a symbol Azumaya algebra of degree \(p\), generalizing the theory developed for \(p=3\). Furthermore, when \(p=5\) and the product is not 1, we show that any quotient division algebra of the Clifford algebra is a cyclic algebra or a tensor product of two cyclic algebras, and every product of two cyclic algebras can be obtained as a quotient. An explicit presentation is given to the Clifford algebra when the form is diagonal.
For a two-dimensional \(p\)-central space, we make the simplifying assumption that one basis element is a sum of two eigenvectors with respect to conjugation by the other. If the product of the eigenvalues is 1 then the Clifford algebra is a symbol Azumaya algebra of degree \(p\), generalizing the theory developed for \(p=3\). Furthermore, when \(p=5\) and the product is not 1, we show that any quotient division algebra of the Clifford algebra is a cyclic algebra or a tensor product of two cyclic algebras, and every product of two cyclic algebras can be obtained as a quotient. An explicit presentation is given to the Clifford algebra when the form is diagonal.
MSC:
| 15A66 | Clifford algebras, spinors |
| 15A69 | Multilinear algebra, tensor calculus |
| 16K20 | Finite-dimensional division rings |
| 11E76 | Forms of degree higher than two |
| 16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |
Keywords:
generalized Clifford algebra; \(p\)-central space; cyclic algebra; eigenvector decomposition; homogeneous binary form; division algebra; symbol Azumaya algebra; tensor productReferences:
| [1] | Dickson, L. E., Linear associative algebras and abelian equations, Trans. Amer. Math. Soc., 15, 1, 31-46 (1914) · JFM 45.0189.03 |
| [2] | Haile, D. E., On the Clifford algebra of a binary cubic form, Amer. J. Math., 106, 6, 1269-1280 (1984) · Zbl 0585.13001 |
| [3] | Haile, D. E., When is the Clifford algebra of a binary cubic form split?, J. Algebra, 146, 2, 514-520 (1992) · Zbl 0758.11022 |
| [4] | Haile, D. E.; Tesser, S., On Azumaya algebras arising from Clifford algebras, J. Algebra, 116, 2, 372-384 (1988) · Zbl 0652.16002 |
| [5] | Heerema, N., An algebra determined by a binary cubic form, Duke Math. J., 21, 423-444 (1954) · Zbl 0056.03002 |
| [6] | Knus, M.-A.; Merkurjev, A.; Rost, M.; Tignol, J.-P., The Book of Involutions, Amer. Math. Soc. Colloq. Publ., vol. 44 (1998), American Mathematical Society · Zbl 0955.16001 |
| [7] | Kulkarni, R. S., On the Clifford algebra of a binary form, Trans. Amer. Math. Soc., 355, 8, 3181-3208 (2003) · Zbl 1035.11015 |
| [8] | Kulkarni, R. S., The extension of the reduced Clifford algebra and its Brauer class, Manuscripta Math., 112, 3, 297-311 (2003) · Zbl 1037.11025 |
| [9] | Lam, T. Y., The Algebraic Theory of Quadratic Forms (1973), W.A. Benjamin, Inc. · Zbl 0259.10019 |
| [10] | E. Matzri, Azumaya Algebras, Masterʼs thesis, Bar-Ilan University, 2004.; E. Matzri, Azumaya Algebras, Masterʼs thesis, Bar-Ilan University, 2004. |
| [11] | Matzri, E.; Rowen, L. H.; Vishne, U., Non-cyclic algebras with \(n\)-central elements, Proc. Amer. Math. Soc., 140, 2, 513-518 (2012) · Zbl 1242.16020 |
| [12] | Revoy, Ph., Algèbres de Clifford et algèbres extérieures, J. Algebra, 46, 1, 268-277 (1977) · Zbl 0373.15012 |
| [13] | Roby, N., Algèbres de Clifford des formes polynomes, C. R. Acad. Sci. Paris Seʼr. I. Math. A, 268, A484-A486 (1969) · Zbl 0206.04904 |
| [14] | Rowen, L. H., Ring Theory, vol. II (1988), Academic Press: Academic Press New York · Zbl 0651.16002 |
| [15] | Tesser, S., Representations of a Clifford algebra that are Azumaya algebras and generate the Brauer group, J. Algebra, 119, 2, 265-281 (1988) · Zbl 0664.10009 |
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