adam chapman
580 California St., Suite 400
San Francisco, CA, 94104
On the generalized Clifford algebra of a monic polynomial
2015, Linear Algebra and its Applications
https://doi.org/10.1016/J.LAA.2014.12.030
Sign up for access to the world's latest research
Abstract
In this paper we study the generalized Clifford algebra defined by Pappacena of a monic (with respect to the first variable) homogeneous polynomial Φ(Z, X 1 , . . . , of degree d in n + 1 variables over some field F . We completely determine its structure in the following cases: n = 2 and d = 3 and either char (F ) = 3, 2 for some r, t, e ∈ F . Except for a few exceptions, this algebra is an Azumaya algebra of rank nine whose center is the coordinate ring of an affine elliptic curve. We also discuss representations of arbitrary generalized Clifford algebras assuming the base field F is algebraically closed of characteristic zero.
References (19)
- M. Artin, F. Rodriguez-Villegas and J. Tate, On the Jacobians of plane cubics, Advances in Math. 198, 366-382, (2005).
- A. Chapman and U. Vishne, Clifford algebras of binary homogeneous forms, J. of Algebra 366, 94-111, (2012).
- M. Ciperiani and D. Krashen, Relative Brauer groups of genus 1 curves, Israel J. Math. 192, 921-949, (2012).
- E. Coskun, R. S. Kulkarni and Y. Mustopa, On representations of Clifford al- gebras of ternary cubic forms, Contemp. Math. 562, 91-99, (2012).
- T. Fisher and R. Newton, Computing the Cassels-Tate Pairing on the 3-Selmer group of an elliptic curve, preprint, (2013).
- D. E. Haile, On the Clifford algebra of a binary cuibc form, Amer. J. Math. 106(6), 1269-1280, (1984).
- D. E. Haile, On the Clifford algebras, conjugate splittings, and function fields of curves, Israel Math. Conf. Proc. 1, 356-361, (1989).
- D. E. Haile, On Clifford algebras and conjugate splittings, J. Algebra 139, 322- 335, (1991).
- D. E. Haile, When is the Clifford algebra of a binary cubic form split?, J. Algebra 146(2), 514-520, (1992).
- D. E. Haile and J.-M. Kuo, Classification of hyperbolic planes in algebras of degree three, J. Pure and Appl. Algebra 217, 1611-1619, (2012).
- D. E. Haile and S. Tesser, On Azumaya algebras arising from Clifford algebras, J. Algebra 116(2), 372-384, (1988).
- N. Heerema, An algebra determined by a binary cubic form, Duke Math. J. 21, 423-444, (1954).
- M.-A. Knus, A. Merkurjev, M. Rost and J.-P. Tignol, "The book of involu- tions", American Mathematical Society Colloquium Publications 44, American Mathematical Society, 1998.
- J.-M. Kuo, On an algebra associated to a ternary cubic curve, J. Algebra 330, 86-102, (2011).
- T. Y. Lam, "The Algebraic Theory of Quadratic Forms", W. A. Benjamin, Inc., 1973.
- C. J. Pappacena, Matrix pencils and a generalized Clifford algebra, Linear Al- gebra and its App. 313, 1-20, (2000).
- Ph. Revoy, Algèbres de Clifford et algèbres extérieures, J. Algebra 46(1), 268- 277, (1977).
- N. Roby, Algèbres de Clifford des formes polynomes, C. R. Acad. Sci. Paris Se'r. I. Math. A 268, A484-A486, (1969).
- M. Van den Bergh, Linearisations of binary and ternary forms, J. Algebra 109, 172-183, (1987).