Abstract
Associated with a nondegenerate hermitian or antithermitian form on a finite-dimensional vector space V over an involutorial division algebra D of finite dimension over its center (of characteristic zero), we define by generators and relations an infinite sequence of finite-dimensional semisimple associative algebras. The representation theory of all these algebras, taken together, is essentially that of the Lie algebra of skew D-endomorphisms of V. The case where D is commutative is presented in detail here; when the form is symetric, the first non-trivial algebra in the sequence is the even Clifford algebra.
Keywords
- High Weight
- Clifford Algebra
- Irreducible Module
- Central Simple Algebra
- Symplectic Space
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Research supported by National Science Foundation grant #MC879-04473, at Yale University. The author expresses his thanks as well to the Institute des Hautes Etudes Scientifiques for its generous hospitality.
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Bibliography
Bourbaki, N., Groupes et algèbres de Lie, chaps. IV–VI. Eléments de Mathématiques, Fasc. XXXIV. Hermann, Paris, 1968.
Cartan, E., Leçons sur la théorie des spineurs, vol. II. Hermann, Paris, 1938. English translation, the theory of spinors, M.I.T. Press, Cambridge(MA), 1967.
Chevalley, C., The algebraic theory of spinors. Columbia University Press, New York, 1954.
Jacobson, N., Basic algebra II. Freeman, San Fransisco, 1980.
_____, Lie algebras. Interscience, New York, 1962. Republished Dover, New York, 1979.
_____, Structure and representations of Jordan algebras, Amer. Math. Soc. Colloq. Publs. Vol. XXXIX. A.M.S., Providence, 1968.
O’Meara, O. T., Introduction to quadratic forms. Academic press-Springer, New York, 1963.
Seligman, G.B., Rational constructions of modules for simple Lie algebras, Contemp. Math., Vol. 5, A.M.S., Providence, 1981.
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© 1983 Springer-Verlag
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Seligman, G.B. (1983). Higher even clifford algebras. In: Malliavin, MP. (eds) Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin. Lecture Notes in Mathematics, vol 1029. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0098931
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DOI: https://doi.org/10.1007/BFb0098931
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