Linear factorization of conical polynomials over certain nonassociative algebras

Author:
J. Lepowsky

Journal:
Trans. Amer. Math. Soc. **216** (1976), 237-248

MSC:
Primary 17B05

MathSciNet review:
0432722

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Abstract: Conical polynomials are defined as certain polynomials in quadratic elements of the universal enveloping algebra of a semisimple symmetric Lie algebra over a field of characteristic zero. These polynomials were used in an earlier paper to describe the conical vectors in certain induced modules. Here it is shown that when the base field is extended to a certain type of nonassociative algebra, the conical polynomials can be factored ``linearly". One such nonassociative algebra is discussed in detail--an (alternative) composition algebra intimately related to the structure of the Lie algebra and studied earlier by B. Kostant in the context of real semisimple Lie algebras. The linear factorization leads in a later paper to an extension of the earlier work on conical vectors in induced modules.

**[1]**Nathan Jacobson,*Structure and representations of Jordan algebras*, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR**0251099****[2,]**Bertram Kostant,*On the existence and irreducibility of certain series of representations*, Bull. Amer. Math. Soc.**75**(1969), 627–642. MR**0245725**, 10.1090/S0002-9904-1969-12235-4**1.**Bertram Kostant,*On the existence and irreducibility of certain series of representations*, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 231–329. MR**0399361****[3]**J. Lepowsky,*On the Harish-Chandra homomorphism*, Trans. Amer. Math. Soc.**208**(1975), 193–218. MR**0376792**, 10.1090/S0002-9947-1975-0376792-7**2.**J. Lepowsky,*Conical vectors in induced modules*, Trans. Amer. Math. Soc.**208**(1975), 219–272. MR**0376786**, 10.1090/S0002-9947-1975-0376786-1**3.**J. Lepowsky,*Existence of conical vectors in induced modules*, Ann. of Math. (2)**102**(1975), no. 1, 17–40. MR**0379613****[4]**Richard D. Schafer,*An introduction to nonassociative algebras*, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966. MR**0210757**

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1976-0432722-1

Keywords:
Conical polynomials,
conical vectors,
semisimple symmetric Lie algebras,
alternative algebras,
composition algebras,
Cayley-Dickson algebras

Article copyright:
© Copyright 1976
American Mathematical Society