Semi-isotopies and the lattice of inner ideals of certain quadratic Jordan algebras

Author:
Jerome M. Katz

Journal:
Trans. Amer. Math. Soc. **199** (1974), 413-427

MSC:
Primary 17A15

DOI:
https://doi.org/10.1090/S0002-9947-1974-0349774-8

MathSciNet review:
0349774

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Abstract: The concept of isotopy plays an extremely important role in the structure theory of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals. We take Koecher's characterization of isotopy and use it as the basis of a definition of semi-isotopy.

It is clear that semi-isotopies induce, in a natural way, automorphisms of the lattice of inner ideals. We concern ourselves with the converse problem; namely, if is a semilinear bijection of a quadratic Jordan algebra such that induces an automorphism of the lattice of inner ideals, is necessarily a semi-isotopy? We answer the above question in the affirmative for a large class of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals (said class includes all such algebras of capacity at least three over fields of characteristic unequal to two). Moreover, we prove that the only such maps which induce the identity automorphism on the lattice are the scalar multiplications.

**[1]**Jean A. Dieudonné,*La géométrie des groupes classiques*, Springer-Verlag, Berlin-New York, 1971 (French). Troisième édition; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 5. MR**0310083****[2]**John R. Faulkner,*Octonion planes defined by quadratic Jordan algebras*, Memoirs of the American Mathematical Society, No. 104, American Mathematical Society, Providence, R.I., 1970. MR**0271180****[3]**N. Jacobson,*Lectures on quadratic Jordan algebras*, Tata Institute of Fundamental Research, Bombay, 1969. Tata Institute of Fundamental Research Lectures on Mathematics, No. 45. MR**0325715****[4]**-,*Some groups and Lie algebras defined by quadratic Jordan algebras*, Bull. Amer. Math. Soc. (to appear).**[5]**N. Jacobson and K. McCrimmon,*Quadratic Jordan algebras of quadratic forms with base points*, J. Indian Math. Soc. (N.S.)**35**(1971), 1–45 (1972). MR**0357531****[6]**J. M. Katz,*Automorphisms of the lattice of inner ideals of certain quadratic Jordan algebras*, Doctoral Dissertation, Yale University, New Haven, Conn., 1972.**[7]**Jerome M. Katz,*Isomorphisms of the lattice of inner ideals of certain quadratic Jordan algebras*, Trans. Amer. Math. Soc.**185**(1973), 309–329. MR**0325716**, https://doi.org/10.1090/S0002-9947-1973-0325716-5**[8]**Kevin McCrimmon,*A general theory of Jordan rings*, Proc. Nat. Acad. Sci. U.S.A.**56**(1966), 1072–1079. MR**0202783****[9]**Kevin McCrimmon,*Inner ideals in quadratic Jordan algebras*, Trans. Amer. Math. Soc.**159**(1971), 445–468. MR**0279145**, https://doi.org/10.1090/S0002-9947-1971-0279145-1

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0349774-8

Keywords:
Quadratic Jordan algebra,
inner ideal,
semi-isotopy,
structure group

Article copyright:
© Copyright 1974
American Mathematical Society