##
**Jordan pairs.**
*(English)*
Zbl 0301.17003

Lecture Notes in Mathematics. 460. Berlin-Heidelberg-New York: Springer-Verlag. xvi, 218 p. DM 23.00 (1975).

Let \(k\) denote a commutative associative ring of scalars. A Jordan pair is a pair \(V= (V^+,V^-)\) of \(k\)-modules, and quadratic maps \(Q_+\colon V^+\rightarrow \operatorname{Hom}_k(V^-,V^+)\) and \(Q_-\colon V^-\rightarrow \operatorname{Hom}_k(V^+,V^-)\) satisfying three technical identities in all scalar extensions. This generalizes the notions of a (unital) quadratic Jordan algebra \((J,U,1)\) (set \(V^+ = V^- = J\), \(Q_+ = Q_- = U)\), and of a Jordan triple system \((T,U)\) (set \(V^+ = V^- = T\), \(Q_+ = Q_- =U)\).

In the case of a quadratic Jordan algebra \((J,U,1)\), there is an isomorphism between the structure group of autotopisms of \(J\) and the automorphism group of \((J,J)\). Thus, theorems which hold for Jordan algebras up to isotopy hold for associated Jordan pairs without this restriction.

In a Jordan pair \((V^+,V^-)\), \(V^+\) and \(V^-\) have homotopic structures as (non-unital) quadratic Jordan algebras, and the condition that \((V^+,V^-)\) be of the form \((J,J)\) is that \(V^-\) contains an invertible element \(v\) (i.e. \(Q_-(v)\) is invertible). Here the unital quadratic Jordan algebra \(J\) is an isotope of \(V^+\), so Jordan pairs containing invertible elements are roughly the same as unital Jordan algebras up to isotopy.

It also turns out that Jordan triple systems are essentially the same as Jordan pairs with involution, involution defined in the appropriate sense.

The author offers the following arguments for studying this generalized concept. Inner automorphisms are naturally defined (in contrast to the other systems). Jordan pairs always contain “enough” idempotents (not so in the other systems). They arise naturally in the Koecher-Tits construction of Lie algebras and associated algebraic groups. As in the theory of Jordan and alternative algebras, the author also discusses alternative pairs, which turn out to have useful connections with Jordan pairs. An alternative pair is a pair \(A = (A^+,A^-)\) of \(k\)-modules and two trilinear maps: \(A^+ \times A^- \times A^+ \rightarrow A^+\) and \(A^- \times A^+ \times A^- \rightarrow A^-\) satisfying three technical identities.

We list the titles of the four chapters and of the 18 sections.

I. Jordan pairs.

1. Definitions and relations with Jordan algebras and triple systems. 2. Identities and Representations. 3. The quasi-inverse. 4. Radicals. 5, Peirce decomposition.

II. Alternative pairs.

6. Basic properties and relations with alternative algebras. 7. The Jordan pair associated with an alternative pair. 8. Imbedding into Jordan pairs. 9. Peirce decomposition.

III. Alternative and Jordan pairs with chain conditions.

10. Inner ideals and chain conditions. 11. Classification of alternative pairs. 12. Classification of Jordan pairs.

IV. Finite-dimensional Jordan pairs.

13. Universal enveloping algebras. 14. Solvability and nilpotence. 15. Cartan subpairs. 16. The generic minimum polynomial. 17. Simple Jordan pairs. 18. Appendix: Polynomial and rational functions.

{The following four corrections were sent by the author to the reviewer. p. 40, line 11 top: read, Nil \(V\) instead of Nil \(J\); p. 101, line 1 top: insert “of 9.3” after “analogue”; p. 159, statement (iv) of 14.14: read “the ideal \(\tilde B\) …”; p. 177, line 2 bottom: read “\(X^{(h,Y)} \ne 0\)”. }

In the case of a quadratic Jordan algebra \((J,U,1)\), there is an isomorphism between the structure group of autotopisms of \(J\) and the automorphism group of \((J,J)\). Thus, theorems which hold for Jordan algebras up to isotopy hold for associated Jordan pairs without this restriction.

In a Jordan pair \((V^+,V^-)\), \(V^+\) and \(V^-\) have homotopic structures as (non-unital) quadratic Jordan algebras, and the condition that \((V^+,V^-)\) be of the form \((J,J)\) is that \(V^-\) contains an invertible element \(v\) (i.e. \(Q_-(v)\) is invertible). Here the unital quadratic Jordan algebra \(J\) is an isotope of \(V^+\), so Jordan pairs containing invertible elements are roughly the same as unital Jordan algebras up to isotopy.

It also turns out that Jordan triple systems are essentially the same as Jordan pairs with involution, involution defined in the appropriate sense.

The author offers the following arguments for studying this generalized concept. Inner automorphisms are naturally defined (in contrast to the other systems). Jordan pairs always contain “enough” idempotents (not so in the other systems). They arise naturally in the Koecher-Tits construction of Lie algebras and associated algebraic groups. As in the theory of Jordan and alternative algebras, the author also discusses alternative pairs, which turn out to have useful connections with Jordan pairs. An alternative pair is a pair \(A = (A^+,A^-)\) of \(k\)-modules and two trilinear maps: \(A^+ \times A^- \times A^+ \rightarrow A^+\) and \(A^- \times A^+ \times A^- \rightarrow A^-\) satisfying three technical identities.

We list the titles of the four chapters and of the 18 sections.

I. Jordan pairs.

1. Definitions and relations with Jordan algebras and triple systems. 2. Identities and Representations. 3. The quasi-inverse. 4. Radicals. 5, Peirce decomposition.

II. Alternative pairs.

6. Basic properties and relations with alternative algebras. 7. The Jordan pair associated with an alternative pair. 8. Imbedding into Jordan pairs. 9. Peirce decomposition.

III. Alternative and Jordan pairs with chain conditions.

10. Inner ideals and chain conditions. 11. Classification of alternative pairs. 12. Classification of Jordan pairs.

IV. Finite-dimensional Jordan pairs.

13. Universal enveloping algebras. 14. Solvability and nilpotence. 15. Cartan subpairs. 16. The generic minimum polynomial. 17. Simple Jordan pairs. 18. Appendix: Polynomial and rational functions.

{The following four corrections were sent by the author to the reviewer. p. 40, line 11 top: read, Nil \(V\) instead of Nil \(J\); p. 101, line 1 top: insert “of 9.3” after “analogue”; p. 159, statement (iv) of 14.14: read “the ideal \(\tilde B\) …”; p. 177, line 2 bottom: read “\(X^{(h,Y)} \ne 0\)”. }

Reviewer: Earl J. Taft (New Brunswick)

### MSC:

17C10 | Structure theory for Jordan algebras |

17C05 | Identities and free Jordan structures |

17C20 | Simple, semisimple Jordan algebras |

17C30 | Associated groups, automorphisms of Jordan algebras |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

17C50 | Jordan structures associated with other structures |

17D05 | Alternative rings |