##
**Jordan algebras.
(Jordan-Algebren.)**
*(German)*
Zbl 0145.26001

Die Grundlehren der mathematischen Wissenschaften. 128. Berlin-Heidelberg-New York: Springer-Verlag. xiv, 357 S. (1966).

This book treats the theory of finite dimensional Jordan algebras. The topics discussed can be seen from the following list of chapters. \(A\) denotes always a (non necessarily associative) algebra over a field \(k\).

I. Einführung (59 pages).

Generalities about nonassociative algebras. Of interest is the treatment of the radical and of semisimplicity. This is based on the notion of associative linear form, which is a linear transformation \(\lambda\) of \(A\) into some \(k\)-vector space \(V\), such that \(\lambda(xy) = \lambda(yx)\), \(\lambda(x(yz)) = \lambda((xy)z)\) (the authors take instead of \(V\) a field extension of \(k\), but this does not seem to be essential). Such a \(\lambda\) is called semi-normal if \(\lambda\) vanishes on all nilpotents of \(A\) and if this remains true after extension of the base field. For any associative \(\lambda\), let \(R_\lambda = \{x\in A\mid \lambda(xy) = 0\) for all \(y\in A\}\) be the radical of the bilinear form \((x, y)\to \lambda(xy)\). Then the radical \(R\) of \(A\) is the intersection of all \(R_\lambda\), \(\lambda\) semi-normal. Semisimplicity being defined in the obvious way, a structure theorem is proved (for flexible, strictly power associative algebras of characteristic \(\neq 2)\). The notion of radical used here, seems to be one which is adapted to the case of Jordan algebras.

II. Strikt potenz-assoziative Algebren mit Einselement (29 p.).

As usual \(A\) is called strictly power associative if it is power associative and if it remains so under base field extension. The chapter contains a discussion of minimum polynomials in such algebras. Moreover the important notion of structure group is introduced here. In the reviewer’s own words, this is as follows. Suppose that \(A\) is commutative and has an identity element \(e\). There is a unique rational map over \(k\) of the affine space defined by \(A\) (in the sense of elementary algebraic geometry), denoted by \(x\mapsto x^{-1}\), which is defined in \(e\) and such that \(x\cdot x^{-1} = e\). Consider the subgroup of \(\mathrm{GL}(A)\times \mathrm{GL}(A)\), consisting of the pairs \((T, T')\) such that \((T x)^{-1} = T'x^{-1}\) (i. e. these two rational maps are equal whenever they are defined). Let \(G = G(A)\) be the projection of this subgroup onto the first factor. \(G\) is the structure group. Clearly \(T\to T'\) is an automorphism of \(G\) of order 2. \(G\) acts in \(A\), as a group of \(k\)-linear transformations.

III. Homogene Algebren (48 p.).

Let \(A\) be strictly power associative, commutative of characteristic \(\neq 2\), with an identity element. One of the interesting facts, proved here, is the characterization of Jordan algebras by means of properties of the structure group \(G\). Namely: \(A\) is a Jordan algebra if and only if for each \(B = A\otimes_k l\), with \(l\) algebraically closed, \(G(B)\) acts transitively on the invertible elements of \(B\). As a by-product, one obtains a simple proof (for finite dimensional algebras) of a basic identity in Jordan algebras (due to Jacobson): let \(P(x)y= 2x(xy) - x^2y\), then \(P(P(x)y) = P(x) P(y) P(x)\). This identity is of very frequent use. The chapter contains a discussion of multiplicative polynomials, i. e. homogeneous polynomial functions on \(A\) which are transformed into themselves by the elements of \(G\), up to a nonzero scalar factor. Using these, one can establish the existence of suitable semi-normal linear forms on central simple Jordan algebras.

IV. Jordan-Algebren (24 p).

Basic facts about Jordan algebras. Peirce decomposition, some results about automorphisms.

V. Mutationen von Jordan-Algebren (17 p.).

If \(A\) is a Jordan algebra of char \(\neq 2\) and if \(f\in A\) is invertible, then \(u_fv = u(vf) + v(uf) - (uv)f\) defines a new multiplication in \(A\), with which \(A\) becomes again a Jordan algebra, denoted by \(A_f\). Its unit element is \(f^{-1}\). \(A_f\) is called a mutation of \(A\). The chapter contains some properties of these.

VI. Beispiele von Jordan-Algebren (30 p.).

The standard examples of special Jordan algebras are discussed (symmetric elements in a simple associative algebra with involution, algebras of Hermitian matrices, Jordan algebras based on a quadratic form). There is also a discussion of \(\omega\)-domains.

VII. Alternative Algebren und nichtspezielle Jordan-Algebren (27 p.).

This chapter treats the Jordan algebras of Hermitian matrices over a composition algebra, as well as some related material (the treatment is rather computational).

VIII. Die Peirce-Zerlegung von Jordan-Algebren in Bezug auf ein vollständiges Orthogonalsystem (39 p.).

Let \(A\) be a commutative Jordan algebra with identity element over a field \(k\), of characteristic \(\neq 2\). Let \(e_1,\dots, e_r\) be a complete orthogonal system of idempotent elements in \(A\). Let \(L(e_i)\) denote multiplication by \(e_i\). There is a direct sum decomposition \(A = \sum_{1\leq i\leq j\leq r} A_{ij}\), where \(A_{ii} = P(e_i)A\), \(A_{ij} = L(e_i) L(e_j)A\) \((i < j)\). In this chapter the properties of this “Peirce decomposition” are discussed as a prelude to the structure theory of simple algebras.

IX. Derivationen von Jordan-Algebren (20 p.).

Discussion of properties of derivations of Jordan algebras. Proof of the theorem that under suitable conditions, all derivations are inner. Finally it is proved, using Killing forms, that in char 0 the Lie algebra of the structure group of a central simple Jordan algebra is reductive.

X. Die Klassifikation der einfachen Jordan-Algebren (18 p.).

Continuing Ch. VIII, the details of the classification of simple Jordan algebras are discussed. This discussion is based on two “isomorphism theorems” (coordinatization theorems in the terminology of Jacobson).

XI. Reelle und komplexe Jordan-Algebren (25 p.).

The main subjects here are the structure theory of formally real Jordan algebras and related results.

The reviewer wants to add a few critical remarks. The authors have given an important role in their book to the structure group, and they have demonstrated convincingly that there is good reason for doing so. Now from its very definition, it is clear that the structure group as defined by the authors, is for an algebraically closed ground field \(k\), a linear algebraic group. Because of this fact, it is rather natural to use the language of algebraic geometry and of algebraic groups. The reviewer believes that in doing so, some matters would be simplified and clarified. Here are a few illustrations of this assertion. \(A\) is always a commutative algebra over (for simplicity) an algebraically closed field \(k\) of char \(\ne 2\), \(G\) is identified with an algebraic group.

(a) The following characterization of Jordan algebras by means of properties of \(G\) is implicit in Ch. III: \(A\) is a Jordan algebra if and only if the orbit \(Ge\) of \(e\) in \(A\) under \(G\) is dense in the Zariski topology of \(A\). The authors’ characterization seems more cumbersome to the reviewer.

(b) Let \(A\) be a Jordan algebra, let \(F\) be a nonzero multiplicative polynomial. By (a), \(Ge\) is dense in \(A\). This implies trivially: if the differential \((dF)_e\) of \(F\) in \(e\) is \(0\), then \((dF)_x\) is \(0\) for all \(x\in A\), hence \(F\) is a \(p\)-th power of another polynomial \((p = \mathrm{char}\, k)\). This fact can be used to derive a number of results of Ch. III (for example Satz 4.2), for which the authors give considerably more complicated proofs.

(c) The result of Ch. IX, stating that in char \(0\) the Lie algebra of \(G\) is reductive if \(A\) is central simple, has a counterpart in arbitrary characteristic, namely that the identity component \(G_0\) of \(G\) is reductive if \(A\) is central simple. The proof of this fact is as follows: using the \(P(x)\) one sees that \(G\) acts irreducibly in \(A\) and then the radical of \(G_0\) consists of the scalar multiplications in \(A\), by a well-known elementary argument (using the Lie-Kolchin theorem). It should be pointed out that in these examples only elementary facts and notions about algebraic groups are required. The reviewer thinks it regrettable that the authors did not bring the algebraic group structure of \(G\) into the picture.

A second remark concerns Ch. III. In that chapter, the authors introduce homogeneous algebras, weakly homogeneous algebras, strongly homogeneous algebras. In the case of main interest all three notions coincide with the notion of a Jordan algebra. The reviewer does not see the reason for the introduction of these different notions and he believes that the exposition would have gained if the authors had restricted themselves to the case where all notions coincide: the case of commutative algebras with identity over a field of char \(\ne 2\), which is really the only case dealt with later on.

In spite of these criticisms the reviewer thinks that the book is very useful, both as a textbook and as a reference. It contains an extensive bibliography.

I. Einführung (59 pages).

Generalities about nonassociative algebras. Of interest is the treatment of the radical and of semisimplicity. This is based on the notion of associative linear form, which is a linear transformation \(\lambda\) of \(A\) into some \(k\)-vector space \(V\), such that \(\lambda(xy) = \lambda(yx)\), \(\lambda(x(yz)) = \lambda((xy)z)\) (the authors take instead of \(V\) a field extension of \(k\), but this does not seem to be essential). Such a \(\lambda\) is called semi-normal if \(\lambda\) vanishes on all nilpotents of \(A\) and if this remains true after extension of the base field. For any associative \(\lambda\), let \(R_\lambda = \{x\in A\mid \lambda(xy) = 0\) for all \(y\in A\}\) be the radical of the bilinear form \((x, y)\to \lambda(xy)\). Then the radical \(R\) of \(A\) is the intersection of all \(R_\lambda\), \(\lambda\) semi-normal. Semisimplicity being defined in the obvious way, a structure theorem is proved (for flexible, strictly power associative algebras of characteristic \(\neq 2)\). The notion of radical used here, seems to be one which is adapted to the case of Jordan algebras.

II. Strikt potenz-assoziative Algebren mit Einselement (29 p.).

As usual \(A\) is called strictly power associative if it is power associative and if it remains so under base field extension. The chapter contains a discussion of minimum polynomials in such algebras. Moreover the important notion of structure group is introduced here. In the reviewer’s own words, this is as follows. Suppose that \(A\) is commutative and has an identity element \(e\). There is a unique rational map over \(k\) of the affine space defined by \(A\) (in the sense of elementary algebraic geometry), denoted by \(x\mapsto x^{-1}\), which is defined in \(e\) and such that \(x\cdot x^{-1} = e\). Consider the subgroup of \(\mathrm{GL}(A)\times \mathrm{GL}(A)\), consisting of the pairs \((T, T')\) such that \((T x)^{-1} = T'x^{-1}\) (i. e. these two rational maps are equal whenever they are defined). Let \(G = G(A)\) be the projection of this subgroup onto the first factor. \(G\) is the structure group. Clearly \(T\to T'\) is an automorphism of \(G\) of order 2. \(G\) acts in \(A\), as a group of \(k\)-linear transformations.

III. Homogene Algebren (48 p.).

Let \(A\) be strictly power associative, commutative of characteristic \(\neq 2\), with an identity element. One of the interesting facts, proved here, is the characterization of Jordan algebras by means of properties of the structure group \(G\). Namely: \(A\) is a Jordan algebra if and only if for each \(B = A\otimes_k l\), with \(l\) algebraically closed, \(G(B)\) acts transitively on the invertible elements of \(B\). As a by-product, one obtains a simple proof (for finite dimensional algebras) of a basic identity in Jordan algebras (due to Jacobson): let \(P(x)y= 2x(xy) - x^2y\), then \(P(P(x)y) = P(x) P(y) P(x)\). This identity is of very frequent use. The chapter contains a discussion of multiplicative polynomials, i. e. homogeneous polynomial functions on \(A\) which are transformed into themselves by the elements of \(G\), up to a nonzero scalar factor. Using these, one can establish the existence of suitable semi-normal linear forms on central simple Jordan algebras.

IV. Jordan-Algebren (24 p).

Basic facts about Jordan algebras. Peirce decomposition, some results about automorphisms.

V. Mutationen von Jordan-Algebren (17 p.).

If \(A\) is a Jordan algebra of char \(\neq 2\) and if \(f\in A\) is invertible, then \(u_fv = u(vf) + v(uf) - (uv)f\) defines a new multiplication in \(A\), with which \(A\) becomes again a Jordan algebra, denoted by \(A_f\). Its unit element is \(f^{-1}\). \(A_f\) is called a mutation of \(A\). The chapter contains some properties of these.

VI. Beispiele von Jordan-Algebren (30 p.).

The standard examples of special Jordan algebras are discussed (symmetric elements in a simple associative algebra with involution, algebras of Hermitian matrices, Jordan algebras based on a quadratic form). There is also a discussion of \(\omega\)-domains.

VII. Alternative Algebren und nichtspezielle Jordan-Algebren (27 p.).

This chapter treats the Jordan algebras of Hermitian matrices over a composition algebra, as well as some related material (the treatment is rather computational).

VIII. Die Peirce-Zerlegung von Jordan-Algebren in Bezug auf ein vollständiges Orthogonalsystem (39 p.).

Let \(A\) be a commutative Jordan algebra with identity element over a field \(k\), of characteristic \(\neq 2\). Let \(e_1,\dots, e_r\) be a complete orthogonal system of idempotent elements in \(A\). Let \(L(e_i)\) denote multiplication by \(e_i\). There is a direct sum decomposition \(A = \sum_{1\leq i\leq j\leq r} A_{ij}\), where \(A_{ii} = P(e_i)A\), \(A_{ij} = L(e_i) L(e_j)A\) \((i < j)\). In this chapter the properties of this “Peirce decomposition” are discussed as a prelude to the structure theory of simple algebras.

IX. Derivationen von Jordan-Algebren (20 p.).

Discussion of properties of derivations of Jordan algebras. Proof of the theorem that under suitable conditions, all derivations are inner. Finally it is proved, using Killing forms, that in char 0 the Lie algebra of the structure group of a central simple Jordan algebra is reductive.

X. Die Klassifikation der einfachen Jordan-Algebren (18 p.).

Continuing Ch. VIII, the details of the classification of simple Jordan algebras are discussed. This discussion is based on two “isomorphism theorems” (coordinatization theorems in the terminology of Jacobson).

XI. Reelle und komplexe Jordan-Algebren (25 p.).

The main subjects here are the structure theory of formally real Jordan algebras and related results.

The reviewer wants to add a few critical remarks. The authors have given an important role in their book to the structure group, and they have demonstrated convincingly that there is good reason for doing so. Now from its very definition, it is clear that the structure group as defined by the authors, is for an algebraically closed ground field \(k\), a linear algebraic group. Because of this fact, it is rather natural to use the language of algebraic geometry and of algebraic groups. The reviewer believes that in doing so, some matters would be simplified and clarified. Here are a few illustrations of this assertion. \(A\) is always a commutative algebra over (for simplicity) an algebraically closed field \(k\) of char \(\ne 2\), \(G\) is identified with an algebraic group.

(a) The following characterization of Jordan algebras by means of properties of \(G\) is implicit in Ch. III: \(A\) is a Jordan algebra if and only if the orbit \(Ge\) of \(e\) in \(A\) under \(G\) is dense in the Zariski topology of \(A\). The authors’ characterization seems more cumbersome to the reviewer.

(b) Let \(A\) be a Jordan algebra, let \(F\) be a nonzero multiplicative polynomial. By (a), \(Ge\) is dense in \(A\). This implies trivially: if the differential \((dF)_e\) of \(F\) in \(e\) is \(0\), then \((dF)_x\) is \(0\) for all \(x\in A\), hence \(F\) is a \(p\)-th power of another polynomial \((p = \mathrm{char}\, k)\). This fact can be used to derive a number of results of Ch. III (for example Satz 4.2), for which the authors give considerably more complicated proofs.

(c) The result of Ch. IX, stating that in char \(0\) the Lie algebra of \(G\) is reductive if \(A\) is central simple, has a counterpart in arbitrary characteristic, namely that the identity component \(G_0\) of \(G\) is reductive if \(A\) is central simple. The proof of this fact is as follows: using the \(P(x)\) one sees that \(G\) acts irreducibly in \(A\) and then the radical of \(G_0\) consists of the scalar multiplications in \(A\), by a well-known elementary argument (using the Lie-Kolchin theorem). It should be pointed out that in these examples only elementary facts and notions about algebraic groups are required. The reviewer thinks it regrettable that the authors did not bring the algebraic group structure of \(G\) into the picture.

A second remark concerns Ch. III. In that chapter, the authors introduce homogeneous algebras, weakly homogeneous algebras, strongly homogeneous algebras. In the case of main interest all three notions coincide with the notion of a Jordan algebra. The reviewer does not see the reason for the introduction of these different notions and he believes that the exposition would have gained if the authors had restricted themselves to the case where all notions coincide: the case of commutative algebras with identity over a field of char \(\ne 2\), which is really the only case dealt with later on.

In spite of these criticisms the reviewer thinks that the book is very useful, both as a textbook and as a reference. It contains an extensive bibliography.

Reviewer: T. A. Springer

### MSC:

17Cxx | Jordan algebras (algebras, triples and pairs) |

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

17C10 | Structure theory for Jordan algebras |

17C05 | Identities and free Jordan structures |

17C20 | Simple, semisimple Jordan algebras |

17C27 | Idempotents, Peirce decompositions |

17C30 | Associated groups, automorphisms of Jordan algebras |

17C55 | Finite-dimensional structures of Jordan algebras |

17A05 | Power-associative rings |

17D05 | Alternative rings |