##
**An introduction to nonassociative algebras.**
*(English)*
Zbl 0145.25601

Pure and Applied Mathematics, 22. New York and London: Academic Press. x, 166 p. (1966).

This book is an expanded version of the lectures held at Oklahoma State University in 1961, and presents in an elementary way some topics in the field of nonassociative algebras, meant for graduate students. The author gives a survey of the most important nonassociative algebras: alternative, Lie and Jordan algebras. Alternative algebras are presented in some detail (Chap. III, p. 27–90), Jordan algebras in a more cursory way (Chap. IV, p. 91–127) and Lie algebras as far as they are related to Jordan and alternative algebras, thus occurring in almost every chapter of the book. There are 5 chapters.

In Chapter I (Introduction) the Lie algebra is defined and the way in which one can construct a Lie algebra \(\mathfrak A^{-}\) after defining the new multiplication \(x\cdot y = [x, y] = xy - yx\) in a given associative algebra \(\mathfrak A\), is indicated. An important example is the derivation algebra \(\mathfrak D(\mathfrak A)\) of an algebra \(\mathfrak A\). The (commutative) Jordan algebra is introduced and the algebra \(\mathfrak A^+\), obtained by replacing the ordinary multiplication (in a given associative algebra \(\mathfrak A\)) by the new product \(x\cdot y = \frac12 (xy + yx)\) turns out to be Jordan. The Birkhoff-Witt theorem, stating that any Lie algebra \(\mathfrak L\) is isomorphic to a subalgebra of an algebra \(\mathfrak A^{-}\), where \(\mathfrak A\) is associative, is mentioned, but the author does not prove the theorem in this book. Next, the alternative algebras are defined and the multiplication table of the most important example, the Cayley algebra \(\mathfrak C\) over \(F\), is given. The rest of the chapter consists of an enumeration of several theorems in the class of associative and Lie algebras which serve as models for generalization and analogy.

In Chapter II arbitrary nonassociative algebras are considered. First of all the associative multiplication algebra \(\mathfrak M(\mathfrak A)\) and the Lie multiplication algebra \(\mathfrak L(\mathfrak A)\) of an algebra \(\mathfrak A\) are defined. The idea of solvability gives rise to the existence of a maximal solvable ideal \(\mathfrak N\) in \(\mathfrak A\), in special cases a suitable equivalent for the radical in the associative case. After mentioning (without proof) that every derivation of a finite-dimensional semisimple Lie algebra of characteristic \(0\) is inner (i.e. belongs to \(\mathfrak L(\mathfrak A))\), Jacobson’s theorem: “Let \(\mathfrak A\) be a finite-dimensional algebra which is a direct sum \(\mathfrak A = \mathfrak S_1\oplus\cdots\oplus\mathfrak S_t\), of simple ideals \(\mathfrak S_i\) over \(F\) of characteristic \(0\) and let \(\mathfrak A\) contain a left (or right) identity, then every derivation \(D\) of \(\mathfrak A\) is inner”, is proved.

After the introduction of the trace form \((x, y)\) the author proves: Let \(\mathfrak A\) be a finite-dimensional algebra over \(F\) with a nondegenerate associative trace form \((x, y)\) and \(\mathfrak B^2\neq 0\) for every ideal \(\mathfrak B\), then \(\mathfrak A = \mathfrak S_1\oplus\cdots\oplus\mathfrak S_t\) where \(\mathfrak S_i\) are simple ideals (Dieudonné).

This chapter is concluded by the definition of the bimodule, connected with the concept of a representation of \(\mathfrak A\).

Chapter III, the most extensive part of this book, deals with alternative algebras. Here we find Artin’s theorem: “The subalgebra generated by any two elements of an alternative algebra \(\mathfrak A\) is associative”. A broad discussion follows with regard to the Peirce decomposition and the existence of the radical, leading to the important result: “A finite-dimensional alternative algebra \(\mathfrak A\) is semisimple if and only if \(\mathfrak A = \mathfrak S_1\oplus\cdots\oplus\mathfrak S_t\) for simple ideals \(\mathfrak S_i\) (Zorn). The Cayley-Dickson process acting on an algebra with involution leads to the construction of quaternion and Cayley algebras. The ideas of trace and norm, appearing in this context, giving rise to the quadratic algebras, make clear that all of the algebras constructed by the Cayley-Dickson process are quadratic algebras over an arbitrary field \(F\). The central simple alternative algebras over \(F\) turn out to be either the Cayley algebras over \(F\) or \((mn)^2\)-dimensional algebras \(\mathfrak D_n=\mathfrak D\otimes F_n\), where \(\mathfrak D\) is a central associative division algebra of degree \(m\) over \(F\). (Kleinfeld’s result, that any simple alternative ring, which is not a nilring and not associative, is a Cayley algebra over its center, is mentioned without proof).

Next, Wedderburn’s principal theorem for alternative algebras follows: \(\mathfrak A=\mathfrak S+\mathfrak N\), where \(\mathfrak A\) is finite-dimensional, \(\mathfrak N\) is the radical, \(\mathfrak A/\mathfrak N\) is separable and \(\mathfrak S\cong\mathfrak A/\mathfrak N\). Furthermore the author proves that the derivation algebra \(\mathfrak D(\mathfrak C)\) of a Cayley algebra \(\mathfrak C\) over \(F\) of characteristic \(\neq 2, 3\) is the exceptional 14-dimensional central simple Lie algebra of type \(G\) over \(F\).

Chapter IV, the Jordan algebras. Albert’s theorem: “Any finite-dimensional Jordan nilalgebra (characteristic \(\neq 2)\) is nilpotent”, is proved and next: The radical of a finite-dimensional Jordan algebra \(\mathfrak J\) (characteristic \(=0)\) is the radical \(\mathfrak J^\bot\) of the trace form \((x, y) = \mathrm{trace}\, R_{xy}\), consequently: Any finite-dimensional semisimple Jordan algebra \(\mathfrak J\) over \(F\) of characteristic 0 is a direct sum \(\mathfrak J = \mathfrak S_1\oplus\cdots\oplus\mathfrak S_t\), of simple ideals \(\mathfrak S_i\). Then the author lists (without proof) all finite-dimensional central simple Jordan algebras of degree \(t\) over \(F\) of characteristic \(\neq 2\). One of them is the exceptional Jordan algebra: the 27-dimensional algebra of self-adjoint elements of the algebra \(\mathfrak C_3\) of all \(3\times 3\) matrices \((\mathfrak C\) is a Cayley algebra) with standard involution \(x\to \bar x'\) and multiplication \(x\cdot y = \frac12 (xy + yx)\).

Finally, if \(\mathfrak J\) is an exceptional central simple Jordan algebra over \(F\) of characteristic \(\neq 2, 3\) then \(\mathfrak D(\mathfrak J)\) is a 52-dimensional central simple Lie algebra of type \(F\) (Chevalley-Schafer) and some more theorems in which the relation between Jordan algebras and exceptional Lie algebras is expressed, conclude this chapter.

In the last Chapter power-associative algebras are investigated. A finite power-associative division ring of characteristic \(\neq 2, 3, 5\) is a field (Albert). Moreover, if \(\mathfrak A\) is a finite-dimensional power-associative algebra over \(F\), with a trace form \((x, y)\), \((e, e)\neq 0\) for any idempotent, \((x,y) = 0\) if \(x\cdot y\) is nilpotent, then the nilradical \(\mathfrak N\) of \(\mathfrak A\) coincides with the nilradical of \(\mathfrak A^+\) and is the radical \(\mathfrak A^\bot\) of the trace form. For \(\mathfrak S= \mathfrak A/\mathfrak N\) we have: \(\mathfrak S = \mathfrak S_1\oplus\cdots\oplus\mathfrak S_t\), for simple \(\mathfrak S_i\) and \(\mathfrak S\) is flexible. The central simple flexible algebras \(\mathfrak A\) over \(F\) such that \(\mathfrak A^+\) is a central simple Jordan algebra are listed (central simple Jordan algebras, quasi-associative central simple algebras and flexible quadratic algebras with nondegenerate norm form).

Then the noncommutative Jordan algebra is introduced. Modulo its nilradical any finite-dimensional noncommutative Jordan algebra of characteristic \(0\) is uniquely expressible as a direct sum of simple ideals, which are (commutative) Jordan or quasi-associative or flexible quadratic. New central algebras appear for characteristic \(p\), the so-called nodal algebras: finite-dimensional power-associative algebras with \(\mathbf 1\) over \(F\), where every element can be written in the form \(\alpha\mathbf 1 + z\), \(\alpha\in F\), \(z\) nilpotent, \(\mathfrak A\) not of the form \(\mathfrak A = F\mathbf 1 +\mathfrak N\) for \(\mathfrak N\) a nilalgebra.

The rest of the chapter deals with some results of Kokoris, who investigated this type of algebra. A rich bibliography concludes the book.

Conclusion: This book satisfies the needs of a first thorough introduction to nonassociative algebras.

In Chapter I (Introduction) the Lie algebra is defined and the way in which one can construct a Lie algebra \(\mathfrak A^{-}\) after defining the new multiplication \(x\cdot y = [x, y] = xy - yx\) in a given associative algebra \(\mathfrak A\), is indicated. An important example is the derivation algebra \(\mathfrak D(\mathfrak A)\) of an algebra \(\mathfrak A\). The (commutative) Jordan algebra is introduced and the algebra \(\mathfrak A^+\), obtained by replacing the ordinary multiplication (in a given associative algebra \(\mathfrak A\)) by the new product \(x\cdot y = \frac12 (xy + yx)\) turns out to be Jordan. The Birkhoff-Witt theorem, stating that any Lie algebra \(\mathfrak L\) is isomorphic to a subalgebra of an algebra \(\mathfrak A^{-}\), where \(\mathfrak A\) is associative, is mentioned, but the author does not prove the theorem in this book. Next, the alternative algebras are defined and the multiplication table of the most important example, the Cayley algebra \(\mathfrak C\) over \(F\), is given. The rest of the chapter consists of an enumeration of several theorems in the class of associative and Lie algebras which serve as models for generalization and analogy.

In Chapter II arbitrary nonassociative algebras are considered. First of all the associative multiplication algebra \(\mathfrak M(\mathfrak A)\) and the Lie multiplication algebra \(\mathfrak L(\mathfrak A)\) of an algebra \(\mathfrak A\) are defined. The idea of solvability gives rise to the existence of a maximal solvable ideal \(\mathfrak N\) in \(\mathfrak A\), in special cases a suitable equivalent for the radical in the associative case. After mentioning (without proof) that every derivation of a finite-dimensional semisimple Lie algebra of characteristic \(0\) is inner (i.e. belongs to \(\mathfrak L(\mathfrak A))\), Jacobson’s theorem: “Let \(\mathfrak A\) be a finite-dimensional algebra which is a direct sum \(\mathfrak A = \mathfrak S_1\oplus\cdots\oplus\mathfrak S_t\), of simple ideals \(\mathfrak S_i\) over \(F\) of characteristic \(0\) and let \(\mathfrak A\) contain a left (or right) identity, then every derivation \(D\) of \(\mathfrak A\) is inner”, is proved.

After the introduction of the trace form \((x, y)\) the author proves: Let \(\mathfrak A\) be a finite-dimensional algebra over \(F\) with a nondegenerate associative trace form \((x, y)\) and \(\mathfrak B^2\neq 0\) for every ideal \(\mathfrak B\), then \(\mathfrak A = \mathfrak S_1\oplus\cdots\oplus\mathfrak S_t\) where \(\mathfrak S_i\) are simple ideals (Dieudonné).

This chapter is concluded by the definition of the bimodule, connected with the concept of a representation of \(\mathfrak A\).

Chapter III, the most extensive part of this book, deals with alternative algebras. Here we find Artin’s theorem: “The subalgebra generated by any two elements of an alternative algebra \(\mathfrak A\) is associative”. A broad discussion follows with regard to the Peirce decomposition and the existence of the radical, leading to the important result: “A finite-dimensional alternative algebra \(\mathfrak A\) is semisimple if and only if \(\mathfrak A = \mathfrak S_1\oplus\cdots\oplus\mathfrak S_t\) for simple ideals \(\mathfrak S_i\) (Zorn). The Cayley-Dickson process acting on an algebra with involution leads to the construction of quaternion and Cayley algebras. The ideas of trace and norm, appearing in this context, giving rise to the quadratic algebras, make clear that all of the algebras constructed by the Cayley-Dickson process are quadratic algebras over an arbitrary field \(F\). The central simple alternative algebras over \(F\) turn out to be either the Cayley algebras over \(F\) or \((mn)^2\)-dimensional algebras \(\mathfrak D_n=\mathfrak D\otimes F_n\), where \(\mathfrak D\) is a central associative division algebra of degree \(m\) over \(F\). (Kleinfeld’s result, that any simple alternative ring, which is not a nilring and not associative, is a Cayley algebra over its center, is mentioned without proof).

Next, Wedderburn’s principal theorem for alternative algebras follows: \(\mathfrak A=\mathfrak S+\mathfrak N\), where \(\mathfrak A\) is finite-dimensional, \(\mathfrak N\) is the radical, \(\mathfrak A/\mathfrak N\) is separable and \(\mathfrak S\cong\mathfrak A/\mathfrak N\). Furthermore the author proves that the derivation algebra \(\mathfrak D(\mathfrak C)\) of a Cayley algebra \(\mathfrak C\) over \(F\) of characteristic \(\neq 2, 3\) is the exceptional 14-dimensional central simple Lie algebra of type \(G\) over \(F\).

Chapter IV, the Jordan algebras. Albert’s theorem: “Any finite-dimensional Jordan nilalgebra (characteristic \(\neq 2)\) is nilpotent”, is proved and next: The radical of a finite-dimensional Jordan algebra \(\mathfrak J\) (characteristic \(=0)\) is the radical \(\mathfrak J^\bot\) of the trace form \((x, y) = \mathrm{trace}\, R_{xy}\), consequently: Any finite-dimensional semisimple Jordan algebra \(\mathfrak J\) over \(F\) of characteristic 0 is a direct sum \(\mathfrak J = \mathfrak S_1\oplus\cdots\oplus\mathfrak S_t\), of simple ideals \(\mathfrak S_i\). Then the author lists (without proof) all finite-dimensional central simple Jordan algebras of degree \(t\) over \(F\) of characteristic \(\neq 2\). One of them is the exceptional Jordan algebra: the 27-dimensional algebra of self-adjoint elements of the algebra \(\mathfrak C_3\) of all \(3\times 3\) matrices \((\mathfrak C\) is a Cayley algebra) with standard involution \(x\to \bar x'\) and multiplication \(x\cdot y = \frac12 (xy + yx)\).

Finally, if \(\mathfrak J\) is an exceptional central simple Jordan algebra over \(F\) of characteristic \(\neq 2, 3\) then \(\mathfrak D(\mathfrak J)\) is a 52-dimensional central simple Lie algebra of type \(F\) (Chevalley-Schafer) and some more theorems in which the relation between Jordan algebras and exceptional Lie algebras is expressed, conclude this chapter.

In the last Chapter power-associative algebras are investigated. A finite power-associative division ring of characteristic \(\neq 2, 3, 5\) is a field (Albert). Moreover, if \(\mathfrak A\) is a finite-dimensional power-associative algebra over \(F\), with a trace form \((x, y)\), \((e, e)\neq 0\) for any idempotent, \((x,y) = 0\) if \(x\cdot y\) is nilpotent, then the nilradical \(\mathfrak N\) of \(\mathfrak A\) coincides with the nilradical of \(\mathfrak A^+\) and is the radical \(\mathfrak A^\bot\) of the trace form. For \(\mathfrak S= \mathfrak A/\mathfrak N\) we have: \(\mathfrak S = \mathfrak S_1\oplus\cdots\oplus\mathfrak S_t\), for simple \(\mathfrak S_i\) and \(\mathfrak S\) is flexible. The central simple flexible algebras \(\mathfrak A\) over \(F\) such that \(\mathfrak A^+\) is a central simple Jordan algebra are listed (central simple Jordan algebras, quasi-associative central simple algebras and flexible quadratic algebras with nondegenerate norm form).

Then the noncommutative Jordan algebra is introduced. Modulo its nilradical any finite-dimensional noncommutative Jordan algebra of characteristic \(0\) is uniquely expressible as a direct sum of simple ideals, which are (commutative) Jordan or quasi-associative or flexible quadratic. New central algebras appear for characteristic \(p\), the so-called nodal algebras: finite-dimensional power-associative algebras with \(\mathbf 1\) over \(F\), where every element can be written in the form \(\alpha\mathbf 1 + z\), \(\alpha\in F\), \(z\) nilpotent, \(\mathfrak A\) not of the form \(\mathfrak A = F\mathbf 1 +\mathfrak N\) for \(\mathfrak N\) a nilalgebra.

The rest of the chapter deals with some results of Kokoris, who investigated this type of algebra. A rich bibliography concludes the book.

Conclusion: This book satisfies the needs of a first thorough introduction to nonassociative algebras.

Reviewer: A. H. Boers (Rijswijk)

### MSC:

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

17D05 | Alternative rings |

17Axx | General nonassociative rings |

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

17Bxx | Lie algebras and Lie superalgebras |