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Papers by Alberto Elduque

The purpose of this notes is to give a presentation of a classical theorem of Nomizu that relates... more The purpose of this notes is to give a presentation of a classical theorem of Nomizu that relates the invariant affine connections on reductive homogeneous spaces and nonassociative algebras. These are the notes of a course to be given at the CIMPA research school "G\'eom\'etrie diff\'erentielle et alg\`ebres non associatives", to be held in Marrakech (April 13-24, 2015).

Proceedings of the American Mathematical Society, 2012

In this work, we consider Lie algebras L containing a subalgebra isomorphic to sl3 and such that ... more In this work, we consider Lie algebras L containing a subalgebra isomorphic to sl3 and such that L decomposes as a module for that sl3 subalgebra into copies of the adjoint module, the natural 3dimensional module and its dual, and the trivial one-dimensional module. We determine the multiplication in L and establish connections with structurable algebras by exploiting symmetry relative to the symmetric group S4.

Communications in Algebra, 2007

Gerstenhaber and Myung in [10] classified all commutative-power associative nilalgebras of dimens... more Gerstenhaber and Myung in [10] classified all commutative-power associative nilalgebras of dimension 4. In [7] Gerstenhaber and Myung's results are generalized by giving a classification of commutative right-nilalgebras of right-nilindex four and dimension at most four, without assuming power-associativity. In this paper we complete this research and give a classification of commutative right-nilalgebras of right-nilindex five and dimension four, without assuming power-associativity, thus completing the classification of commutative right-nilalgebras of dimension at most four.

Communications in Algebra, 1987

Algebra and Logic, 1994

ABSTRACT

Contemporary Mathematics, 1995

Contemporary Mathematics, 2015

A survey of some properties of Okubo algebras is presented here. Emphasis is put on automorphisms... more A survey of some properties of Okubo algebras is presented here. Emphasis is put on automorphisms and derivations of these algebras, especially in characteristic three, where the situation is more involved and interesting. In this case, the Okubo algebra is closely related to a nodal noncommutative Jordan algebra. These latter algebras have appeared several times in the work of Helmut Strade.

Proceedings of the Royal Society of Edinburgh Section A Mathematics

Let F be a field of characteristic different from two and A a central simple non-Lie Malcev algeb... more Let F be a field of characteristic different from two and A a central simple non-Lie Malcev algebra over F. Our purpose is to determine the maximal subalgebras of A (with respect to inclusion). Maximal subalgebras of simple Lie algebras over an algebraically closed field have been studied by Dynkin, [3, 41, and maximal subalgebras of central simple associative, alternative and special Jordan algebras have been studied by Racine [ 121.

Contemporary Mathematics, 1999

Publicacions Matemàtiques, 2008

The classical Tits construction provides models of the exceptional simple Lie algebras in terms o... more The classical Tits construction provides models of the exceptional simple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group S4 on this construction are given. By means of these actions, the models provided by the Tits construction are related to models of the exceptional Lie algebras obtained from two different types of structurable algebras. Some models of exceptional Lie superalgebras are discussed too.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2009

Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are ... more Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are considered, and their decomposition into a direct sum of irreducible modules for the given action is studied.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1995

ABSTRACT

The purpose of this notes is to give a presentation of a classical theorem of Nomizu that relates... more The purpose of this notes is to give a presentation of a classical theorem of Nomizu that relates the invariant affine connections on reductive homogeneous spaces and nonassociative algebras. These are the notes of a course to be given at the CIMPA research school "G\'eom\'etrie diff\'erentielle et alg\`ebres non associatives", to be held in Marrakech (April 13-24, 2015).

Proceedings of the American Mathematical Society, 2012

In this work, we consider Lie algebras L containing a subalgebra isomorphic to sl3 and such that ... more In this work, we consider Lie algebras L containing a subalgebra isomorphic to sl3 and such that L decomposes as a module for that sl3 subalgebra into copies of the adjoint module, the natural 3dimensional module and its dual, and the trivial one-dimensional module. We determine the multiplication in L and establish connections with structurable algebras by exploiting symmetry relative to the symmetric group S4.

Communications in Algebra, 2007

Gerstenhaber and Myung in [10] classified all commutative-power associative nilalgebras of dimens... more Gerstenhaber and Myung in [10] classified all commutative-power associative nilalgebras of dimension 4. In [7] Gerstenhaber and Myung's results are generalized by giving a classification of commutative right-nilalgebras of right-nilindex four and dimension at most four, without assuming power-associativity. In this paper we complete this research and give a classification of commutative right-nilalgebras of right-nilindex five and dimension four, without assuming power-associativity, thus completing the classification of commutative right-nilalgebras of dimension at most four.

Communications in Algebra, 1987

Algebra and Logic, 1994

ABSTRACT

Contemporary Mathematics, 1995

Contemporary Mathematics, 2015

A survey of some properties of Okubo algebras is presented here. Emphasis is put on automorphisms... more A survey of some properties of Okubo algebras is presented here. Emphasis is put on automorphisms and derivations of these algebras, especially in characteristic three, where the situation is more involved and interesting. In this case, the Okubo algebra is closely related to a nodal noncommutative Jordan algebra. These latter algebras have appeared several times in the work of Helmut Strade.

Proceedings of the Royal Society of Edinburgh Section A Mathematics

Let F be a field of characteristic different from two and A a central simple non-Lie Malcev algeb... more Let F be a field of characteristic different from two and A a central simple non-Lie Malcev algebra over F. Our purpose is to determine the maximal subalgebras of A (with respect to inclusion). Maximal subalgebras of simple Lie algebras over an algebraically closed field have been studied by Dynkin, [3, 41, and maximal subalgebras of central simple associative, alternative and special Jordan algebras have been studied by Racine [ 121.

Contemporary Mathematics, 1999

Publicacions Matemàtiques, 2008

The classical Tits construction provides models of the exceptional simple Lie algebras in terms o... more The classical Tits construction provides models of the exceptional simple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group S4 on this construction are given. By means of these actions, the models provided by the Tits construction are related to models of the exceptional Lie algebras obtained from two different types of structurable algebras. Some models of exceptional Lie superalgebras are discussed too.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2009

Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are ... more Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are considered, and their decomposition into a direct sum of irreducible modules for the given action is studied.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1995

ABSTRACT