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Papers by duong thanh

Revista De La Union Matematica Argentina, 2014

We give an expansion of two notions of double extension and T∗-extension for quadratic and odd qu... more We give an expansion of two notions of double extension and T∗-extension for quadratic and odd quadratic Lie superalgebras. Also, we provide a classification of quadratic and odd quadratic Lie superalgebras up to dimension 6. This classification is considered up to isometric isomorphism, mainly in the solvable case, and the obtained Lie superalgebras are indecomposable.

First, we study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vec... more First, we study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra. We give an isomorphic characterization of 2-step nilpotent pseudo-Euclidean Jordan algebras. Next, we find a Jordan-admissible condition for a Novikov algebra N\NN. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.

Journal of Algebra, 2014

In this paper, we generalize some results on quadratic Lie algebras to quadratic Lie superalgebra... more In this paper, we generalize some results on quadratic Lie algebras to quadratic Lie superalgebras, by applying graded Lie algebras tools. We establish a one-toone correspondence between non-Abelian quadratic Lie superalgebra structures and nonzero even super-antisymmetric 3-forms satisfying a structure equation. An invariant number of quadratic Lie superalgebras is then defined, called the dup-number. Singular quadratic Lie superalgebras (i.e. those with nonzero dupnumber) are studied. We show that their classification follows the classifications of O(m)-adjoint orbits of o(m) and Sp(2n)-adjoint orbits of sp(2n). An explicit formula for the quadratic dimension of singular quadratic Lie superalgebras is also provided. Finally, we discuss a class of 2-nilpotent quadratic Lie superalgebras associated to a particular super-antisymmetric 3-form.

Algebras and Representation Theory, 2011

We define a new invariant of quadratic Lie algebras and give a complete study and classification ... more We define a new invariant of quadratic Lie algebras and give a complete study and classification of singular quadratic Lie algebras, i.e. those for which the invariant does not vanish. The classification is related to O(n)-adjoint orbits in o(n). 0. INTRODUCTION Let g be a non-Abelian quadratic Lie algebra equipped with a bilinear form B. We can associate to (g, B) a canonical non-zero 3-form I ∈ 3 (g) g defined by I(X ,Y, Z) := B([X ,Y ], Z), ∀ X ,Y, Z ∈ g. Let {•, •} be the super-Poisson bracket on (g). The 3-form I satisfies (see [PU07]): {I, I} = 0. Conversely, given a quadratic vector space (g, B) and a non-zero 3-form I ∈ 3 (g) such that {I, I} = 0, there is a non-Abelian quadratic Lie algebra structure on g such that I is the canonical 3-form associated to g ([PU07]). Let Q(n) be the set of non-Abelian quadratic Lie algebra structures on the quadratic vector space C n. We identify Q(n) ↔ I ∈ 3 (C n) | {I, I} = 0 and Q(n) is an affine variety in 3 (C n) (Proposition 2.8). The dup-number of a non-Abelian quadratic Lie algebra g is defined by dup(g) := dim ({α ∈ g * | α ∧ I = 0}) , where I is the 3-form associated to g. It measures the decomposability of the 3form I and its range is {0, 1, 3} (Proposition 1.1). For instance, I is decomposable if, and only if, dup(g) = 3 and the corresponding quadratic Lie algebras are classified in [PU07], up to i-isomorphism (i.e. isometric isomorphism). It is easy to check that the dup-number of g is invariant by i-isomorphism, that is, two iisomorphic quadratic Lie algebras have the same dup-number (Lemma 2.1). We shall prove in this paper, a much stronger result: the dup-number of g is invariant by isomorphism.

Review of Economics and Statistics, 2000

In this paper, we classify solvable quadratic Lie algebras up to dimension 6. In dimensions small... more In this paper, we classify solvable quadratic Lie algebras up to dimension 6. In dimensions smaller than 6, we use the Witt decomposition given in [Bou59] and a result in [PU07] to obtain two non-Abelian indecomposable solvable quadratic Lie algebras. In the case of dimension 6, by applying the method of double extension given in [Kac85] and [MR85] and the classification result of singular quadratic Lie algebras in [DPU], we have three families of solvable quadratic Lie algebras which are indecomposable and not isomorphic.

Revista De La Union Matematica Argentina, 2014

We give an expansion of two notions of double extension and T∗-extension for quadratic and odd qu... more We give an expansion of two notions of double extension and T∗-extension for quadratic and odd quadratic Lie superalgebras. Also, we provide a classification of quadratic and odd quadratic Lie superalgebras up to dimension 6. This classification is considered up to isometric isomorphism, mainly in the solvable case, and the obtained Lie superalgebras are indecomposable.

First, we study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vec... more First, we study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra. We give an isomorphic characterization of 2-step nilpotent pseudo-Euclidean Jordan algebras. Next, we find a Jordan-admissible condition for a Novikov algebra N\NN. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.

Journal of Algebra, 2014

In this paper, we generalize some results on quadratic Lie algebras to quadratic Lie superalgebra... more In this paper, we generalize some results on quadratic Lie algebras to quadratic Lie superalgebras, by applying graded Lie algebras tools. We establish a one-toone correspondence between non-Abelian quadratic Lie superalgebra structures and nonzero even super-antisymmetric 3-forms satisfying a structure equation. An invariant number of quadratic Lie superalgebras is then defined, called the dup-number. Singular quadratic Lie superalgebras (i.e. those with nonzero dupnumber) are studied. We show that their classification follows the classifications of O(m)-adjoint orbits of o(m) and Sp(2n)-adjoint orbits of sp(2n). An explicit formula for the quadratic dimension of singular quadratic Lie superalgebras is also provided. Finally, we discuss a class of 2-nilpotent quadratic Lie superalgebras associated to a particular super-antisymmetric 3-form.

Algebras and Representation Theory, 2011

We define a new invariant of quadratic Lie algebras and give a complete study and classification ... more We define a new invariant of quadratic Lie algebras and give a complete study and classification of singular quadratic Lie algebras, i.e. those for which the invariant does not vanish. The classification is related to O(n)-adjoint orbits in o(n). 0. INTRODUCTION Let g be a non-Abelian quadratic Lie algebra equipped with a bilinear form B. We can associate to (g, B) a canonical non-zero 3-form I ∈ 3 (g) g defined by I(X ,Y, Z) := B([X ,Y ], Z), ∀ X ,Y, Z ∈ g. Let {•, •} be the super-Poisson bracket on (g). The 3-form I satisfies (see [PU07]): {I, I} = 0. Conversely, given a quadratic vector space (g, B) and a non-zero 3-form I ∈ 3 (g) such that {I, I} = 0, there is a non-Abelian quadratic Lie algebra structure on g such that I is the canonical 3-form associated to g ([PU07]). Let Q(n) be the set of non-Abelian quadratic Lie algebra structures on the quadratic vector space C n. We identify Q(n) ↔ I ∈ 3 (C n) | {I, I} = 0 and Q(n) is an affine variety in 3 (C n) (Proposition 2.8). The dup-number of a non-Abelian quadratic Lie algebra g is defined by dup(g) := dim ({α ∈ g * | α ∧ I = 0}) , where I is the 3-form associated to g. It measures the decomposability of the 3form I and its range is {0, 1, 3} (Proposition 1.1). For instance, I is decomposable if, and only if, dup(g) = 3 and the corresponding quadratic Lie algebras are classified in [PU07], up to i-isomorphism (i.e. isometric isomorphism). It is easy to check that the dup-number of g is invariant by i-isomorphism, that is, two iisomorphic quadratic Lie algebras have the same dup-number (Lemma 2.1). We shall prove in this paper, a much stronger result: the dup-number of g is invariant by isomorphism.

Review of Economics and Statistics, 2000

In this paper, we classify solvable quadratic Lie algebras up to dimension 6. In dimensions small... more In this paper, we classify solvable quadratic Lie algebras up to dimension 6. In dimensions smaller than 6, we use the Witt decomposition given in [Bou59] and a result in [PU07] to obtain two non-Abelian indecomposable solvable quadratic Lie algebras. In the case of dimension 6, by applying the method of double extension given in [Kac85] and [MR85] and the classification result of singular quadratic Lie algebras in [DPU], we have three families of solvable quadratic Lie algebras which are indecomposable and not isomorphic.