Frobenius Lie Algebras Research Articles

Symplectic Form yang Berkaitan Dengan Satu-form Suatu Aljabar Lie Berdimensi Rendah

In this paper, we study symplectic form on low dimensional real Lie algebra. A symplectic form is very important in classifying of Lie algebra types. Based on their dimension and certain conditions, there are two types of Lie algebras. A lie algebra with odd dimension endowed with one-form such that is called a contact Lie algebra, while a Lie algebra whose dimension is even and it is endowed with zero index is called a Frobenius Lie algebra. The research aimed to give explicit formula of a symplectic form of low dimensional contact Lie algebras and Frobenius Lie algebras. We established that a one-form associated to simplectic form determine a type of a Lie algebra whether a contact or a Frobenius Lie algebras.To clearer the main results, we give some examples of one-form and symplectic form of Frobenius and contact Lie algebras.

AN EXACT SYMPLECTIC STRUCTURE OF LOW DIMENSIONAL 2-STEP SOLVABLE LIE ALGEBRAS

In this paper, we study a Lie algebra equipped by an exact symplectic structure. This condition implies that the Lie algebra has even dimension. The research aims to identify and to contruct 2-step solvable exact symplectic Lie algebras of low dimension with explicit formulas for their one-forms and symplectic forms. For case of four-dimensional, we found that only one class among three classes is 2-step solvable exact symplectic Lie algebra. Furthermore, we also give more examples for case six and eight dimensional of Lie algebras with exact symplectic forms which is included 2-step solvable exact sympletic Lie algebras. Moreover, it is well known that a 2-step solvable Lie algebra equipped by an exact symplectic form is nothing but it is called a 2-step solvable Frobenius Lie algebra.

Struktur Simplektik pada Aljabar Lie Affine aff(2,R)

In this research, we studied the affine Lie algebra aff(2,R). The aim of this research is to determine the 1-form in affine Lie algebra aff(2,R) which is associated with its symplectic structure so that affine Lie algebra aff(2,R) is a Frobenius Lie algebra. Realized the elements of the affine Lie algebra aff(2,R) in matrix form, then calculated the Lie brackets and formed the structure matrix of the affine Lie algebra aff(2,R). 1-form of the affine Lie algebra aff(2,R) is obtained from the determinant of the structure matrix of the affine Lie algebra aff(2,R). Furthermore, proved that the 2-form is symplectic and related to the 1-form. The result obtained is that the affine Lie algebra aff(2,R) has 1-form α=ε_12^*+ε_23^* on aff(2,R)^* which is related to its symplectic structure, β=ε_11^*∧ε_12^*+ε_12^*∧ε_22^*+ε_21^*∧ε_13^*+ε_22^*∧ε_23^* such that the affine Lie algebra aff(2,R) is a Frobenius Lie algebra. For further research, it can be developed into an affine Lie algebra with dimensions n(n+1).

On Properties of Five-dimensional Nonstandard Filiform Lie algebra

In this paper, we study the five-dimensional nonstandard Filiform Lie algebra and their basis elements representations. The aim of this research is to determine the basis elements of five-dimensional nonstandard Filiform Lie algebras representation in the form of real matrices. The method used in this study is by following Ceballos, Núñez, and Tenorio’s work. The results of this study are five real matrices as the realization of the basis elements of the five-dimensional nonstandard Filiform Lie algebra. We also discuss some results relate to five-dimensional nonstandard Filiform Lie algebra’s properties. The five-dimensional nonstandard Filiform Lie algebra is always nilpotent. For further research, it can be extended to five classes of Filiform Lie algebra, both standard and nonstandard with six dimensions. Moreover, it can be computed their split torus such that their direct sums are Frobenius Lie algebras.

Open orbits and primitive zero ideals for solvable Lie algebras

Abstract The aim of the paper is to provide a characterization criterion of exponential solvable Frobenius Lie algebras (having open coadjoint orbits), in terms of primitive ideals of the associated enveloping algebra. In the case of complex solvable Lie algebras, we also show that an algebraic adjoint orbit is open if and only if the associated primitive ideal through the Dixmier map is trivial.

Index of seaweed subalgebras of classical Lie algebras

We generalize the results in [3] giving a reduction algorithm allowing to compute the index of seaweed subalgebras of classical simple Lie algebras. We thus are able to obtain the index of some interesting families of seaweed subalgebras and to give new examples of large classes of Frobenius Lie algebras among them.

Left-symmetric algebra structures on contact Lie algebras

The aim of this work is to give necessary and sufficient conditions to extend an LSA structure on a Lie algebra to a Lie algebra that either contains as a subalgebra, or is a central extension of it. We also study conditions under which a Lie algebra together with an LSA structure admit a Frobenius or a contact form. Furthermore, we describe the properties of the Reeb vector associated with the codimension one contact ideal with respect to the LSA structure on the Frobenius Lie algebra.

Levi Decomposition of Frobenius Lie Algebra of Dimension 6

In this paper, we study notion of the Lie algebra of dimension 6. The finite dimensional Lie algebra can be expressed in terms of decomposition between Levi subalgebra and the maximal solvable ideal. This form of decomposition is called Levi decomposition. The work aims to obtain Levi decomposition of Frobenius Lie algebra of dimension 6. To achieve this aim, we compute Levi subalgebra and the maximal solvable ideal (radical) of with respect to its basis. To obtain Levi subalgebra and the maximal solvable ideal, we apply literature reviews about Lie algebra and decomposition Levi in Dagli result. For future research, decomposition Levi for higher dimension of Frobenius Lie algebra is still an open problem.

THE NON-DEGENERACY OF THE SKEW-SYMMETRIC BILINEAR FORM OF THE FINITE DIMENSIONAL REAL FROBENIUS LIE ALGEBRA

A Frobenius Lie algebra is recognized as the Lie algebra whose stabilizer at a Frobenius functional is trivial. This condition is equivalent to the existence of a skew-symmetric bilinear form which is non-degenerate. On the other hand, the Lie algebra is Frobenius as well if its orbit on the dual vector space is open. In this paper, we study the skew-symmetric bilinear form of finite dimensional Frobenius Lie algebra corresponding to its Frobenius functional. The work aims to prove that a Lie algebra of dimension is Frobenius if and only if the -th derivation of the Frobenius functional is not equal to zero. Indeed, this condition implies that the skew-symmetric bilinear form is non-degenerate and vice versa. In addition, some properties of Frobenius functionals are obtained. Furthermore, the computations are given using the coadjoint orbits and the structure matrix. As a discussion, we can investigate these results in the algebra case whether giving rise to a left-invariant K hler structure of a Frobenius Lie group or not.

On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications

In this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections between these cohomology with the corresponding ordinary cohomology and cohomology of algebraic groups. Let G be a semisimple and simply connected algebraic group G over an algebraically closed field of characteristic p>h, where h is a Coxeter number. Denote the first Frobenius kernel and Lie algebra of G by G1 and g, respectively. First, we calculate the restricted cohomology of g with coefficients in simple modules for two families of restricted simple modules. Since in the restricted region the restricted cohomology of g is equivalent to the corresponding cohomology of G1, we describe them as the cohomology of G1 in terms of the cohomology for G1 with coefficients in dual Weyl modules. Then, we give a necessary and sufficient condition for the isomorphisms Hn(G1,V)≅Hn(G,V) and Hn(g,V)≅Hn(G,V), and a necessary condition for the isomorphism Hn(g,V)≅Hn(G1,V), where V is a simple module with highest restricted weight. Using these results, we obtain all non-trivial isomorphisms between the cohomology of G, G1, and g with coefficients in the considered simple modules.

A Left-Symmetric Structure on The Semi-Direct Sum Real Frobenius Lie Algebra of Dimension 8

Let be the Lie algebra of the semi-direct sum of the real vector space and the Lie algebra of the sets of all real matrices. In this paper, a Frobenius functional is constructed in order for the Lie algebra to be the real Frobenius Lie algebra of dimension 8. Moreover, a bilinear form corresponding to this Frobenius functional is symplectic. Then the obtained symplectic bilinear form induces the left-symmetric algebra structures on . In other words, the Lie algebra is the left-symmetric algebra. In particular, we give the formulas of its left-symmetric algebra structure explicitely. The left-symmetric algebra structures for case of higher dimension of this Lie algebra type are still an open problem to be investigated.

The evolution of the spectrum of a Frobenius Lie algebra under deformation

The category of Frobenius Lie algebras is stable under deformation, and here we examine explicit infinitesimal deformations of four and six dimensional Frobenius Lie algebras to illustrate that the spectrum of a Frobenius Lie algebra can evolve under deformation.

The Index and Spectrum of Lie Poset Algebras of Types B, C, and D

We define posets of types B, C, and D. These posets encode the matrix forms of certain Lie algebras which lie between the algebras of upper-triangular and diagonal matrices. Our primary concern is the index and spectral theories of such type-B, C, and D Lie poset algebras. For an important restricted class, we develop combinatorial index formulas and, in particular, characterize posets corresponding to Frobenius Lie algebras. In this latter case we show that the spectrum is binary; that is, consists of an equal number of 0's and 1's. Interestingly, type-B, C, and D Lie poset algebras can be related to Reiner's notion of a parset.

KILLING FORM ALJABAR LIE FROBENIUS BERDIMENSI ≤4 UNTUK MENENTUKAN KESEMISEDERHANAANNYA

We study the notion of the Killing form for Frobenius Lie algebras of dimension . The Killing form is a symmetric bilinear form on a finite dimensional Lie algebra over a field defined by where is denoted the trace and is an adjoint representation of . A Lie algebras is said to be semisimple if it has the nondegenerate Killing form. The research aims to consider the criterion for semisimplicity of Frobenius Lie algebras of dimension by using the Killing form. The results show that each Frobenius Lie algebra of dimension and is not semisimple since the the Killing form is degenerate or in other words, a determinant of a representation matrix of the Killing form is equal to zero. For the future research, it is still an open problem to consider the general formulas of the Killing form for higher dimensional Frobenius Lie algebra whether degenerate or nondegenerate such that the semisimplicity of a Lie algebra can be considered. We conjecture that each finite dimensional real Frobenius Lie algebra is not semisimple.

Quasi-Associative Algebras on the Frobenius Lie Algebra M_3 (R)⊕gl_3 (R)

In this paper, we study the quasi-associative algebra property for the real Frobenius Lie algebra of dimension 18. The work aims to prove that is a quasi-associative algebra and to compute its formulas explicitly. To achieve this aim, we apply the literature reviews method corresponding to Frobenius Lie algebras, Frobenius functionals, and the structures of quasi-associative algebras. In the first step, we choose a Frobenius functional determined by direct computations of a bracket matrix of and in the second step, using an induced symplectic structure, we obtain the explicit formulas of quasi-associative algebras for . As the results, we proved that has the quasi-associative algebras property, and we gave their formulas explicitly. For future research, the case of the quasi-associative algebras on is still an open problem to be investigated. Our result can motivate to solve this problem.

The Existence of Affine Structures on the Borel Subalgebra of Dimension 6

The notion of affine structures arises in many fields of mathematics, including convex homogeneous cones, vertex algebras, and affine manifolds. On the other hand, it is well known that Frobenius Lie algebras correspond to the research of homogeneous domains. Moreover, there are 16 isomorphism classes of 6-dimensional Frobenius Lie algebras over an algebraically closed field. The research studied the affine structures for the 6-dimensional Borel subalgebra of a simple Lie algebra. The Borel subalgebra was isomorphic to the first class of Csikós and Verhóczki’s classification of the Frobenius Lie algebras of dimension 6 over an algebraically closed field. The main purpose was to prove that the Borel subalgebra of dimension 6 was equipped with incomplete affine structures. To achieve the purpose, the axiomatic method was considered by studying some important notions corresponding to affine structures and their completeness, Borel subalgebras, and Frobenius Lie algebras. A chosen Frobenius functional of the Borel subalgebra helped to determine the affine structure formulas well. The result shows that the Borel subalgebra of dimension 6 has affine structures which are not complete. Furthermore, the research also gives explicit formulas of affine structures. For future research, another isomorphism class of 6-dimensional Frobenius Lie algebra still needs to be investigated whether it has complete affine structures or not.

On Frobenius functionals of the Lie algebra M3(ℝ) ⊕ gl3(ℝ)

In the present paper, we study the Lie algebra written in the semi-direct sum formula of the vector space M3 (ℝ) and the Lie algebra gl3 (ℝ) whose both contain 3×3 real matrices. We denote it by g3 : = M3 (ℝ) ⊕ gl3 (ℝ). The aim of this research is to study the existence of a linear functional such that g3 is the Frobenius Lie algebra of dimension 18. Such the linear functional is called the Frobenius functional. We applied the literature reviews to achieve this result, particularly we study the notion of Frobenius Lie algebra in Ooms and Rais results. The main result of our research is the proof that g3 is Frobenius Lie algebra. For the future research, the existence of a Frobenius functional is still an open problem to study for higher dimensional Lie algebras.

The construction of real Frobenius Lie algebras from non-commutative nilpotent Lie algebras of dimension

In this present paper, we study real Frobenius Lie algebras constructed from non-commutative nilpotent Lie algebras of dimension ≤ 4. The main purpose is to obtain Frobenius Lie algebras of dimension ≤ 6. Particularly, for a given non-commutative nilpotent Lie algebras N of dimension ≤ 4 we show that there exist commutative subalgebras of dimension ≤ 2 such that the semi-direct sums ɡ = N⊕T is Frobenius Lie algebras. Moreover, T is called a split torus which is a commutative subalgebra of derivation of N and it depends on the given N. To obtain this split torus, we apply Ayala’s formulas of a Lie algebra derivation by taking a diagonal matrix of a standard representation matrix of the Lie algebra derivation of N. The discussion of higher dimension of Frobenius Lie algebras obtained from non-commutative nilpotent Lie algebras is still an open problem.

Sur l'indice des sous-algèbres biparaboliques de [formula omitted

We give some properties on the index of seaweed subalgebras of the complex Lie algebra gl(n) which allow to obtain formulas for the index of some interesting classes of this family and to give new families of Frobenius Lie algebras.

On Properties of the (2n+1)-Dimensional Heisenberg Lie Algebra

In the present paper, we study some properties of the Heisenberg Lie algebra of dimension . The main purpose of this research is to construct a real Frobenius Lie algebra from the Heisenberg Lie algebra of dimension . To achieve this, we exhibit how to compute the derivation of the Heisenberg Lie algebra by following Oom’s result. In this research, we use a literature review method to some related papers corresponding to a derivation of a Lie algebra, Frobenius Lie algebras, and Plancherel measure. Determining a conjecture of a real Frobenius Lie algebra is obtained. As the main result, we prove that conjecture. Namely, for the given the Heisenberg Lie algebra, there exists a commutative subalgebra of dimension one such that its semi direct sum is a real Frobenius Lie algebra of dimension . Futhermore, in the notion of the Lie group of the Heisenberg Lie algebra which is called the Heisenberg Lie group, we compute the generalized character of its group and we determine the Plancherel measure of the unitary dual of the Heisenberg Lie group. As our contributions, we complete some examples of Frobenius Lie algebras obtained from a nilpotent Lie algebra and we also give alternative computations to find the Plancherel measure of the Heisenberg Lie group.