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Journal : CAUCHY: Jurnal Matematika Murni dan Aplikasi

On Square-Integrable Representations of A Lie Group of 4-Dimensional Standard Filiform Lie Algebra Kurniadi, Edi
CAUCHY Vol 6, No 2 (2020): CAUCHY: Jurnal Matematika Murni dan Aplikasi
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | Full PDF (606.026 KB) | DOI: 10.18860/ca.v6i2.9094

Abstract

In this paper, we study irreducible unitary representations of a real standard filiform Lie group with dimension equals 4 with respect to its basis. To find this representations we apply the orbit method introduced by Kirillov. The corresponding orbit of this representation is genereric orbits of dimension 2. Furthermore, we show that obtained representation of this group is square-integrable. Moreover, in such case , we shall consider its Duflo-Moore operator as multiple of scalar  identity operator. In our case  that scalar is equal to one.
A Left-Symmetric Structure on The Semi-Direct Sum Real Frobenius Lie Algebra of Dimension 8 Kurniadi, Edi; Gusriani, Nurul; Subartini, Betty
CAUCHY Vol 7, No 2 (2022): CAUCHY: Jurnal Matematika Murni dan Aplikasi (May 2022) (Issue in Progress)
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v7i2.13462

Abstract

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.
Levi Decomposition of Frobenius Lie Algebra of Dimension 6 Henti, Henti; Kurniadi, Edi; Carnia, Ema
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 7, No 3 (2022): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v7i3.15656

Abstract

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.
On Properties of Five-dimensional Nonstandard Filiform Lie algebra Putra, Ricardo Eka; Kurniadi, Edi; Carnia, Ema
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 8, No 2 (2023): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v8i2.21018

Abstract

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.