Struktur Aljabar Koszul pada Aljabar Lie M_(3,1) (R)⋊〖gl〗_3 (R)

Nur Hafizhah; Edi Kurniadi; Ema Carnia

doi:10.21831/pythagoras.v17i1.39713

Struktur Aljabar Koszul pada Aljabar Lie M_(3,1) (R)⋊〖gl〗_3 (R)

Nur Hafizhah, Departemen Matematika, FMIPA, Universitas Padjadjaran, Indonesia
Edi Kurniadi, Departemen Matematika, FMIPA, Universitas Padjadjaran, Indonesia
Ema Carnia, Departemen Matematika, FMIPA, Universitas Padjadjaran, Indonesia

10.21831/pythagoras.v17i1.39713

Abstract

Dalam penelitian ini dipelajari aljabar Lie affine aff(3) berdimensi 12 yang merupakan jumlah semi langsung dari ruang vektor matriks berukuran 3x1 dan aljabar Lie matriks berukuran 3x3 . Tujuan penelitian ini adalah untuk membuktikan eksistensi dan struktur aljabar koszul pada aljabar Lie aff(3). Aljabar Lie tersebut adalah aljabar Lie Frobenius. Oleh karena itu, terdapat suatu fungsional linear yang mengakibatkan nilai fungsional linear pada matriks strukturnya tidak sama dengan nol. Fungsional linear yang demikian ini disebut fungsional Frobenius. Dalam penelitian ini diberikan juga bagaimana mendapatkan matriks struktur, menghitung determinannya serta memilih fungsional Frobenius yang tepat. Hasil yang diperoleh dalam penelitian ini adalah rumus eksplisit struktur aljabar koszul pada aljabar Lie affine berdimensi 12 melalui induksi pada bentuk simplektik dari fungsional Frobeniusnya. Sebagai bahan diskusi untuk penelitian selanjutnya, hasil yang diperoleh dapat dikembangkan untuk menentukan struktur aljabar koszul pada aljabar Lie affine berdimensi n(n+1).

Structure of Koszul Algebra in Lie Algebra M_(3,1) (R)⋊〖gl〗_3 (R)

Abstract

In this research, we study the affine Lie algebra aff(3) of 12 dimension which is the semi-direct sum of the vector space of a matrix of 3x1 and Lie algebra of a matrix of 3x3. The research aims to prove the existence and structure of koszul algebras on the affine Lie algebra aff(3) . Since its Lie algebra is Frobenius then there exists a linear functional whose values in the matrix structure are not equal to zero. Such a linear functional is called a Frobenius functional. Furthermore, in this study, it is also given how to obtain the structure matrix, to calculate its determinants, and to choose the right Frobenius functional. The results obtained in this study are explicit formulas for the structure of the koszul algebra on 12-dimensional Lie affine algebra through induction in the symplectic form of its Frobenius functional. As a discussion material for further research, the results obtained can be developed to determine the structure of koszul algebra in affine Lie algebra of dimension n(n+1).

Keywords

Aljabar Lie; aljabar Lie Frobenius; aljabar Koszul

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DOI: https://doi.org/10.21831/pythagoras.v17i1.39713