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All Journal Pythagoras: Jurnal Matematika dan Pendidikan Matematika JURMA YUSTISI
Hafizhah, Nur
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Education Law, Crime, Criminology & Criminal Justice Mathematics

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Struktur Aljabar Koszul pada Aljabar Lie M_(3,1) (R)⋊〖gl〗_3 (R) Hafizhah, Nur; Kurniadi, Edi; Carnia, Ema
PYTHAGORAS Jurnal Matematika dan Pendidikan Matematika Vol. 17 No. 1: June 2022
Publisher : Department of Mathematics Education, Faculty of Mathematics and Natural Sciences, UNY

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 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)AbstractIn 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).
Legal Responsibility Of Bpom In Monitoring Illegal Cosmetic Products (Case Study: Salmon Dna Product By Dr. Richard Lee) Hafizhah, Nur
JURNAL MAHASISWA YUSTISI Vol. 3 No. 2 (2025)
Publisher : Universitas Ibn Khaldun Bogor

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.32832/jurmayustisi.v3i2.1560

Abstract

This study examines the legal liability of the Indonesian Food and Drug Authority (BPOM) in supervising the circulation of illegal cosmetics in Indonesia, focusing on a case study of the Salmon DNA product by Dr. Richard Lee. The background of this research is based on the widespread distribution of illegal cosmetics without marketing authorization, which endangers consumer safety, along with the rising trend of influencer-driven promotions on social media. The objective of this study is to analyze the forms of legal protection provided to consumers and to evaluate the supervisory role of BPOM in addressing such violations. This research employs a normative juridical method with a qualitative approach through literature review, analysis of statutory regulations, legal literature, and BPOM reports. The findings reveal that BPOM possesses sufficient legal instruments and authority to carry out both preventive and repressive supervision; however, its effectiveness is still hindered by limited resources, rapid technological advancements that accelerate the distribution of illegal products, and low consumer awareness. The Salmon DNA case study exposes violations related to the marketing authorization concerning the use of injection methods that do not comply with regulations. The recommendations proposed include the need to enhance public education, strengthen cross-sectoral coordination, and enforce stricter legal measures to minimize consumer risks and reinforce legal protection. This research is expected to contribute valuable input for the development of cosmetic supervision policies in Indonesia.
Co-Authors Edi Kurniadi Ema Carnia Kusumaningrum, Tanjung Anitasari Indah