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


  • Edi Kurniadi Mathematics Department of FMIPA of Universitas Padjadjaran
  • Ema Carnia Mathematics Department of FMIPA of Unpad
  • Herlina Napitupulu Mathematics Department of FMIPA of Unpad



affine structures, Borel subalgebras, Frobenius Lie algebras.


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.


Plum Analytics


Alvarez, M. A., Rodríguez-Vallarte, M. C., & Salgado, G. (2018). Contact and Frobenius solvable Lie algebras with Abelian Nilradical. Communications in Algebra, 46(10), 4344-4354.

Burde, D. (2006). Left-symmetric algebras, or pre-Lie algebras in geometry and physics. Central European Journal of Mathematics, 4(3), 323-357.

Burde, D. (1998). Simple left-symmetric algebras with ¶ solvable Lie algebra. Manuscripta Mathematica, 95(3), 397-411.

Csikós, B., & Verhóczki, L. (2007). Classification of Frobenius Lie algebras of dimension ≤ 6. Publicationes Mathematicae-Debrecen, 70(3-4), 427-451.

Diatta, A., Manga, B., & Mbaye, A. (2020). On systems of commuting matrices, Frobenius Lie algebras and Gerstenhaber's Theorem. ArXiv:2002.08737.

Diatta, A., & Manga, B. (2014). On properties of principal elements of Frobenius Lie algebras. Journal of Lie Theory, 24(3), 849-864.

Fujiwara, H., & Ludwig, J. (2015). Harmonic analysis on exponential solvable Lie groups. Springer.

Gerstenhaber, M., & Giaquinto, A. (2009). The principal element of a Frobenius Lie algebra. Letters in Mathematical Physics, 88, 333-341.

Hilgert, J., & Neeb, K. H. (2011). Structure and geometry of Lie groups. Springer Science & Business Media.

Kurniadi, E. (2019). Harmonic analysis for finite dimensional real Frobenius Lie algebras (Doctoral dissertation). Nagoya University.

Kurniadi, E., & Ishi, H. (2019). Harmonic analysis for 4-dimensional real Frobenius Lie algebras. In International Conference on Geometric and Harmonic Analysis on Homogeneous Spaces and Applications (pp. 95-109).

Ooms, A. I. (1980). On Frobenius Lie algebras. Communications in Algebra, 8, 13-52.

Segal, D. (1992). The structure of complete left-symmetric algebras. Mathematische Annalen, 293(1), 569-578.






Abstract 272  .
PDF downloaded 221  .