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Keywords = Lie algebras of cartan-type

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14 pages, 290 KiB  
Article
Biderivations of Simple Modular Lie Algebras of Cartan-Type
by Wei Bai and Wende Liu
Mathematics 2025, 13(16), 2596; https://doi.org/10.3390/math13162596 - 13 Aug 2025
Viewed by 230
Abstract
Assume that L is a simple Lie algebra of Cartan-type over an algebraically closed field with a characteristic p>3. We demonstrate that all symmetric biderivations vanish by using weight space decompositions relative to a suitable torus and the standard Z [...] Read more.
Assume that L is a simple Lie algebra of Cartan-type over an algebraically closed field with a characteristic p>3. We demonstrate that all symmetric biderivations vanish by using weight space decompositions relative to a suitable torus and the standard Z-grading structures of L. We then conclude that every biderivation of L is inner, based on a general result concerning skew-symmetric biderivations. As the direct applications, we determine the linear commuting maps and commutative post-Lie algebra structures on L completely. Full article
14 pages, 731 KiB  
Article
Space, Matter and Interactions in a Quantum Early Universe. Part II: Superalgebras and Vertex Algebras
by Piero Truini, Alessio Marrani, Michael Rios and Klee Irwin
Symmetry 2021, 13(12), 2289; https://doi.org/10.3390/sym13122289 - 1 Dec 2021
Cited by 2 | Viewed by 1742
Abstract
In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra gu that extends e9. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds algebra. We turn gu into a Lie superalgebra [...] Read more.
In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra gu that extends e9. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds algebra. We turn gu into a Lie superalgebra sgu with no superpartners, in order to comply with the Pauli exclusion principle. There is a natural action of the Poincaré group on sgu, which is an automorphism in the massive sector. We introduce a mechanism for scattering that includes decays as particular resonant scattering. Finally, we complete the model by merging the local sgu into a vertex-type algebra. Full article
(This article belongs to the Special Issue Modified Gravity, Supergravity and Cosmological Applications)
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