Consider the special linear Lie algebra sln(K) over an infinite field of characteristic different from 2. We prove that for any nonzero nilpotent X there exists a nilpotent Y such that the matrices X ...and Y generate the Lie algebra sln(K).
We present LieART 2.0 which contains substantial extensions to the Mathematica application LieART (LieAlgebras and Representation Theory) for computations frequently encountered in Lie algebras and ...representation theory, such as tensor product decomposition and subalgebra branching of irreducible representations. The basic procedure is unchanged—it computes root systems of Lie algebras, weight systems and several other properties of irreducible representations, but new features and procedures have been included to allow the extensions to be seamless. The new version of LieART continues to be user friendly. New extended tables of properties, tensor products and branching rules of irreducible representations are included in the supplementary material for use without Mathematica software. LieART 2.0 now includes the branching rules to special subalgebras for all classical and exceptional Lie algebras up to and including rank 15.
Program Title: LieART 2.0
CPC Library link to program files:http://dx.doi.org/10.17632/8vm7j67bwt.1
Licensing provisions: GNU Lesser General Public License
Programming language: Mathematica
External routines/libraries: Wolfram Mathematica 8–12
Nature of problem: The use of Lie algebras and their representations is widespread in physics, especially in particle physics. The description of nature in terms of gauge theories requires the assignment of fields to representations of compact Lie groups and their Lie algebras. Mass and interaction terms in the Lagrangian give rise to the need for computing tensor products of representations of Lie algebras. The mechanism of spontaneous symmetry breaking leads to the application of subalgebra decomposition. This computer code was designed for the purpose of Grand Unified Theory (GUT) model building, (where compact Lie groups beyond the U(1), SU(2) and SU(3) of the Standard Model of particle physics are needed), but it has found use in a variety of other applications. Tensor product decomposition and subalgebra decomposition have been implemented for all classical Lie groups SU(N), SO(N) and Sp(2N) and all the exceptional groups E6, E7, E8, F4 and G2. This includes both regular and irregular (special) subgroup decomposition of all Lie groups up through rank 15, and many more.
Solution method: LieART generates the weight system of an irreducible representation (irrep) of a Lie algebra by exploiting the Weyl reflection groups, which is inherent in all simple Lie algebras. Tensor products are computed by the application of Klimyk’s formula, except for SU(N)’s, where the Young-tableaux algorithm is used. Subalgebra decomposition of SU(N)’s are performed by projection matrices, which are generated from an algorithm to determine maximal subalgebras as originally developed by Dynkin 1,2. We generate projection matrices by the Dynkin procedure, i.e., removing dots from the Dynkin or extended Dynkin diagram, for regular subalgebras, and we implement explicit projection matrices for special subalgebras.
Restrictions: Internally irreps are represented by their unique Dynkin label. LieART’s default behavior in TraditionalForm is to print the dimensional name, which is the labeling preferred by physicist. Most Lie algebras can have more than one irrep of the same dimension and different irreps with the same dimension are usually distinguished by one or more primes (e.g. 175 and 175′ of A4). To determine the need for one or more primes of an irrep a brute-force loop over other irreps must be performed to search for irreps with the same dimensionality. Since Lie algebras have an infinite number of irreps, this loop must be cut off, which is done by limiting the maximum Dynkin digit in the loop. In rare cases for irreps of high dimensionality in high-rank algebras the used cutoff is too low and the assignment of primes is incorrect. However, this only affects the display of the irrep. All computations involving this irrep are correct, since the internal unique representation of Dynkin labels is used.
Abstract-In this paper, we first introduce the notion of twisted Rota-Baxter operators on 3-Hom-Lie algebras and define a cohomology of twisted Rota-Baxter operators on 3-Hom-Lie algebras with ...coefficients in a suitable representation. Furthermore, we introduce and study 3-Hom-NS-Lie-algebras as the underlying structure of twisted Rota-Baxter operators on 3-Hom-Lie algebras. Finally, we investigate twisted Rota-Baxter operators on 3-Hom-Lie algebras induced by Hom-Lie algebras.
A classification of p-nilpotent 5-dimensional restricted Lie algebras over algebraically closed fields of characteristic p>3 is provided. This is achieved by employing a natural restricted analogue ...of the known method by Skjelbred and Sund for classifying ordinary nilpotent Lie algebras as central extensions of Lie algebras of smaller dimension.
We prove that every 2-local automorphism on a finite-dimensional semi-simple Lie algebra L over an algebraically closed field of characteristic zero is an automorphism. We also show that each ...finite-dimensional nilpotent Lie algebra L with dimL≥2 admits a 2-local automorphism which is not an automorphism.
On the structure tensor of sln Bari, Kashif K.
Linear algebra and its applications,
11/2022, Letnik:
653
Journal Article
Recenzirano
The structure tensor of sln, denoted Tsln, is the tensor arising from the Lie bracket bilinear operation on the set of traceless n×n matrices over C. This tensor is intimately related to the well ...studied matrix multiplication tensor. Studying the structure tensor of sln may provide further insight into the complexity of matrix multiplication and the “hay in a haystack” problem of finding explicit sequences tensors with high rank or border rank. We aim to find new bounds on the rank and border rank of this structure tensor in the case of sl3 and sl4. We additionally provide bounds in the case of the lie algebras so4 and so5. The lower bounds on the border ranks were obtained via various recent techniques, namely Koszul flattenings, border substitution, and border apolarity. Upper bounds on the rank of Tsl3 are obtained via numerical methods that allowed us to find an explicit rank decomposition.
In this article, we study Nijenhuis operators on Hom-Lie algebras. We construct a graded Lie algebra (via the Hom-analog of the Frölicher-Nijenhuis bracket) whose Maurer-Cartan elements are given by ...Nijenhuis operators. This allows us to define the cohomology associated to a Nijenhuis operator. As an application, we study formal deformations of Nijenhuis operators that are generated by the above-defined cohomology. Finally, we introduce Hom-NS-Lie algebras as an algebraic structure behind Nijenhuis operators on Hom-Lie algebras. We provide various examples of Hom-NS-Lie algebras.
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