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Dietrich Burde - One of the best experts on this subject based on the ideXlab platform.
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commutative post Lie Algebra structures and linear equations for nilpotent Lie Algebras
Journal of Algebra, 2019Co-Authors: Dietrich Burde, Karel Dekimpe, Wolfgang Alexander MoensAbstract:Abstract We show that for a given nilpotent Lie Algebra g with Z ( g ) ⊆ [ g , g ] all commutative post-Lie Algebra structures, or CPA-structures, on g are complete. This means that all left and all right multiplication operators in the Algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie Algebras F g , c and discover a strong relationship to solving systems of linear equations of type [ x , u ] + [ y , v ] = 0 for generator pairs x , y ∈ F g , c . We use results of Remeslennikov and Stohr concerning these equations to prove that, for certain g and c, the free-nilpotent Lie Algebra F g , c has only central CPA-structures.
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rota baxter operators and post Lie Algebra structures on semisimple Lie Algebras
arXiv: Rings and Algebras, 2018Co-Authors: Dietrich Burde, Vsevolod GubarevAbstract:Rota--Baxter operators $R$ of weight $1$ on $\mathfrak{n}$ are in bijective correspondence to post-Lie Algebra structures on pairs $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is complete. We use such Rota--Baxter operators to study the existence and classification of post-Lie Algebra structures on pairs of Lie Algebras $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is semisimple. We show that for semisimple $\mathfrak{g}$ and $\mathfrak{n}$, with $\mathfrak{g}$ or $\mathfrak{n}$ simple, the existence of a post-Lie Algebra structure on such a pair $(\mathfrak{g},\mathfrak{n})$ impLies that $\mathfrak{g}$ and $\mathfrak{n}$ are isomorphic, and hence both simple. If $\mathfrak{n}$ is semisimple, but $\mathfrak{g}$ is not, it becomes much harder to classify post-Lie Algebra structures on $(\mathfrak{g},\mathfrak{n})$, or even to determine the Lie Algebras $\mathfrak{g}$ which can arise. Here only the case $\mathfrak{n}=\mathfrak{sl}_2(\Bbb{C})$ was studied. In this paper we determine all Lie Algebras $\mathfrak{g}$ such that there exists a post-Lie Algebra structure on $(\mathfrak{g},\mathfrak{n})$ with $\mathfrak{n}=\mathfrak{sl}_2(\Bbb{C})\oplus \mathfrak{sl}_2(\Bbb{C})$.
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commutative post Lie Algebra structures on Lie Algebras
Journal of Algebra, 2016Co-Authors: Dietrich Burde, Wolfgang Alexander MoensAbstract:Abstract We show that any CPA-structure (commutative post-Lie Algebra structure) on a perfect Lie Algebra is trivial. Furthermore we give a general decomposition of inner CPA-structures, and classify all CPA-structures on complete Lie Algebras. As a special case we obtain the CPA-structures of parabolic subAlgebras of semisimple Lie Algebras.
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post Lie Algebra structures on pairs of Lie Algebras
Journal of Algebra, 2016Co-Authors: Dietrich Burde, Karel DekimpeAbstract:Abstract We study post-Lie Algebra structures on pairs of Lie Algebras ( g , n ) , which describe simply transitive nil-affine actions of Lie groups. We prove existence results for such structures depending on the interplay of the Algebraic structures of g and n . We consider the classes of simple, semisimple, reductive, perfect, solvable, nilpotent, abelian and unimodular Lie Algebras. Furthermore we consider commutative post-Lie Algebra structures on perfect Lie Algebras. Using Lie Algebra cohomology we can classify such structures in several cases. We also study commutative structures on low-dimensional Lie Algebras and on nilpotent Lie Algebras.
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commutative post Lie Algebra structures on Lie Algebras
arXiv: Rings and Algebras, 2015Co-Authors: Dietrich Burde, Wolfgang Alexander MoensAbstract:We show that any CPA-structure (commutative post-Lie Algebra structure) on a perfect Lie Algebra is trivial. Furthermore we give a general decomposition of inner CPA-structures, and classify all CPA-structures on parabolic subAlgebras of simple Lie Algebras.
Luca Romano - One of the best experts on this subject based on the ideXlab platform.
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Lie Algebra expansion and integrability in superstring sigma models
Journal of High Energy Physics, 2020Co-Authors: Andrea Fontanella, Luca RomanoAbstract:Lie Algebra expansion is a technique to generate new Lie Algebras from a given one. In this paper, we apply the method of Lie Algebra expansion to superstring σ-models with a ℤ4 coset target space. By applying the Lie Algebra expansion to the isometry Algebra, we obtain different σ-models, where the number of dynamical fields can change. We reproduce and extend in a systematic way actions of some known string regimes (flat space, BMN and non-relativistic in AdS5×S5). We define a criterion for the Algebra truncation such that the equations of motion of the expanded action of the new σ-model are equivalent to the vanishing curvature condition of the Lax connection obtained by expanding the Lax connection of the initial model.
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Lie Algebra expansion and integrability in superstring sigma models
arXiv: High Energy Physics - Theory, 2020Co-Authors: Andrea Fontanella, Luca RomanoAbstract:Lie Algebra expansion is a technique to generate new Lie Algebras from a given one. In this paper, we apply the method of Lie Algebra expansion to superstring $\sigma$-models with a $\mathbb{Z}_4$ coset target space. By applying the Lie Algebra expansion to the isometry Algebra, we obtain different $\sigma$-models, where the number of dynamical fields can change. We reproduce and extend in a systematic way actions of some known string regimes (flat space, BMN and non-relativistic in AdS$_5 \times$S$^5$). We define a criterion for the Algebra truncation such that the equations of motion of the expanded action of the new $\sigma$-model are equivalent to the vanishing curvature condition of the Lax connection obtained by expanding the Lax connection of the initial model.
Karel Dekimpe - One of the best experts on this subject based on the ideXlab platform.
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commutative post Lie Algebra structures and linear equations for nilpotent Lie Algebras
Journal of Algebra, 2019Co-Authors: Dietrich Burde, Karel Dekimpe, Wolfgang Alexander MoensAbstract:Abstract We show that for a given nilpotent Lie Algebra g with Z ( g ) ⊆ [ g , g ] all commutative post-Lie Algebra structures, or CPA-structures, on g are complete. This means that all left and all right multiplication operators in the Algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie Algebras F g , c and discover a strong relationship to solving systems of linear equations of type [ x , u ] + [ y , v ] = 0 for generator pairs x , y ∈ F g , c . We use results of Remeslennikov and Stohr concerning these equations to prove that, for certain g and c, the free-nilpotent Lie Algebra F g , c has only central CPA-structures.
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post Lie Algebra structures on pairs of Lie Algebras
Journal of Algebra, 2016Co-Authors: Dietrich Burde, Karel DekimpeAbstract:Abstract We study post-Lie Algebra structures on pairs of Lie Algebras ( g , n ) , which describe simply transitive nil-affine actions of Lie groups. We prove existence results for such structures depending on the interplay of the Algebraic structures of g and n . We consider the classes of simple, semisimple, reductive, perfect, solvable, nilpotent, abelian and unimodular Lie Algebras. Furthermore we consider commutative post-Lie Algebra structures on perfect Lie Algebras. Using Lie Algebra cohomology we can classify such structures in several cases. We also study commutative structures on low-dimensional Lie Algebras and on nilpotent Lie Algebras.
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post Lie Algebra structures on pairs of Lie Algebras
arXiv: Rings and Algebras, 2015Co-Authors: Dietrich Burde, Karel DekimpeAbstract:We study post-Lie Algebra structures on pairs of Lie Algebras $(\mathfrak{g},\mathfrak{n})$, motivated by nil-affine actions of Lie groups. We prove existence results for such structures depending on the interplay of the Algebraic structures of $\mathfrak{g}$ and $\mathfrak{n}$. We consider the classes of simple, semisimple, reductive, perfect, solvable, nilpotent, abelian and unimodular Lie Algebras. Furthermore we consider commutative post-Lie Algebra structures on perfect Lie Algebras. Using Lie Algebra cohomology we prove that such structures are trivial in several cases. We classify commutative structures on low-dimensional Lie Algebras, and study the case of nilpotent Lie Algebras.
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affine actions on Lie groups and post Lie Algebra structures
Linear Algebra and its Applications, 2012Co-Authors: Dietrich Burde, Karel Dekimpe, Kim VercammenAbstract:Abstract We introduce post-Lie Algebra structures on pairs of Lie Algebras ( g , n ) defined on a fixed vector space V. Special cases are LR-structures and pre-Lie Algebra structures on Lie Algebras. We show that post-Lie Algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtain several results on the existence of post-Lie Algebra structures, in terms of the Algebraic structure of the two Lie Algebras g and n . One result is, for example, that if there exists a post-Lie Algebra structure on ( g , n ) , where g is nilpotent, then n must be solvable. Furthermore special cases and examples are given. This includes a classification of all complex, two-dimensional post-Lie Algebras.
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post Lie Algebra structures and generalized derivations of semisimple Lie Algebras
arXiv: Rings and Algebras, 2011Co-Authors: Dietrich Burde, Karel DekimpeAbstract:We study post-Lie Algebra structures on pairs of Lie Algebras (g,n), and prove existence results for the case that one of the Lie Algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie Algebra structures on (g,n). For semisimple n and certain solvable g we construct canonical post-Lie Algebra structures. On the other hand we prove that there are no post-Lie Algebra structures for semisimple n and solvable, unimodular g. We also determine the generalized $(\al,\be,\ga)$-derivations of n in the semisimple case. As an application we classify post-Lie Algebra structures induced by generalized derivations.
Janerik Roos - One of the best experts on this subject based on the ideXlab platform.
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the homotopy Lie Algebra of a complex hyperplane arrangement is not necessarily finitely presented
Experimental Mathematics, 2008Co-Authors: Janerik RoosAbstract:We present a theory that produces several examples in which the homotopy Lie Algebra of a complex hyperplane arrangement is not finitely presented. We also present examples of hyperplane arrangements in which the enveloping Algebra of this Lie Algebra has an irrational Hilbert series. This answers two questions of Denham and Suciu.
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the homotopy Lie Algebra of a complex hyperplane arrangement is not necessarily finitely presented
arXiv: Algebraic Topology, 2006Co-Authors: Janerik RoosAbstract:We present a theory that produces several examples where the homotopy Lie Algebra of a complex hyperplane arrangement is not finitely presented. We also present examples of hyperplane arrangements where the enveloping Algebra of this Lie Algebra has an irrational Hilbert series. This answers two questions of Denham and Suciu.
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a computer aided study of the graded Lie Algebra of a local commutative noetherian ring
Journal of Pure and Applied Algebra, 1994Co-Authors: Janerik RoosAbstract:We initiate a systematic study of the homotopy Lie Algebra gR of a local commutative noetherian ring R. Particular emphasis is put on the sub Lie Algebra ηitR, generated by elements of degree 1. The computer is used in an essential way to detect new phenomena and new structure results, which are then proved by hand calculations. In Appendix B (by Clas Lofwall) it is proved that in many cases (including many examples encountered in this paper) gitR is a nice semi-direct product of ηitR by means of a free graded Lie Algebra.
Wolfgang Alexander Moens - One of the best experts on this subject based on the ideXlab platform.
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commutative post Lie Algebra structures and linear equations for nilpotent Lie Algebras
Journal of Algebra, 2019Co-Authors: Dietrich Burde, Karel Dekimpe, Wolfgang Alexander MoensAbstract:Abstract We show that for a given nilpotent Lie Algebra g with Z ( g ) ⊆ [ g , g ] all commutative post-Lie Algebra structures, or CPA-structures, on g are complete. This means that all left and all right multiplication operators in the Algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie Algebras F g , c and discover a strong relationship to solving systems of linear equations of type [ x , u ] + [ y , v ] = 0 for generator pairs x , y ∈ F g , c . We use results of Remeslennikov and Stohr concerning these equations to prove that, for certain g and c, the free-nilpotent Lie Algebra F g , c has only central CPA-structures.
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commutative post Lie Algebra structures on Lie Algebras
Journal of Algebra, 2016Co-Authors: Dietrich Burde, Wolfgang Alexander MoensAbstract:Abstract We show that any CPA-structure (commutative post-Lie Algebra structure) on a perfect Lie Algebra is trivial. Furthermore we give a general decomposition of inner CPA-structures, and classify all CPA-structures on complete Lie Algebras. As a special case we obtain the CPA-structures of parabolic subAlgebras of semisimple Lie Algebras.
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commutative post Lie Algebra structures on Lie Algebras
arXiv: Rings and Algebras, 2015Co-Authors: Dietrich Burde, Wolfgang Alexander MoensAbstract:We show that any CPA-structure (commutative post-Lie Algebra structure) on a perfect Lie Algebra is trivial. Furthermore we give a general decomposition of inner CPA-structures, and classify all CPA-structures on parabolic subAlgebras of simple Lie Algebras.
