2015
DOI: 10.4171/ggd/305
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Graphs and two-step nilpotent Lie algebras

Abstract: We consider a method popular in the literature of associating a two-step nilpotent Lie algebra with a finite simple graph. We prove that the two-step nilpotent Lie algebras associated with two graphs are Lie isomorphic if and only if the graphs from which they arise are isomorphic. Mathematics Subject Classification. Primary: 22E25

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Cited by 17 publications
(17 citation statements)
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References 21 publications
(35 reference statements)
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“…In this section we define a Lie algebra associated to a graph, which extends the definitions in [5,12]. Recall that for a Lie algebra g we have two descending series of subalgebras:…”
Section: Cliques and Solvable Lie Algebrasmentioning
confidence: 99%
See 2 more Smart Citations
“…In this section we define a Lie algebra associated to a graph, which extends the definitions in [5,12]. Recall that for a Lie algebra g we have two descending series of subalgebras:…”
Section: Cliques and Solvable Lie Algebrasmentioning
confidence: 99%
“…One such example is [5], where Dani and Mainkar defined a 2-step nilpotent Lie algebra associated to any simple graph. Later, Mainkar [12] showed that such algebras are isomorphic if and only if the corresponding graphs are isomorphic. The construction has been used to study various geometric properties of the Lie algebras of graph type and, in particular, the existence of symplectic structures [14], possible nilsolitons and Einstein solvmanifolds defined by solvable extensions [10,9,7].…”
Section: Introductionmentioning
confidence: 99%
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“…There are many standard bases – any one of them can be obtained from a particular one by an automorphism of frakturn (for the description of the derivation algebra see Section 2). However, by the result of [7] the algebra frakturn determines the graph scriptG uniquely, up to isomorphism. A Lie algebra associated with a graph is two‐step nilpotent (and is abelian if and only if E=).…”
Section: Introductionmentioning
confidence: 99%
“…Motivated by this fact we study in this paper an important class of two-step nilpotent Lie groups, the nilpotent Lie groups associated with graphs. For an up-to-date account on geometry and dynamics of such groups we refer the reader to [2,3,7].…”
Section: Introductionmentioning
confidence: 99%