93 research outputs found
Contact seaweeds II: type C
This paper is a continuation of earlier work on the construction of contact
forms on seaweed algebras. In the prequel to this paper, we show that every
index-one seaweed subalgebra of is contact by
identifying contact forms that arise from Dougherty's framework. We extend this
result to include index-one seaweed subalgebras of .
Our methods are graph-theoretic and combinatorial
Classification of contact seaweeds
A celebrated result of Gromov ensures the existence of a contact structure on
any connected, non-compact, odd dimensional Lie group. In general, such
structures are not invariant under left translation. The problem of finding
which Lie groups admit a left-invariant contact structure resolves to the
question of determining when a Lie algebra is contact; that is,
admits a one-form such that
In full generality, this remains an open question; however we settle it for
the important category of the evocatively named seaweed algebras by showing
that an index-one seaweed is contact precisely when it is quasi-reductive.
Seaweeds were introduced by Dergachev and Kirillov who initiated the
development of their index theory -- since completed by Joseph, Panyushev,
Yakimova, and Coll, among others. Recall that a contact Lie algebra has index
one -- but not characteristically so. Leveraging recent work of Panyushev,
Baur, Moreau, Duflo, Khalgui, Torasso, Yakimova, and Ammari, who collectively
classified quasi-reductive seaweeds, our equivalence yields a full
classification of contact seaweeds. We remark that since type-A and type-C
seaweeds are de facto quasi-reductive (by a result of Panyushev), in these
types index one alone suffices to ensure the existence of a contact form
Structure of Colored Complete Graphs Free of Proper Cycles
For a fixed integer m, we consider edge colorings of complete graphs which contain no properly edge colored cycle Cm as a subgraph. Within colorings free of these subgraphs, we establish global structure by bounding the number of colors that can induce a spanning and connected subgraph. In the case of smaller cycles, namely C4,C5, and C6, we show that our bounds are sharp
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