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 g is contact; that is,
admits a one-form Οβgβ such that
Οβ§(dΟ)kξ =0.
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