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Deformation Theory Seminar

Monday, April 25, 2022 - 2:00pm

Nick and Vince Coll



University of Pennsylvania

passcode 967922


A ($2k+1$)$-$dimensional Lie algebra $\mathfrak{g}$ is called \textit{contact} if there exists a one-form $\varphi\in\mathfrak{g}^*$ such that $\varphi  \wedge (d\varphi)^k\ne 0$; such a $\varphi$ is called a \textit{contact form} and defines a \textit{contact structure} on an underlying Lie group $G$ -- which may not be unique. Moreover, $\varphi  \wedge (d\varphi)^k$ is a \textit{volume form} on $G.$ Here, our overall -- but overbroad -- goal is to classify contact Lie algebras among two categories of combinatorially defined algebraic Lie algebras: seaweed Lie algebras (``seaweeds") and Lie poset algebras. For seaweeds, we provide a complete classification by establishing that an index-one seaweed $\mathfrak{s}$ is contact if and only if it is quasi-reductive; that is, admits a one-form whose coadjoint stabilizer has center consisting of semisimple elements of $\mathfrak{s}.$ As a result, we show that every contact seaweed admits a contact form with a semisimple Reeb vector.


The contact analysis of Lie poset algebras is more subtle, and our results only concern Lie poset algebras in type A. In this setting, we construct an infinite family of contact, type-A Lie poset algebras via an inductive gluing procedure which yields both the defining ``contact toral" poset of the Lie algebra and an associated contact form. We conjecture that every contact, type-A Lie poset algebra with a trivial center corresponds to a contact toral poset, from which it would follow that every centerless, contact, type-A Lie poset algebra and its underlying Lie group are rigid. Moreover, we also conjecture that -- as with seaweeds -- every contact, type-A Lie poset algebra admits a contact form with a semisimple Reeb vector.