A new class of Lie algebras, called thin Lie algebras, is shown to have finite-dimensional global Weyl modules, with the Hamiltonian vector fields on the plane as the motivating example.
The three graces in the Tits--Kantor--Koecher category
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abstract
A metaphor of Loday describes Lie, associative, and commutative associative algebras as ``the three graces'' of the operad theory. In this article, we study the three graces in the category of $\mathfrak{sl}_2$-modules that are sums of copies of the trivial and the adjoint representation. That category is not symmetric monoidal, and so one cannot apply the wealth of results available for algebras over operads. Motivated by a recent conjecture of the second author and Mathieu, we embark on the exploration of the extent to which that category ``pretends'' to be symmetric monoidal. To that end, we examine various homological properties of free associative algebras and free associative commutative algebras, and study the Lie subalgebra generated by the generators of the free associative algebra.
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Global Weyl modules for thin Lie algebras are finite-dimensional
A new class of Lie algebras, called thin Lie algebras, is shown to have finite-dimensional global Weyl modules, with the Hamiltonian vector fields on the plane as the motivating example.