A compendium of Lie structures on tensor products

Abstract

We demonstrate how a simple linear-algebraic technique used earlier to compute low-degree cohomology of current Lie algebras, can be utilized to compute other kinds of structures on such Lie algebras, and discuss further generalizations, applications, and related questions. While doing so, we touch upon such seemingly diverse topics as associative algebras of infinite representation type, Hom-Lie structures, Poisson brackets of hydrodynamic type, Novikov algebras, simple Lie algebras in small characteristics, and Koszul dual operads.

v.2: fixed minor typos

References in corpus (9)

• Deformations of Lie Algebras using $\sigma$-derivations
• Low-dimensional cohomology of current Lie algebras and analogs of the Riemann tensor for loop manifolds
• Lie algebras of smooth sections
• Invariants of Lie algebras extended over commutative algebras without unit
• Novikov operad is not Koszul
• Codimension growth of a variety of Novikov algebras
• Nonexistence of invariant symmetric forms on generalized Jacobson-Witt algebras revisited
• Lie algebra deformations in characteristic 2
• A converse to the Whitehead Theorem

Cited by in corpus (6)

• Hom-Lie structures on Kac-Moody algebras
• Deformations of current Lie algebras. I. Small algebras in characteristic 2
• On contact brackets on the tensor product
• Lie algebras and around: selected questions
• Commutative post-Lie algebra structures on Kac--Moody algebras
• Lie structures and chain ideal lattices