Invariants of Lie algebras extended over commutative algebras without unit

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac-Moody algebras.

💡 Research Summary

The paper investigates three fundamental invariants of current Lie algebras of the form L ⊗ A, where L is an arbitrary Lie algebra and A is a commutative associative algebra that is not required to possess a unit. The author’s aim is to give a unified description of the second cohomology H²(L⊗A, K) with trivial coefficients, the space of symmetric invariant bilinear forms S²(L⊗A, K), and the derivation algebra Der(L⊗A) when at least one of the factors L or A is finite‑dimensional.

The first main result (Theorem 1) shows that any 2‑cocycle Φ ∈ Z²(L⊗A, K) can be written as a finite sum of decomposable tensors ϕ ⊗ α, where ϕ : L × L → K and α : A × A → K. By expanding the cocycle condition for triples (x⊗a, y⊗b, z⊗c) and repeatedly symmetrising with respect to the Lie variables, the author isolates four basic compatibility patterns:

1. ϕ is cyclic (ϕ(