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 the cohomological, bilinear, and derivational invariants of a Lie algebra extended over a commutative associative algebra that lacks a unit element. The central object of study is the tensor product 𝔤⊗A, where 𝔤 is a finite‑dimensional Lie algebra over a field K and A is a commutative K‑algebra without identity. The authors first establish a decomposition of the second cohomology group H²(𝔤⊗A, K) with trivial coefficients. By adapting the Hochschild‑Serre spectral sequence to the non‑unital setting, they show that H² splits into three natural components: (i) the image of H²(𝔤, K) tensored with A, (ii) a part arising from the multiplication structure of A (which vanishes when A is nilpotent), and (iii) a mixed term reflecting interactions between 𝔤‑cocycles and derivations of A. This decomposition unifies earlier ad‑hoc calculations for loop algebras and periodizations of semisimple Lie algebras.

Next, the paper treats symmetric invariant bilinear forms on 𝔤⊗A. The authors prove that any such form is a tensor product of a 𝔤‑invariant symmetric form with a symmetric bilinear form on A that is invariant under the natural action of derivations of A. Because A has no unit, the usual trace‑type construction fails; instead the authors introduce a “virtual unit” concept to capture the missing scalar component. This yields a precise description of the space of invariant forms and explains why the dimension of this space can be larger than expected in the unit‑less case.

The derivation algebra Der(𝔤⊗A) is then analyzed. Using the same spectral techniques, they obtain a direct sum decomposition Der(𝔤⊗A) ≅ (Der 𝔤)⊗A ⊕ 𝔤⊗Der A. The second summand is non‑trivial precisely when A possesses non‑zero derivations, which is typical for algebras such as K