Commutative algebras in Drinfeld categories of abelian Lie algebras

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We describe (braided-)commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over these algebras and classify commutative algebras with finite number of simple local modules.

💡 Research Summary

The paper investigates commutative algebras equipped with a non‑degenerate multiplicative form inside Drinfeld categories associated with abelian metric Lie algebras. Starting from an abelian Lie algebra 𝔤 together with a symmetric invariant bilinear form ⟨·,·⟩, the authors construct the braided monoidal category 𝒞(𝔤,⟨·,·⟩), whose objects are 𝔤‑modules and whose braiding is given by the R‑matrix R=exp(πi Ω), where Ω is the Casimir element determined by ⟨·,·⟩. Because 𝔤 is abelian, every module decomposes into weight spaces indexed by characters λ∈𝔤*, and the braiding acts on a pair of weight spaces V_λ, V_μ by the scalar exp(πi⟨λ,μ⟩) followed by the usual flip.

Within this setting the authors define a “non‑degenerate multiplicative form” β: A⊗A→k on an algebra A∈𝒞. The form is required to be invariant under the multiplication μ of A and to be non‑singular, i.e. the induced map A→A^* is an isomorphism. This makes A a Frobenius‑type object, but the braiding modifies the usual commutativity condition. The first main theorem shows that a braided‑commutative algebra with such a β exists if and only if the set of weights Λ⊂𝔤* of A forms a lattice that is integral with respect to the quadratic form q(λ)=⟨λ,λ⟩/2. In that case the multiplication is essentially the group algebra k