Homology of Free-Lie Current Algebras over Square-Zero Extensions

Definitions and mathematical statement

Let $E$ and $V$ be arbitrary finite-dimensional complex vector spaces. Define the commutative unital complex algebra $A={\mathbb{C}}\oplus E$ by \[ (\lambda,e)(\mu,f)=(\lambda\mu,\lambda f+\mu e), \] so that $E$ is an ideal with $E^2=0$. Let $\mathfrak g=\operatorname{Lie}(V)$ be the free complex Lie algebra generated by $V$: every complex-linear map from $V$ to a complex Lie algebra extends uniquely to a Lie algebra homomorphism from $\mathfrak g$. Form the current Lie algebra \[ L(E,V)=\mathfrak g\otimes_{{\mathbb{C}}}A, \qquad [x\otimes a,y\otimes b]=[x,y]\otimes ab. \] Use ordinary Chevalley--Eilenberg homology with trivial coefficients ${\mathbb{C}}$. Explicitly, let $C_n(L)=\bigwedge^n_{{\mathbb{C}}}L$, with differential \[ d_n(x_1\wedge\cdots\wedge x_n) =\sum_{1\leq i<j\leq n}(-1)^{i+j+1}[x_i,x_j]\wedge x_1\wedge\cdots\wedge\widehat{x_i}\wedge\cdots\wedge\widehat{x_j}\wedge\cdots\wedge x_n, \] where a hat denotes omission and $d_0=0$. Write \[ H_n(L;{\mathbb{C}})=\ker(d_n)/\operatorname{im}(d_{n+1}), \qquad H_*(L;{\mathbb{C}})=\bigoplus_{n\geq0}H_n(L;{\mathbb{C}}). \] The coefficient choice is the usual interpretation of the supplied notation $H_*(L)$; the source record does not explicitly specify coefficients.

Problem. Determine $H_n(L(E,V);{\mathbb{C}})$ for every integer $n\geq0$ and every pair of finite-dimensional complex vector spaces $E,V$, preferably by a decomposition into a direct sum of polynomial functors in $E$ and $V$, natural under complex-linear maps in both variables. Here a polynomial functor means a functor whose induced maps depend polynomially on the matrix entries of the input linear maps. The source does not specify polynomial degrees, multiplicities, an irreducible decomposition, or whether a finer grading is intended; these remain qualifications of the preferred form of the answer.

Short English statement

For arbitrary finite-dimensional complex vector spaces $E,V$, compute all homology groups of the free Lie algebra on $V$ tensored with the square-zero algebra ${\mathbb{C}}\oplus E$. Preferably express the answer naturally as a sum of polynomial functors of $E,V$. Trivial complex coefficients are the conventional interpretation used here; the supplied record leaves them implicit.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003713$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The local registry locations are data/unsolvedmath/problems.json and the supplementary catalog/problems.json; the supplied record is the evidence used for this draft.
• Supplied report citation: Topology of Arrangements and Representation Stability (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/2, pp. 43--123. https://doi.org/10.4171/owr/2018/2. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. It assigns proposed year 2018 and supplies no individual proposer.
• Supplied literature reference: Zusmanovich, current Lie algebras. https://arxiv.org/abs/0808.0217.
• Supplied literature reference: Oberwolfach Report 2/2018. https://doi.org/10.4171/owr/2018/2.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that general formulas exist for important low-degree cases of current Lie-algebra homology, but that no general computation of $H_*(\mathfrak g\otimes A)$ for the stated broad class is available. This is supplied dated literature triage derived from the supplied OWR curation, not an independently verified assessment. Time-sensitive status and cited claims require independent verification.
• The supplied registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. This is a supplied editorial qualification, not mathematical verification of this draft.