[Paper Review] L-infinity Formality check for the Hochschild Complex of Certain Universal Enveloping Algebras

Martin Bordemann, Olivier Elchinger+2 more|arXiv (Cornell University)|Jul 9, 2018

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper investigates $L_∞u0020$-formality for the Hochschild complex of universal enveloping algebras of certain Lie algebras, including Cartan-3-regular quadratic Lie algebras (e.g., $\mathfrak{so}(3)$) and free Lie algebras. Using the characteristic $3$-class obstruction and homotopy transfer, it shows formality fails in these cases but restores $L_\infty$-quasi-isomorphism by explicitly computing a single non-trivial $d_3$ bracket of order 3.

ABSTRACT

We study the L-infinity-formality problem for the Hochschild complex of the universal enveloping algebra of some examples of Lie algebras such as Cartan-3-regular quadratic Lie algebras (for example semisimple Lie algebras and in more detail so(3)), and free Lie algebras generated by a vector space of dimension at least 2. We show that for these examples formality in Kontsevich's sense does NOT hold, although some of them allow unconditioned deformability. We compute the L-infinity-structure on the cohomology given by homotopy transfer in certain cases.

Motivation & Objective

• To determine whether the Hochschild complex of universal enveloping algebras of specific Lie algebras admits $L_\infty$-formality in Kontsevich's sense.
• To investigate the obstruction to $L_\infty$-formality using the characteristic $3$-class $c_3$ in graded Chevalley-Eilenberg cohomology.
• To compute higher-order $L_\infty$ brackets on cohomology via homotopy transfer when formality fails.
• To explicitly construct the $d_3$ bracket restoring $L_\infty$-quasi-isomorphism in non-formal cases, particularly for $\mathfrak{so}(3)$ and free Lie algebras.

Proposed method

• Use the characteristic $3$-class $c_3$ as the primary obstruction to $L_\infty$-formality in the graded Chevalley-Eilenberg cohomology of the Hochschild cohomology graded Lie algebra.
• Apply Kontsevich’s formality map to relate the Hochschild complex of $U(\mathfrak{g})$ to the Chevalley-Eilenberg complex of $\mathfrak{g}$ with values in $\mathcal{S}\mathfrak{g}$.
• Compute Schouten brackets of the quadratic Casimir and the Cartan $3$-cocycle in the case of quadratic Lie algebras to evaluate $c_3$.
• For free Lie algebras, exploit the isomorphism $U(\mathfrak{f}(V)) \cong \mathcal{T}^\bullet V$ and compute a complement to inner derivations in the space of derivations.
• Employ the $L_\infty$-form of the homotopy perturbation lemma to reconstruct higher brackets on cohomology when formality fails.
• Explicitly construct the $d_3$ bracket of order 3 to restore $L_\infty$-quasi-isomorphism in both $\mathfrak{so}(3)$ and free Lie algebra cases.

Experimental results

Research questions

• RQ1 Does the Hochschild complex of the universal enveloping algebra of a nonabelian reductive Lie algebra admit $L_\infty$-formality?
• RQ2 What is the nature of the obstruction to $L_\infty$-formality for $\mathfrak{so}(3)$ and free Lie algebras?
• RQ3 Can the $L_\infty$-quasi-isomorphism be restored by adding a single higher-order bracket $d_3$ in non-formal cases?
• RQ4 How does the characteristic $3$-class $c_3$ detect failure of formality in the Hochschild complex of $U(\mathfrak{g})$?
• RQ5 What is the explicit structure of the $d_3$ bracket restoring $L_\infty$-quasi-isomorphism for $\mathfrak{so}(3)$ and free Lie algebras?

Key findings

• The Hochschild complex of the universal enveloping algebra of a nonabelian reductive Lie algebra, including $\mathfrak{so}(3)$, is not $L_\infty$-formal.
• The failure of formality is obstructed by a nontrivial characteristic $3$-class $c_3$ in the graded Chevalley-Eilenberg cohomology of the Hochschild cohomology graded Lie algebra.
• For $\mathfrak{so}(3)$, the $L_\infty$-quasi-isomorphism is restored by adding a single explicit $d_3$ bracket of order 3.
• For free Lie algebras generated by a vector space of dimension $\geq 2$, formality also fails, and the $L_\infty$-quasi-isomorphism is restored by adding a single $d_3$ bracket of order 3.
• Despite the lack of formality, the deformation problem for these associative algebras is always solvable due to their rigidity, showing formality is not necessary for integrability.
• The $d_3$ bracket is explicitly computed in both cases using homotopy transfer and the structure of derivations and inner derivations in the free algebra case.