[Paper Review] Lie superalgebras of differential operators

Janusz Grabowski, Alexei Kotov+1 more|arXiv (Cornell University)|Nov 8, 2010

Advanced Topics in Algebra6 references3 citations

TL;DR

This paper classifies Lie superalgebra isomorphisms between first-order superdifferential operator algebras on smooth supermanifolds, proving they are induced by supermanifold diffeomorphisms and automorphisms linked to even superdivergences. The automorphism group is shown to be a semi-direct product of diffeomorphism-induced automorphisms and divergence-induced automorphisms, with superdivergences proven to exist globally on any supermanifold via Berezinian densities.

ABSTRACT

We describe explicitly Lie superalgebra isomorphisms between the Lie superalgebras of first-order superdifferential operators on supermanifolds, showing in particular that any such isomorphism induces a diffeomorphism of the supermanifolds. We also prove that the group of automorphisms of such a Lie superalgebra is a semi-direct product of the subgroup induced by the supermanifold diffeomorphisms and another subgroup which consists of automorphisms determined by even superdivergences. These superdivergences are proven to exist on any supermanifold and their local form is explicitly described as well

Motivation & Objective

• To classify Lie superalgebra isomorphisms between the Lie superalgebras of first-order superdifferential operators on smooth supermanifolds.
• To determine the structure of the automorphism group of such Lie superalgebras.
• To prove the existence of even superdivergences on any smooth supermanifold.
• To characterize the unique maximal super Lie ideal of ad-nilpotent elements in the first-order superdifferential operator algebra.
• To relate automorphisms of the superdifferential operator algebra to those of the supervector field algebra via cohomological data.

Proposed method

• The authors use the canonical splitting $\mathcal{D}^1(\mathcal{M}) = \mathcal{X}(\mathcal{M}) \oplus \mathcal{A}(\mathcal{M})$ to decompose automorphisms into diffeomorphism-induced and divergence-induced components.
• They characterize $\mathcal{A}(\mathcal{M})$, the algebra of smooth functions, as the unique maximal super Lie ideal of ad-nilpotent elements in $\mathcal{D}^1(\mathcal{M})$ when $\dim\mathcal{M} \neq 0|1$, ensuring automorphisms preserve this ideal.
• The automorphisms are shown to be of the form $X + f \mapsto X + f + c(X)$, where $c: \mathcal{X}(\mathcal{M}) \to \mathcal{A}(\mathcal{M})$ is a 1-cocycle, linking automorphisms to Lie superalgebra cohomology.
• The existence of superdivergences is established via Berezinian densities: a global nowhere-vanishing section $\sigma$ of the 1-density sheaf $\mathfrak{D}_1 = \mathrm{Ber} \otimes \mathrm{or}(M)$ induces a divergence $\gamma_\sigma(X) = (\mathcal{L}_X \sigma)\sigma^{-1}$.
• The cohomology of $\mathcal{X}(\mathcal{M})$ with values in $\mathcal{A}(\mathcal{M})$ is computed, showing every 1-cocycle is a combination of a closed even 1-form and a fixed divergence.
• The construction relies on the fact that every supermanifold is diffeomorphic to $\Pi V$ for a vector bundle $V$, allowing a global identification of $\mathcal{A}(\mathcal{M}) \simeq \Gamma(\Lambda^\bullet V^*)$.

Experimental results

Research questions

• RQ1 What is the structure of the automorphism group of the Lie superalgebra of first-order superdifferential operators on a smooth supermanifold?
• RQ2 How are automorphisms of $\mathcal{D}^1(\mathcal{M})$ related to automorphisms of the supervector field algebra $\mathcal{X}(\mathcal{M})$?
• RQ3 Can every automorphism of $\mathcal{D}^1(\mathcal{M})$ be decomposed into a diffeomorphism-induced part and a divergence-induced part?
• RQ4 What is the role of superdivergences in classifying automorphisms of $\mathcal{D}^1(\mathcal{M})$?
• RQ5 Does every smooth supermanifold admit a globally defined even superdivergence?

Key findings

• The automorphism group of $\mathcal{D}^1(\mathcal{M})$ is isomorphic to a semi-direct product of the group of supermanifold diffeomorphisms and the group of automorphisms induced by even superdivergences.
• Any automorphism of $\mathcal{D}^1(\mathcal{M})$ preserves the subalgebra $\mathcal{A}(\mathcal{M})$ of smooth functions, which is the unique maximal super Lie ideal of ad-nilpotent elements when $\dim\mathcal{M} \neq 0|1$.
• Automorphisms of $\mathcal{D}^1(\mathcal{M})$ are completely determined by a supermanifold diffeomorphism and a 1-cocycle $c: \mathcal{X}(\mathcal{M}) \to \mathcal{A}(\mathcal{M})$, with only those induced by diffeomorphisms extending from $\mathcal{X}(\mathcal{M})$ to $\mathcal{D}^1(\mathcal{M})$.
• Every smooth supermanifold admits a globally defined even superdivergence, constructed via a nowhere-vanishing section of the 1-density sheaf $\mathfrak{D}_1 = \mathrm{Ber} \otimes \mathrm{or}(M)$.
• The first cohomology $H^1(\mathcal{X}(\mathcal{M}), \mathcal{A}(\mathcal{M}))$ is isomorphic to the direct sum of the space of closed even 1-forms and the space of constant functions, reflecting the decomposition of 1-cocycles.
• The divergence $\gamma_\sigma$ associated with a global section $\sigma$ of $\mathfrak{D}_1$ is well-defined and independent of the sign of $\sigma$, with $\gamma_\sigma(X) = (\mathcal{L}_X \sigma)\sigma^{-1}$.