[Paper Review] Differential operators on Lie and graded Lie algebras

G. Sardanashvily|arXiv (Cornell University)|Apr 1, 2010

Advanced Topics in Algebra13 references3 citations

TL;DR

This paper introduces a novel framework for differential operators on Lie and graded Lie algebras by treating their multiplications as derivations—hence first-order differential operators. It defines higher-order operators as compositions of first-order ones and establishes a Chevalley–Eilenberg-type differential calculus on Lie algebras, showing equivalence to the standard complex when the algebra has trivial center and all derivations are inner. The approach generalizes to finite-dimensional Lie algebras, Poisson algebras, vector fields, and canonical commutation/anticommutation relation algebras.

ABSTRACT

Theory of differential operators on associative algebras is not extended to the non-associative ones in a straightforward way. We consider differential operators on Lie algebras. A key point is that multiplication in a Lie algebra is its derivation. Higher order differential operators on a Lie algebra are defined as composition of the first order ones. The Chevalley--Eilenberg differential calculus over a Lie algebra is defined. Examples of finite-dimensional Lie algebras, Poisson algebras, algebras of vector fields, and algebras of canonical commutation relations are considered. Differential operators on graded Lie algebras are defined just as on the Lie ones.

Motivation & Objective

• To extend the theory of differential operators from commutative and associative algebras to non-associative Lie and graded Lie algebras, where standard definitions fail due to non-module structures of derivations.
• To resolve the foundational issue that derivations in non-associative algebras do not satisfy the standard higher-order differential operator condition, by redefining higher-order operators as compositions of first-order ones.
• To construct a Chevalley–Eilenberg-type differential calculus on Lie algebras, generalizing the classical complex when the algebra has trivial center and all derivations are inner.
• To provide concrete examples across finite-dimensional Lie algebras, Poisson algebras, vector fields, and canonical commutation/anticommutation relation algebras, demonstrating the applicability of the framework.
• To extend the theory to graded Lie algebras and modules, including graded manifolds and Grassmann algebras, showing that derivations act as first-order differential operators on these structures.

Proposed method

• Defining differential operators on a Lie algebra $ A $ via the iterated action $ ilde{ abla}_{a_0} ilde{ abla}_{a_1} ilde{ abla}_{a_k} abla = 0 $, where $ ilde{ abla}_a abla(b) = a abla(b) - abla(ab) $, generalizing the standard commutative ring definition.
• Establishing that multiplication in a Lie algebra is a derivation, hence a first-order differential operator, which serves as the foundation for higher-order operators.
• Constructing the Chevalley–Eilenberg differential calculus on a Lie algebra $ A $ by defining a differential complex based on the Lie algebra cohomology structure.
• Proving that when $ A $ has trivial center and all derivations are inner, the Chevalley–Eilenberg calculus coincides with the standard Chevalley–Eilenberg complex.
• Extending the framework to graded Lie algebras by defining graded derivations and using the graded Lie bracket $ [u, u'] = u hd u' - (-1)^{|u||u'|} u' hd u $, ensuring consistency with supergeometry.
• Applying the formalism to physical examples: algebras of vector fields, Poisson algebras, and the CAR algebra (canonical anticommutation relations), showing that first-order differential operators on modules arise naturally from derivations.

Experimental results

Research questions

• RQ1 How can differential operators be consistently defined on non-associative Lie algebras, given that derivations do not form modules and standard definitions fail?
• RQ2 Can higher-order differential operators on Lie algebras be meaningfully defined without relying on the standard commutative ring framework?
• RQ3 Under what conditions does the proposed Chevalley–Eilenberg differential calculus on a Lie algebra reduce to the classical Chevalley–Eilenberg complex?
• RQ4 How do derivations of Lie algebras, such as those in the CAR algebra, act as first-order differential operators on modules like Grassmann algebras?
• RQ5 To what extent can this framework be generalized to graded Lie algebras and their modules, particularly in the context of graded manifolds and supergeometry?

Key findings

• The multiplication operation in any Lie algebra is a derivation, hence a first-order differential operator, providing a natural starting point for defining higher-order operators.
• Higher-order differential operators on Lie algebras are defined as compositions of first-order operators, circumventing the failure of standard definitions in non-associative settings.
• The Chevalley–Eilenberg differential calculus on a Lie algebra $ A $ is constructed via the standard cohomological complex, and it coincides with the classical Chevalley–Eilenberg complex when $ A $ has trivial center and all derivations are inner.
• For finite-dimensional Lie algebras, the Chevalley–Eilenberg calculus describes matrix geometry, providing a geometric interpretation of cohomology in terms of differential operators.
• In the case of the CAR algebra (canonical anticommutation relations), generic derivations are given by $ abla w = (M_1 heta + O_1 heta', M_2 heta' + O_2 heta, C_1 heta + C_2 heta') $, with symmetry constraints on the endomorphisms.
• First-order differential operators on the Grassmann algebra $ igwedge ext{span}igig(cigig) $, viewed as a $ W $-module, take the form $ abla = (C_2 - M c) rac{ abla}{ abla c} + C_1 c + u $, satisfying $ abla(wh) = ( abla w)h + w abla(h) $, where $ abla w $ is a derivation of $ W $.