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Universal enveloping algebras of Lie-Rinehart algebras: crossed products, connections, and curvature

Xavier Bekaert, Niels Kowalzig, Paolo Saracco

TL;DR

The paper extends Blattner–Cohen–Montgomery theory to left Hopf algebroids, enabling crossed/smash product decompositions of universal enveloping algebras of Lie–Rinehart algebras in the projective setting. For a short exact sequence $0\to\mathfrak n\to\mathfrak g\to\mathfrak h\to0$ of Lie–Rinehart algebras that are projective as $A$-modules, it proves $\,\mathcal U_A(\mathfrak g)\simeq U_A(\mathfrak n)\varhash_\sigma\mathcal U_A(\mathfrak h)$ with a Hopf $2$-cocycle $\sigma$, and gives smash-product decompositions in cases of curved or flat connections. An alternative description via a Lie cocycle $\tau$ and the equivalence to a crossed product $R\times_\tau\mathcal U_A(\mathfrak h)$ is provided, together with a geometric interpretation in terms of invariants on principal bundles and foliations. The framework is illustrated through transformation Lie algebroids, Atiyah algebroids of principal and vector bundles, and foliations, highlighting how Ehresmann connections induce a factorisation of the associative algebra generated by invariant vector fields as a product of vertical/foliation operators and base differential operators. Overall, the results unify algebraic and geometric decompositions of enveloping algebras in the Lie–Rinehart setting and extend classical splitting phenomena to curved and flat connection contexts.

Abstract

We extend a theorem, originally formulated by Blattner-Cohen-Montgomery for crossed products arising from Hopf algebras weakly acting on noncommutative algebras, to the realm of left Hopf algebroids. Our main motivation is an application to universal enveloping algebras of projective Lie-Rinehart algebras: for any given curved (resp. flat) connection, that is, a linear (resp. Lie-Rinehart) splitting of a Lie-Rinehart algebra extension, we provide a crossed (resp. smash) product decomposition of the associated universal enveloping algebra, and vice versa. As a geometric example, we describe the associative algebra generated by the invariant vector fields on the total space of a principal bundle as a crossed product of the algebra generated by the vertical ones and the algebra of differential operators on the base.

Universal enveloping algebras of Lie-Rinehart algebras: crossed products, connections, and curvature

TL;DR

The paper extends Blattner–Cohen–Montgomery theory to left Hopf algebroids, enabling crossed/smash product decompositions of universal enveloping algebras of Lie–Rinehart algebras in the projective setting. For a short exact sequence of Lie–Rinehart algebras that are projective as -modules, it proves with a Hopf -cocycle , and gives smash-product decompositions in cases of curved or flat connections. An alternative description via a Lie cocycle and the equivalence to a crossed product is provided, together with a geometric interpretation in terms of invariants on principal bundles and foliations. The framework is illustrated through transformation Lie algebroids, Atiyah algebroids of principal and vector bundles, and foliations, highlighting how Ehresmann connections induce a factorisation of the associative algebra generated by invariant vector fields as a product of vertical/foliation operators and base differential operators. Overall, the results unify algebraic and geometric decompositions of enveloping algebras in the Lie–Rinehart setting and extend classical splitting phenomena to curved and flat connection contexts.

Abstract

We extend a theorem, originally formulated by Blattner-Cohen-Montgomery for crossed products arising from Hopf algebras weakly acting on noncommutative algebras, to the realm of left Hopf algebroids. Our main motivation is an application to universal enveloping algebras of projective Lie-Rinehart algebras: for any given curved (resp. flat) connection, that is, a linear (resp. Lie-Rinehart) splitting of a Lie-Rinehart algebra extension, we provide a crossed (resp. smash) product decomposition of the associated universal enveloping algebra, and vice versa. As a geometric example, we describe the associative algebra generated by the invariant vector fields on the total space of a principal bundle as a crossed product of the algebra generated by the vertical ones and the algebra of differential operators on the base.
Paper Structure (29 sections, 42 theorems, 314 equations)

This paper contains 29 sections, 42 theorems, 314 equations.

Table of Contents

  1. Preliminaries and first results
  2. Lie-Rinehart algebras
  3. Lie-Rinehart actions, connections, and curvature
  4. Curved semi-direct sum of Lie-Rinehart algebras
  5. Universal enveloping algebras
  6. A symmetrisation map
  7. Left bialgebroids
  8. Left Hopf algebroids
  9. Left module algebras and the smash product
  10. Right comodule algebras
  11. A generalised Blattner-Cohen-Montgomery theorem for left Hopf algebroids
  12. Weak actions, cocycles, and crossed products
  13. Crossed products from coring split exact sequences
  14. Smash and crossed product decomposition of universal enveloping algebras
  15. Short exact sequences of Lie-Rinehart algebras and left Hopf kernels
  16. ...and 14 more sections

Key Result

Theorem 1

If $0 \to \mathfrak{n} \stackrel{\iota}{\hookrightarrow} \mathfrak{g} \stackrel{\pi}{\twoheadrightarrow} \mathfrak{h} \to 0$ is a short exact sequence of Lie-Rinehart algebras which are projective as left $A$-modules, then we have an isomorphism of $A$-rings and right $\mathcal{U}_{{A}}(\mathfrak{h})$-comodule algebras, where $\sigma$ is a suitable $U_{{{A}}}(\mathfrak{n})$-valued Hopf $2$-cocycl

Theorems & Definitions (119)

  • Theorem : Theorem \ref{['thm:mainthmA']}, Corollary \ref{['cor:mainthmB']}, and Theorem \ref{['thm:mainthmC']}
  • Proposition : Proposition \ref{['prop:anothersplitting']}
  • Theorem : Theorem \ref{['thm:sigmatwisted']}
  • Theorem : Theorem \ref{['rain&fog']}
  • Example 1.1: Lie algebroid
  • Definition 1.2: Lie-Rinehart ideal
  • Example 1.3: Kernel of a Lie-Rinehart algebra morphism
  • Definition 1.4: Representations of Lie-Rinehart algebras Huebschmann
  • Proposition 1.5
  • Corollary 1.6
  • ...and 109 more