[Paper Review] Twisting on associative algebras and Rota-Baxter type operators

Kyousuke Uchino|ArXiv.org|Oct 23, 2007

Advanced Topics in Algebra28 references3 citations

TL;DR

This paper introduces a twisting operation on associative algebras inspired by Drinfeld's twisting in Lie theory, using derived bracket constructions on the Hochschild complex. It establishes that Rota-Baxter operators and Nijenhuis operators arise as solutions to Maurer-Cartan equations in a differential graded Lie algebra structure, providing a unified framework for deformation theory and operator algebras in associative settings.

ABSTRACT

We will introduce an operation "twisting" on Hochschild complex by analogy with Drinfeld's twisting operations. By using the twisting and derived bracket construction, we will study differential graded Lie algebra structures associated with bi-graded Hochschild complex. We will show that Rota-Baxter type operators are solutions of Maurer-Cartan equations. As an application of twisting, we will give a construction of associative Nijenhuis operators.

Motivation & Objective

• To extend Drinfeld's twisting operations from Lie to associative algebras using Hochschild cohomology.
• To construct a differential graded Lie algebra structure on the Hochschild complex of a twilled algebra.
• To show that Rota-Baxter operators and Nijenhuis operators emerge as solutions to Maurer-Cartan equations in this framework.
• To provide a cohomological characterization of associative Nijenhuis operators via twisting and derived bracket constructions.

Proposed method

• Define a bigraded Lie algebra structure on the Hochschild complex $ C^*( A_1 \oplus \t A_2) $ using Gerstenhaber's bracket.
• Introduce a twisting operation via a 1-cochain $ H: \t A_2 \to \t A_1 $, modeled on Hamiltonian vector fields in Poisson geometry.
• Decompose the associative structure $ \theta $ into four components: $ \hat{\phi}_1, \hat{\mu}_1, \hat{\mu}_2, \hat{\phi}_2 $, and derive transformation rules under twisting.
• Use the derived bracket construction to induce a dg-Lie algebra structure on $ C^*(\t A_2, \t A_1) $, enabling deformation theory.
• Show that Rota-Baxter operators of weight zero satisfy the generalized Rota-Baxter identity and yield solutions to the Maurer-Cartan equation.
• Construct Nijenhuis operators as compositions of integral operators and derivations, proving they satisfy the Nijenhuis condition via explicit verification.

Experimental results

Research questions

• RQ1 How can Drinfeld's twisting operation be generalized from Lie to associative algebras using Hochschild cohomology?
• RQ2 What is the role of the derived bracket construction in linking associative twilled algebras to differential graded Lie algebras?
• RQ3 Under what conditions do Rota-Baxter operators on an algebra correspond to solutions of the Maurer-Cartan equation in the induced dg-Lie algebra?
• RQ4 How do strong Maurer-Cartan operators induce Nijenhuis operators on associative algebras?
• RQ5 What is the cohomological significance of the generalized Rota-Baxter identity in the context of twisting and deformation theory?

Key findings

• Rota-Baxter operators of weight zero on an associative algebra induce solutions to the Maurer-Cartan equation in the dg-Lie algebra $ C^*(\t A_2, \t A_1) $, establishing a cohomological link to deformation theory.
• The twisting operation on associative algebras is fully determined by transformation rules of the four components $ \hat{\phi}_1, \hat{\mu}_1, \hat{\mu}_2, \hat{\phi}_2 $, with explicit formulas provided in Theorem 4.5.
• A strong Maurer-Cartan operator $ \Omega $, defined as a derivation from $ \mathcal{A} $ to $ M_\pi $, induces a Nijenhuis operator $ N = \int \Omega $, which satisfies the Nijenhuis condition.
• In the example of $ C^1([0,1]) $, the integral operator $ \pi(f)(x) = \int_0^x f(t)dt $ and a derivation $ \Omega(f)(x) = \omega(x)f'(x) $ yield a Nijenhuis operator $ N(f)(x) = \int_0^x \omega(t)f'(t)dt $, explicitly verified to satisfy the Nijenhuis identity.
• In noncommutative settings, such as formal power series algebras or the Weyl algebra, the formal integral operator is a Rota-Baxter operator of weight zero, and the composition $ N = \int \Omega $ yields a projection-like Nijenhuis operator.
• The Nijenhuis operator $ N $ on the Weyl algebra $ W\langle x, \partial_x \rangle $ acts as a projection onto elements of the form $ k_{ij} \partial_x^i * x^j $ with $ j \neq 0 $, confirming its geometric and algebraic significance.