Controlling graded Lie algebras and cohomologies of double Lie algebras
Abstract
We construct a graded Lie algebra structure on the space of cyclically skew-symmetric cochains whose Maurer-Cartan elements characterize double Lie algebra structures. Twisting by a fixed double Lie bracket yields a cochain complex that is isomorphic to the positive arity part of the complex introduced by Fairon and Valeri after reindexing and degreewise sign adjustment. Using the Koszul resolution of the double Lie properad, we identify the resulting differential graded Lie algebra with the convolution differential graded Lie algebra on the deformation complex of the fixed double Lie algebra. We also construct a graded Lie algebra governing skew-symmetric Rota-Baxter operators on a finite-dimensional symmetric Frobenius algebra. For a finite-dimensional vector space V and A=en(V), we lift the known correspondence between double Lie algebra structures on V and skew-symmetric Rota-Baxter operators on A to an isomorphism of their governing graded Lie algebras. After twisting by corresponding Maurer-Cartan elements, this isomorphism becomes an isomorphism of differential graded Lie algebras and hence induces an isomorphism between the associated cohomology groups. We further prove that the cohomology of a skew-symmetric Rota-Baxter operator on a symmetric Frobenius algebra can be seen as the cyclic cohomology of its descendent associative algebra. Thus the cohomology of a double Lie algebra is identified with the cyclic cohomology of the descendent associative algebra associated with the skew-symmetric Rota-Baxter operator on the matrix algebra.
Related
- Deformations and the controlling L∞-structure of extended Rota-Baxter Lie algebras
- Deformation theory and the controlling L∞-structure of extended Rota-Baxter algebras
- Graded Betti numbers of graded Möbius algebras of uniform matroids
- Graded identities for matrix algebras of order two over a finite field
- The associated graded algebras of biserial fractional Brauer graph algebras
BibTeX
@misc{xu2026controlling,
title = {{Controlling graded Lie algebras and cohomologies of double Lie algebras}},
author = {Zhongyin Xu and Chengming Bai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27420},
eprint = {2609.27420},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27420},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Controlling graded Lie algebras and cohomologies of double Lie algebras
AU - Zhongyin Xu
AU - Chengming Bai
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27420
UR - https://arxiv.org/abs/2609.27420
AB - We construct a graded Lie algebra structure on the space of cyclically skew-symmetric cochains whose Maurer-Cartan elements characterize double Lie algebra structures. Twisting by a fixed double Lie bracket yields a cochain complex that is isomorphic to the positive arity part of the complex introduced by Fairon and Valeri after reindexing and degreewise sign adjustment. Using the Koszul resolution of the double Lie properad, we identify the resulting differential graded Lie algebra with the convolution differential graded Lie algebra on the deformation complex of the fixed double Lie algebra. We also construct a graded Lie algebra governing skew-symmetric Rota-Baxter operators on a finite-dimensional symmetric Frobenius algebra. For a finite-dimensional vector space V and A=en(V), we lift the known correspondence between double Lie algebra structures on V and skew-symmetric Rota-Baxter operators on A to an isomorphism of their governing graded Lie algebras. After twisting by corresponding Maurer-Cartan elements, this isomorphism becomes an isomorphism of differential graded Lie algebras and hence induces an isomorphism between the associated cohomology groups. We further prove that the cohomology of a skew-symmetric Rota-Baxter operator on a symmetric Frobenius algebra can be seen as the cyclic cohomology of its descendent associative algebra. Thus the cohomology of a double Lie algebra is identified with the cyclic cohomology of the descendent associative algebra associated with the skew-symmetric Rota-Baxter operator on the matrix algebra.
ER -