nLab derived bracket

Idea

Given a differential graded Lie algebra $(V, \mathrm{d}, [-,-], )$, its derived bracket (Kosmann-Schwarzbach 1996) is the composition

$$\big[ \mathrm{d}(-) , - \big] \;\colon\; V \otimes V \longrightarrow V$$

of the Lie bracket with the differential .

Related concepts

• Lie bracket

References

The concept originates with:

• Yvette Kosmann-Schwarzbach: From Poisson algebras to Gerstenhaber algebras, Annales de l’institut Fourier, tome 46, no 5 (1996) 1243–1274 [numdam:AIF_1996__46_5_1243_0]

• Yvette Kosmann-Schwarzbach: Derived brackets, Lett. Math. Phys. 69 (2004) 61–87 [doi:10.1007/s11005-004-0608-8, arXiv:math/0312524]

Generalization:

• Theodore Voronov: Higher derived brackets and homotopy algebras, J. Pure and Appl. Algebra. 202 1–3 (2005) 133–153 [doi:10.1016/j.jpaa.2005.01.010, arXiv:math/0304038]