For converting light energy into electricity, an optical pendulum generator was designed by combining photomechanical movement of liquid-crystalline actuator (LCA) with Faraday's law of electromagnetic induction. Bilayer cantilever actuators were first fabricated with LDPE and LCA. Their photomechanical movement drove the attached copper coils to cut magnetic line of force generating electricity. The output electricity was proportional to the changing rate of the magnetic flux, which was greatly influenced by light intensity, film thickness, and sample size. Continuous electrical output was also achieved. This simple strategy may expand applications of photoactive materials in the capture and storage of light energy.

We determine the $$L_\infty $$ L ∞ -algebra that controls deformations of a relative Rota–Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying $$\mathsf {Lie}\mathsf {Rep}$$ Lie Rep  pair by the dg Lie algebra controlling deformations of the relative Rota–Baxter operator. Consequently, we define the cohomology of relative Rota–Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota–Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a homotopy relative Rota–Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota–Baxter Lie algebras is intimately related to pre-Lie$$_\infty $$ ∞ -algebras.

Bilayer photomechanical films are fabricated by depositing one layer of molecular azobenzene chromophores onto flexible low-density polyethylene substrates. The photoinduced bending and unbending behavior of five azobenzene derivatives including azobenzene, 4-hydroxy-azobenzene, 4-((4-hydroxyphenyl)diazenyl)bezoitrile, 4-((4-methoxyph-enyl)diazenyl)phenol, and 4-(phenyldiazenyl)phenol is systematically studied by considering the incident light intensity and the thickness of the coated chromophore layers. Precise control of photoinduced curling of the bilayer film is successfully achieved upon irradiation with two beams of UV light, and the curled films can be recovered by thermal relaxation in the dark. The easily fabricated bilayer films show fast photomechanical response, strong photoinduced stress, and stability similar to crosslinked polymeric films.

The main object of study of this paper is the notion of a LieDer pair, i.e. a Lie algebra with a derivation. We introduce the concept of a representation of a LieDer pair and study the corresponding cohomologies. We show that a LieDer pair is rigid if the second cohomology group is trivial, and a deformation of order n is extensible if its obstruction class, which is defined to be an element is the third cohomology group, is trivial. We classify central extensions of LieDer pairs using the second cohomology group with the coefficient in the trivial representation. For a pair of derivations, we define its obstruction class and show that it is extensible if and only if the obstruction class is trivial. Finally, we classify Lie2Der pairs using the third cohomology group of a LieDer pair.