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We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the "odd laplacian", ∆, of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples include generators of the Schouten bracket of multivectors on a manifold (the supermanifold being the cotangent bundle where the coordinates in the fibres are odd) and generators of the Koszul-Schouten bracket of forms on a Poisson manifold (the supermanifold being the tangent bundle, with odd coordinates on the fibres).

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Sexual selection has been invoked as a major force in the evolution of secondary sexual traits, including sexually dimorphic colorations. For example, previous studies have shown that display complexity and elaborate ornamentation in lizards are associated with variables that reflect the intensity of intrasexual selection. However, these studies have relied on techniques of colour analysis based on human -rather than lizard -visual perception. Here, we use reflectance spectrophotometry and visual modelling to quantify sexual dichromatism considering the overall colour patterns of lacertids, a lizard clade in which visual signalling has traditionally been underrated. These objective methods of colour analysis reveal a large, previously unreported, degree of sexual dichromatism in lacertids. Using a comparative phylogenetic approach, we further demonstrate that sexual dichromatism is positively associated with body size dimorphism (an index of intrasexual selection), suggesting that conspicuous coloration in male lacertids has evolved to improve opponent assessment under conditions of intense malemale competition. Our findings provide the first evidence for the covariation of sexual dichromatism and sexual size dimorphism in lacertids and suggest that the prevalent role of intrasexual selection in the evolution of ornamental coloration is not restricted to the iguanian lineage, but rather may be a general trend common to many diurnal lizards.

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