Journal of Homotopy and Related Structures
We show that a double Lie algebroid, together with a chosen decomposition, is equivalent to a pair of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the double Lie algebroids arising from the tangent bundle of a Lie algebroid and the cotangent bundle of a Lie bialgebroid.
Double Lie algebroids, Representations up to homotopy, Matched pairs
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Gracia-Saz, A.; Lean, M. Jotz; Mackenzie, K. C. H.; and Mehta, Rajan Amit, "Double Lie Algebroids and Representations up to Homotopy" (2018). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.