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Given an oriented surface  Σ  with base point *∈∂ Σ , we introduce for all  N ≥1, a canonical quasi-Poisson bracket on the space of  N -dimensional linear representations of  π  1 ( Σ ,*). Our bracket extends the well-known Poisson bracket on GL  N  -invariant functions on this space. Our main tool is a natural structure of a quasi-Poisson double algebra (in the sense of M. Van den Bergh) on the group algebra of  π  1 ( Σ ,*).

Quasi-Poisson Structures on Representation Spaces of Surfaces