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A homology cylinder of a surface induces an automorphism of the completedgroup ring of the fundamental group of the surface. We introduce a new methodof computing the automorphism by using the Goldman Lie algebra of the surfaceor some skein algebra. In particular, we give a refinement of a formula by Kunoand Massuyeau.

The total Johnson homomorphism on the homology cylinder and the bracket-quantization HOMFLY-PT skein algebra