By A. M. Naveira
Lawsuits of the Intl convention held to honor the sixtieth birthday of A.M. Naveira. convention was once held July 8-14, 2002 in Valencia, Spain. For graduate scholars and researchers in differential geometry.
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Extra resources for Differential Geometry: Proceedings of the International Symposium Held at Peniscola, Spain, October 3-10, 1982
65) on an arbitrary 1-form µ ∈ 1 (G). The left-hand side gives J ∗ µ, [ρM (v), π (α)] = d(J ∗ µ)(ρM (v), π (α)) + LρM (v) J ∗ µ, π (α) − Lπ (α) J ∗ µ, ρM (v) ∗ ∗ (α)) − LρM (v) µ, (σ ∨ )∗ ρM (α) = −(dµ)(ρ(v), (σ ∨ )∗ ρM + L(σ ∨ )∗ ρM∗ (α) µ, ρ(v) . 67) Evaluating µ on the right-hand side, we get − LρM (v) (α), ρM σ ∨ µ = −LρM (v) α, ρM σ ∨ µ + α, [ρM (v), ρM σ ∨ µ] . Now, using [ρM (v), ρM (v)] ˜ = ρM ([v, v]) ˜ + ρM LρM (v) (v) ˜ for v˜ = σ ∨ µ ∈ ∞ C (M, g), we get ∗ ∗ ∗ −LρM (v) ρM (α), σ ∨ µ + ρM (α), [v, σ ∨ µ] + ρM (α), LρM (v) (σ ∨ µ) .
For any manifold M equipped with an inﬁnitesimal action ρM : g −→ T M, and any g-equivariant map J : M −→ G, the operator 1 C = 1 − ρM ρ ∨ (dJ ) : T M −→ T M 4 and its dual C ∗ : T ∗ M −→ T ∗ M satisfy the formulas ρM σ ∨ σ = CρM , and J ∗ σ σ ∨ = C ∗ J ∗ . 16 follows from the next two propositions, each one describing explicitly one direction of the asserted one-to-one correspondence. 19. Let M be a quasi-Poisson g-manifold, and let A = T ∗ M ⊕ g be its associated Lie algebroid, with anchor r.
The second identity is immediate from the ﬁrst and the last ones. To prove the last identity, we evaluate j (U (α, v)). The ﬁrst component gives 1 1 ∗ −J ∗ − (ρ ∨ )∗ ρM (α) + σ (v) = −J ∗ σ (v) + α − 1 − (ρM ρ ∨ J )∗ α 4 4 ∗ ∗ = −(J σ (v) + C (α)) + α. 59) The second component is 1 ∗ σ ∨ − (ρ ∨ )∗ ρM α + σ (v) . 22). 60) is 1 v − ρ ∨ dJ (ρM (v) + π (α)). , j ◦ U + i ◦ (r, s) = Id. 19. 20. Let J : (M, L) → (G, LG ) be a Dirac realization. Identifying LG with g G, we know that there is an induced action of g on M, denoted by ρM .