By Robert M.F. Moss, Charles B. Thomas

Shipped from united kingdom, please permit 10 to 21 enterprise days for arrival. Algebraic K-Theory and its Geometric purposes, paperback, Lecture Notes in arithmetic 108. 86pp. 25cm. ex. lib.

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10), in analogy with the notation for “f -related’’ bivector ﬁelds on a manifold. Similarly, f : N → M is a backward Dirac map, or simply b-Dirac, if LN = {(Y, df ∗ α) | Y ∈ T N, α ∈ T ∗ M and (df (Y ), α) ∈ LM }. 11) If LM and LN are associated with twisted presymplectic structures ωM and ωN , then a b-Dirac map is just a map satisfying f ∗ ωM = ωN . As before, we will write LN = f ∗ LM to denote that f is a b-Dirac map. Note that f ∗ LM is always a well-deﬁned, though not necessarily smooth, subbundle of T N, in contrast with f∗ LN , which may not be well deﬁned at all.

S. 67). Comparing the two ∗ α replaced by any formulas, we see that the resulting equation makes sense for ρM element in C ∞ (M, g∗ ). On the other hand, since the equation is C ∞ (M)-linear with respect to this element, we may assume that the element is a constant a ∈ g∗ (and the remaining appearances of ρM become ρ). s. 68), becomes −(dµ)(ρ(v), (σ ∨ )∗ a) − Lρ(v) µ, (σ ∨ )∗ a + L(σ ∨ )∗ a µ, ρ(v) = a, [v, σ ∨ µ] , Dirac structures, momentum maps, and quasi-Poisson manifolds or, equivalently, −µ([ρ(v), (σ ∨ )∗ a]) = a, [v, σ ∨ µ] .

65). 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) .