# Algebraic K-Theory and its Geometric Applications by Robert M.F. Moss, Charles B. Thomas

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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Example text

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) .