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By Tucker R.W., Calmet J., Seiler W.M.
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Example text
N=1 k=0 In the same way, we can speak about an A∞-coalgebra as a dg-module that has to verify the dual properties (in this case the operations are denoted by ∆i). Thanks to the perturbation theory, to speak about an A∞ -(co)algebra M is equivalent to give a reduction from a dg-(co)algebra to the dg-module M. Let us recall how it is possible. The key in the homological perturbation theory is the Basic Perturbation Lemma (briefly, BPL [2]), which is an algorithm whose input is a reduction of dg–modules c : {N, M, f, g, φ} and a perturbation datum δ of dN whose output is a new reduction cδ .
Consider a system E, which describes automorphisms of a given geometric structure. The corresponding symbolic system is g ⊂ ST ∗ ⊗ T . The automorphism group has maximal dimension iff the system is formally integrable. Consider the examples, when the geometric structure is symplectic, complex or Riemannian (all these structures are of the first order). Let at first g be generated by g1 = sp(n) ⊂ T ∗ ⊗ T . Our tangent space T = TxM is equipped with a symplectic structure ω, and we can ω identify T ∗ ≃ T and we get g1 = S 2 T ∗ ⊂ T ∗ ⊗ T ∗ .
1) are denoted as X = (x1, . . 1): 0 0 0 A= 0 0 0 ˜ ∗ (X) with n = 6, where X ∗ = AX ∗ + Φ 0 0 0 0 0 0 (c − 1)p0 0 0 −1 0 0 0 1/c 0 . 5) a = σp20 + 1 + 1/c σ, b = 1/c + σp20 (1/c − 1) . 6) where Eq. 5) has two zero roots and twin roots λ1 = λ2 = 0, λ3 = −λ4 , λ5 = −λ6 . 7) ˜ σ be the part of the family Sσ with p2 ∈ R. Evidently S ˜ σ ⊃ Re Sσ . Let S 0 ˜ σ only. 7′) is represented in Fig. 1. 7′) into five sets D1, D2 , D3, D4, D5 . 618). 45 GIFT 2006 ¯3, In the set D1 eigenvalues λ3 , λ4, λ5 , λ6 are complex: λ4 = −λ3 , λ5 = λ ¯ 3 ; in sets D2 and D3 two of them are real and another two are λ6 = −λ pure imaginary, in D4 they are pure imaginary, in D5 they are real.