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Homotopy theory of the master equation package applied to algebra and geometry: a sketch of two interlocking programs

100%
Sullivan D.
Banach Center Publications
|
2009
|
tom 85
|
nr 1
297-305
EN
Using the algebraic theory of homotopies between maps of dga's we obtain a homotopy theory for algebraic structures defined by collections of multiplications and comultiplications. This is done by expressing these structures and resolved versions of them in terms of dga maps. This same homotopy theory of dga maps applies to extract invariants beyond homological periods from systems of moduli spaces that determine systems of chains that satisfy master equations like dX + X*X = 0. Minimal models of these objects resemble Postnikov decompositions in the homotopy theory of spaces and maps.
2
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A formula for topology/deformations and its significance

63%
Lawrence R., Sullivan D.
Fundamenta Mathematicae
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2014
|
tom 225
|
nr 1
229-242
EN
The formula is $∂e = (ad_e)b + ∑_{i=0}^{∞} (B_i)/i! (ad_e)^{i}(b-a)$, with ∂a + 1/2 [a,a] = 0 and ∂b + 1/2 [b,b] = 0, where a, b and e in degrees -1, -1 and 0 are the free generators of a completed free graded Lie algebra L[a,b,e]. The coefficients are defined by $x/(e^{x} - 1) = ∑_{n=0}^{∞} Bₙ/n! xⁿ$. The theorem is that ∙ this formula for ∂ on generators extends to a derivation of square zero on L[a,b,e]; ∙ the formula for ∂e is unique satisfying the first property, once given the formulae for ∂a and ∂b, along with the condition that the "flow" generated by e moves a to b in unit time. The immediate significance of this formula is that it computes the infinity cocommutative coalgebra structure on the chains of the closed interval. It may be derived and proved using the geometrical idea of flat connections and one-parameter groups or flows of gauge transformations. The deeper significance of such general DGLAs which want to combine deformation theory and rational homotopy theory is proposed as a research problem.
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