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MAKE IT YOURSELF STRONG HOMOTOPY STRUCTURE

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MAKE IT YOURSELF STRONG HOMOTOPY STRUCTURE OLGA KRAVCHENKO Abstract This are notes for the talk at the conference GAP III, in Perugia. We talk about a general definition of a strong homotopy structure as a solution of a Maurer-Cartan equation on a corresponding governing Lie algebra. We apply this philosophy to construct strong homotopy Lie bialgebras. 1. Basic example: strong homotopy associative algebra Why is it called homotopy? In fact, the following general definition of chain homotopic maps is used over and over again. Definition 1. Two maps ? and ? between complexes (C, d) = · · · ? A ? B ? C ? · · · and (C?, d?) = · · · ? A? ? B? ? C ? ? · · · are called chain homotopic if there is a map h : C ? C[?1] such that ?? ? = hd+ d?h. We could draw it diagrammatically as follows: (1) · · · d // A d // ?? ?? B d // ?? ?? h ~~||| ||| || C d // ?? ?? h ~~||| ||| || · · · · · · d ? // A? d ? // B? d ? // C ? d ? // · · · This is a useful definition in many instances: for example to show that a complex is acyclic one could look for a homotopy of the

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Abstract
MAKE IT YOURSELF STRONG HOMOTOPY STRUCTURE
OLGA KRAVCHENKO
This are notes for the talk at the conference GAP III, in Perugia. We talk about a general definition of a strong homotopy structure as a solution of a Maurer-Cartan equation on a corresponding governing Lie algebra. We apply this philosophy to construct strong homotopy Lie bialgebras.
1.strong homotopy associative algebraBasic example:
Why is it called homotopy? In fact, the following general definition of chain homotopic maps is used over and over again. Definition 1.Two mapsαandβbetween complexes (C, d) =∙ ∙ ∙ ABC→ ∙ ∙ ∙and 0 0 0 0 0 (C, d) =∙ ∙ ∙ ABC∙ ∙→ ∙ are called chain homotopic if there is a maph:C → C[1] such that 0 αβ=hd+d h. We could draw it diagrammatically as follows:
d d d d ////// // (1)∙ ∙ ∙A B C∙ ∙ ∙ | | | | | | h h β α|β α|β α | | | | | | 0²²~~|0²² ~~|0²²0 d d d d //0//0//0// ∙ ∙ ∙ ∙ ∙ ∙ A B C This is a useful definition in many instances: for example to show that a complex is acyclic one could look for a homotopy of the identity map to a zero map. LetA= (Ai, d) be a graded differential space, in other words a complex with a differential d:AiAi+1,(an elementaAiif its degreea=i.) Define a grading on the tensor powers of n A:Aas follows n X (2) (a1+a2+∙ ∙ ∙+an) =ajn+ 1. j=1 Consider a mapµwhich acts fromAk+1Al+1Ak+l+1,this map is of degree 0 with respect to the chosen grading. Indeed, the degree of elements fromAk+1Al+1is (k+1)+(l+1)2+1 =k+l+1, 2 soAk+1Al+1(A)k+l+1. In particular, ifk=l= 0,an example of such a map will be a usual multiplication on an ordinary ungraded vector space. We say that the productµ:AAAis associative up-to homotopy if the mapsµ(µId) : 33 AAandµ(Idµ) :AAare homotopic in the sense of (1). That is, if there is a mapµ3 of degree1 which provides a homotopy ofµ(µId)µ(Idµ) to a zero map:
(3)
d d d d 333 // // // ∙ ∙ ∙(A)i1(A)i(A)i+1 £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ µ(µId)µ(Idµ)£ µ3£µ £3 £ £ £ £ £ £ £ £ £ £ £ ²²¡¡£²² ¡¡£²² d d d d // // // ∙ ∙ ∙Ai1AiAi+1
Date: July 25, 2005.
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