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$b_\infty$-algebra structure in homology of a homotopy Gerstenhaber algebra
T. V. Kadeishvili A. Razmadze Mathematical Institute, Georgian Academy of Sciences
Abstract:
The minimality theorem states, in particular, that on cohomology
$H(A)$ of a dg algebra there exists sequence of operations
$m_i:H(A)^{\otimes i}\to H(A)$,
$i=2,3,\dots$, which form a
minimal
$A_\infty$-algebra
$(H(A),\{m_i\})$. This structure
defines on the bar construction
$BH(A)$ a correct differential
$d_m$ so that the bar constructions
$(BH(A),d_m)$ and
$BA$ have
isomorphic homology modules. It is known that if
$A$ is equipped
additionally with a structure of homotopy Gerstenhaber algebra,
then on
$BA$ there is a multiplication which turns it into a dg
bialgebra. In this paper, we construct algebraic operations
$E_{p,q}:H(A)^{\otimes p}\otimes H(A)^{\otimes q}\to H(A)$,
$p,q=0,1,2,\dots$, which turn
$(H(A),\{m_i\},\{E_{p,q}\})$ into
a
$B_\infty$-algebra. These operations determine on
$BH(A)$
correct multiplication, so that
$(BH(A),d_m)$ and
$BA$ have
isomorphic homology algebras.
UDC:
512.7
Language: English
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