Notations
When a function 
depends on 
linearly, we will write:
depends on linearly, we will write:
|
to stress linearity.
The cone of the Lie superalgebra 
is:
is:
|
where 
stands for semidirect sum of Lie superalgebras, with arrow pointing towards the ideal,
and 
is suspension, of degree 
. (We consider as a graded vector space, all having
degree zero, then 
all has degree .)
stands for semidirect sum of Lie superalgebras, with arrow pointing towards the ideal,
and is suspension, of degree . (We consider as a graded vector space, all having
degree zero, then all has degree .)
It may seem strange to assign degree
to
, instead of degree
. In string theory context, we want the grade to be the ghost number. At the same time, we do not want to replace
, because we want to agree with the notations of LodayVallette.
Throughout the paper we will follow
the notations
of LodayVallette.
From a vector space 
over a field 
(for us 
) we construct
an algebra 
, which consists of tensors of modulo some quadratic relations 
.
The coalgebra 
consists of tensors of 
, such that the tensor product of
any pair of neighbor fall into 
.
The cobar construction of is denoted 
.
over a field (for us ) we construct
an algebra , which consists of tensors of modulo some quadratic relations .
The coalgebra consists of tensors of , such that the tensor product of
any pair of neighbor fall into .
The cobar construction of is denoted .One motivation for using the formalism of quadratic algebras is technical, as it automates keeping track of signs. We will translate into “more elementary” language of Alekseev:2010gr in 〚Simpler notations〛.
PDF = pseudo-differential form.
Hyperlinks to web content are highlighted blue.