Notations
When a function depends on linearly, we will write:
to stress linearity.
The cone of the Lie superalgebra 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 .)
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 with , 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 .
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〛.
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