(Here will be a group (a discrete one for the moment).)
A crossed -algebra is a type of -graded algebra, with an inner product and a ‘crossed’ or ‘twisted’ multiplication. They arise as the analogues of Frobenius algebras for 2d-HQFTs, equivariant TQFTs? and in slight generality in the study of symmetries of singularities. They were introduced by Turaev in 1998. The generalised structures have been studied by R. Kaufmann?.
We will lead up to the definition of crossed -algebra through various stages.
We need a particular form of graded algebra in which the summands are projective modules, so we give that form first.
A graded -algebra or -algebra over a field (or more generally a commutative ring), is an associative algebra, , over with a decomposition,
as a direct sum of projective -modules of finite type such that
(i) for any (so, if is graded , and is graded , then is graded ),
(ii) has a unit for 1, the identity element of .
(i) The group algebra, , has an obvious -algebra structure in which each summand of the decomposition is free of dimension 1.
(ii) For any associatve -algebra, , the algebra, , is also -algebra. Multiplication in is given by for , , in the obvious notation.
(iii) If is the trivial group, then a -graded algebra is just an algebra (of finite type), of course.
A Frobenius -algebra is a -algebra, , together with a symmetric -bilinear form,
(i) if ;
(ii) the restriction of to is non-degenerate for each ;
(iii) for any .
We note that (ii) implies that , the dual of .
(i) The group algebra, , is a Frobenius -algebra with if , and 0 otherwise, and then extending linearly. (Here we write both for the element of labelling the summand , and the basis element generating that summand.)
(iii) For trivial, a Frobenius 1-algebra is a Frobenius algebra.
Finally the notion of crossed -algebra combines the above with an action of on , explicitly:
A crossed -algebra over is a Frobenius -algebra, , over together with a group homomorphism,
(i) if and we write for the corresponding automorphism of , then preserves , (i.e., ) and
for all ;
(ii) for all ;
(iii) (twisted or crossed commutativity) for any , , , ;
(iv) for any and ,
where denotes the -valued trace of the endomorphism. (The homomorphism sends to , whilst for . This is sometimes called the ‘torus condition’.)
a) We note that the usage of terms differs between Turaev’s book (2010) and here, as we have taken ‘crossed -algebra’ to include the Frobenius condition. We thus follow Turaev’s original convention (preprint 1999) in this.
b) The useful terminology ‘twisted sector’ in a crossed -algebra refers to a summand, , for and index, , which is not the identity element of , of course, then is called the ‘untwisted sector’
(To be added)