(Here $G$ will be a group (a discrete one for the moment).)
A crossed $G$-algebra is a type of $G$-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?.
Moore and Segal (see references below) do not like the term ‘crossed algebra’ and suggest the alternative name ‘Turaev algebra’.
We will lead up to the definition of crossed $G$-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 $G$-algebra or $G$-algebra over a field (or more generally a commutative ring), $\mathbb{k}$ is an associative algebra, $L$, over $\mathbb{k}$ with a decomposition,
as a direct sum of projective $\mathbb{k}$-modules of finite type such that
(i) $L_g L_h \subseteq L_{gh}$ for any $g,h \in G$ (so, if $\ell_1$ is graded $g$, and $\ell_2$ is graded $h$, then $\ell_1\ell_2$ is graded $gh$),
and
(ii) $L$ has a unit $1 = 1_L\in L_1$ for 1, the identity element of $G$.
(i) The group algebra, $\mathbb{k}[G]$, has an obvious $G$-algebra structure in which each summand of the decomposition is free of dimension 1.
(ii) For any associatve $\mathbb{k}$-algebra, $A$, the algebra, $A[G]= A\otimes_\mathbb{k}\mathbb{k}[G]$, is also $G$-algebra. Multiplication in $A[G]$ is given by $(ag)(bh) = (ab)(gh)$ for $a,b \in A$, $g,h \in G$, in the obvious notation.
(iii) If $G$ is the trivial group, then a $G$-graded algebra is just an algebra (of finite type), of course.
A Frobenius $G$-algebra is a $G$-algebra, $L$, together with a symmetric $\mathbb{k}$-bilinear form,
such that
(i) $\rho(L_g\otimes L_h) = 0$ if $h \neq g^{-1}$;
(ii) the restriction of $\rho$ to $L_g \otimes L_{g^{-1}}$ is non-degenerate for each $g\in G$;
and
(iii) $\rho(ab,c) = \rho(a,bc)$ for any $a,b,c \in L$.
\
We note that (ii) implies that $L_{g^{-1}} \cong L_g^*$, the dual of $L_g$.
(i) The group algebra, $L = \mathbb{k}[G]$, is a Frobenius $G$-algebra with $\rho(g,h) = 1$ if $gh = 1$, and 0 otherwise, and then extending linearly. (Here we write $g$ both for the element of $G$ labelling the summand $L_g$, and the basis element generating that summand.)
(iii) For $G$ trivial, a Frobenius 1-algebra is a Frobenius algebra.
Finally the notion of crossed $G$-algebra combines the above with an action of $G$ on $L$, explicitly:
A crossed $G$-algebra over $\mathbb{k}$ is a Frobenius $G$-algebra, $L$, over $\mathbb{k}$ together with a group homomorphism,
satisfying:
(i) if $g\in G$ and we write $\varphi_g = \varphi(g)$ for the corresponding automorphism of $L$, then $\varphi_g$ preserves $\rho$, (i.e., $\rho(\varphi_ga,\varphi_gb) = \rho(a,b)$) and
for all $h\in G$;
(ii) $\varphi_g|_{L_g} = id$ for all $g\in G$;
(iii) (twisted or crossed commutativity) for any $g,h \in G$, $a\in L_g$, $b\in L_h$, $\varphi_h(a)b = ba$;
(iv) for any $g,h \in G$ and $c \in L_{ghg^{-1}h^{-1}}$,
where $Tr$ denotes the $\mathbb{k}$-valued trace of the endomorphism. (The homomorphism $c\varphi_h$ sends $a\in L_g$ to $c\varphi_h(a) \in L_g$, whilst $(\varphi_{g^{-1}}c)(b) = \varphi_{g^{-1}}(cb)$ for $c \in L_h$. 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 $G$-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 $G$-algebra refers to a summand, $L_g$, for and index, $g$, which is not the identity element of $G$, of course, then $L_1$ is called the ‘untwisted sector’
(To be added)
twisted G-algebra?
V. Turaev, Homotopy Quantum Field Theory , EMS Tracts in Math.10, European Math. Soc. Publ. House, Zurich 2010. (for the detailed development and the links with low dimensional topology and HQFTs).
Greg Moore, Graeme Segal, D-branes and K-theory in 2D topological field theory (arXiv hep-th 0609042)