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\begin{document}

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\section*{crossed G-algebra}

\hypertarget{crossed_galgebra}{}\section*{{Crossed G-algebra}}\label{crossed_galgebra}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{nomenclature}{Nomenclature}\dotfill \pageref*{nomenclature} \linebreak
\noindent\hyperlink{definitions}{Definitions:}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{graded_algebra}{Graded $G$-algebra}\dotfill \pageref*{graded_algebra} \linebreak
\noindent\hyperlink{example}{Example:}\dotfill \pageref*{example} \linebreak
\noindent\hyperlink{frobenius_algebra}{Frobenius $G$-algebra}\dotfill \pageref*{frobenius_algebra} \linebreak
\noindent\hyperlink{examples_continued}{Examples continued:}\dotfill \pageref*{examples_continued} \linebreak
\noindent\hyperlink{crossed_algebra}{Crossed $G$-algebra}\dotfill \pageref*{crossed_algebra} \linebreak
\noindent\hyperlink{note}{Note:}\dotfill \pageref*{note} \linebreak
\noindent\hyperlink{operations_on_crossed_algebras}{Operations on crossed $G$-algebras}\dotfill \pageref*{operations_on_crossed_algebras} \linebreak
\noindent\hyperlink{related_notions}{Related notions}\dotfill \pageref*{related_notions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

(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]].

\hypertarget{nomenclature}{}\subsection*{{Nomenclature}}\label{nomenclature}

[[Greg Moore|Moore]] and [[Graeme Segal|Segal]] (see references below) do not like the term `crossed algebra' and suggest the alternative name `Turaev algebra'.

\hypertarget{definitions}{}\subsection*{{Definitions:}}\label{definitions}

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.

\hypertarget{graded_algebra}{}\subsubsection*{{Graded $G$-algebra}}\label{graded_algebra}

A \textbf{graded $G$-algebra} or \textbf{$G$-algebra} over a field (or more generally a commutative ring), $\mathbb{k}$ is an associative algebra, $L$, over $\mathbb{k}$ with a decomposition,

\begin{displaymath}
L = \bigoplus_{g\in G} L_g,
\end{displaymath}
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 $G$.

\hypertarget{example}{}\subsubsection*{{Example:}}\label{example}

(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.

\hypertarget{frobenius_algebra}{}\subsubsection*{{Frobenius $G$-algebra}}\label{frobenius_algebra}

A \textbf{Frobenius $G$-algebra} is a $G$-algebra, $L$, together with a symmetric $\mathbb{k}$-bilinear form,

\begin{displaymath}
\rho : L\otimes L \to \mathbb{k}
\end{displaymath}
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 $a,b,c \in L$.

$\backslash$

We note that (ii) implies that $L_{g^{-1}} \cong  L_g^*$, the dual of $L_g$.

\hypertarget{examples_continued}{}\subsubsection*{{Examples continued:}}\label{examples_continued}

(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]].

\hypertarget{crossed_algebra}{}\subsubsection*{{Crossed $G$-algebra}}\label{crossed_algebra}

Finally the notion of crossed $G$-algebra combines the above with an action of $G$ on $L$, explicitly:

A \textbf{crossed $G$-algebra} over $\mathbb{k}$ is a Frobenius $G$-algebra, $L$, over $\mathbb{k}$ together with a group homomorphism,

\begin{displaymath}
\varphi: G \to Aut(L)
\end{displaymath}
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

\begin{displaymath}
\varphi_g(L_h) \subseteq L_{ghg^{-1}}
\end{displaymath}
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 $c \in L_{ghg^{-1}h^{-1}}$,

\begin{displaymath}
Tr(c\varphi_h : L_g \to L_g) = Tr(\varphi_{g^{-1}}c : L_h \to L_h),
\end{displaymath}
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'.)

\hypertarget{note}{}\subsubsection*{{Note:}}\label{note}

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'

\hypertarget{operations_on_crossed_algebras}{}\subsection*{{Operations on crossed $G$-algebras}}\label{operations_on_crossed_algebras}

(To be added)

\hypertarget{related_notions}{}\subsection*{{Related notions}}\label{related_notions}

\begin{itemize}%
\item crossed $C$-algebra for a [[crossed module]] of groups;

\end{itemize}
\emph{[[twisted G-algebra]]}

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item V. [[Turaev]], \emph{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 [[HQFT]]s).


\item [[Greg Moore]], [[Graeme Segal]], \emph{D-branes and K-theory in 2D topological field theory} (\href{http://arxiv.org/abs/hep-th/0609042}{arXiv hep-th 0609042})



\end{itemize}
[[!redirects crossed G-algebras]] [[!redirects Turaev algebra]]



\end{document}