\documentclass[12pt,twoside]{article}   %Marcel Dekker 2004% ===> this file was generated automatically by noweave --- better not edit it
\usepackage{latexsym,amssymb,amsmath,amsfonts,amscd,amsthm,amstext}
\usepackage[all]{xy}  %epsfig,graphics
%\setcounter{page}{1}
\unitlength=1mm
\def\emline#1#2#3#4#5#6{\put(#1,#2){\special{em:point 1}} %{\special{em:moveto}}
 \put(#4,#5){\special{em:point 2}}\special{em:line 1,2}}  %,1.3pt}}  %0.4pt}}
\def\newpic#1{}\thicklines\special{em:linewidth 1.3pt}
%\linethickness{1.3pt}

\def\guy{\begin{picture}(3,3)\emline{1}{0}{1}{1}{2}{2}
\emline{1}{2}{3}{0}{3}{4}\emline{1}{2}{5}{2}{3}{6}\end{picture}}
\def\guc{\begin{picture}(3,3)\emline{1}{3}{1}{1}{1}{2}
\emline{1}{1}{3}{0}{0}{4}\emline{1}{1}{5}{2}{0}{6}\end{picture}}
\def\guo{\begin{picture}(3,4)\put(1,2){\circle{2}}
\emline{1}{1}{1}{1}{0}{2}\emline{1}{3}{3}{1}{4}{4}\end{picture}}
\def\guw{\begin{picture}(4,3)\put(2,0){\oval(2,2)[t]}
\put(2,3){\oval(2,2)[b]}\emline{2}{2}{1}{2}{1}{2}\end{picture}\,}
\def\gud{\begin{picture}(6,4)\put(2,2){\oval(2,2)[t]}
\put(4,2){\oval(2,2)[b]}
\emline{5}{2}{1}{5}{4}{2}\emline{2}{4}{3}{2}{3}{4}
\emline{1}{0}{5}{1}{2}{6}\emline{4}{0}{7}{4}{1}{8}\end{picture}\;}
\def\gue{\begin{picture}(6,4)\put(2,2){\oval(2,2)[b]}
\put(4,2){\oval(2,2)[t]}
\emline{1}{4}{1}{1}{2}{2}\emline{2}{0}{3}{2}{1}{4}
\emline{4}{4}{5}{4}{3}{6}\emline{5}{0}{7}{5}{2}{8}\end{picture}\;}

\swapnumbers
       \theoremstyle{definition}
\newtheorem{NR}{Definition}[section]
\newtheorem{Leibniz}{Definition}[section]
\newtheorem{Gerst}[Leibniz]{Definition}
\newtheorem{commutator}[Leibniz]{Graded commutator}
\newtheorem{FNalg}{Fr\"olicher-Nijenhuis Lie $M^\wedge$-module}[section]
\newtheorem{exp1}[FNalg]{Example}
\newtheorem{com1}[FNalg]{Comment}
\newtheorem{Zero}{Zero grade derivation}[section]
\newtheorem{Angular}[Zero]{Angular rotation}
\newtheorem{trace}{Trace is a co-unit}[section]
\newtheorem{atom}{Definition}[section]
      \theoremstyle{plain}
\newtheorem{lem1}[Leibniz]{Lemma}
\newtheorem{NR2}[Leibniz]{Lemma}
\newtheorem{universal}{Universal property}[section]
\newtheorem{FN1}[universal]{Theorem}
\newtheorem{lem2}[FNalg]{Lemma}
\newtheorem{thmoz}[FNalg]{Theorem}
\newtheorem{thm2}[Zero]{Theorem}
\newtheorem{Bianchi}[Zero]{Bianchi identity}
\newtheorem{thm4}[atom]{Theorem}
\newtheorem{thm5}[atom]{Theorem}
\newtheorem{thm3}{Theorem}[section]
%{Hypothesis}{Exercise}{Proposition}
%   \theoremstyle{remark}Note, Conjecture, Corollary, Challenge}

\numberwithin{equation}{section}
\newcommand{\bk}{\Bbbk}\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}\newcommand{\Q}{\mathbb{Q}}
\newcommand{\La}{\mathcal{L}}\newcommand{\F}{\mathcal{F}}
\newcommand{\ie}{\textit{i.e.}\,}
\newcommand{\alg}{\operatorname{alg}}
\newcommand{\Mod}{\operatorname{mod}}
\newcommand{\obj}{\operatorname{obj}}
\newcommand{\iso}{\operatorname{iso}}
\newcommand{\cat}{\text{cat}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\cotr}{\operatorname{cotr}}
\newcommand{\ev}{\text{ev}}
\newcommand{\coev}{\operatorname{coev}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\End}{\operatorname{End\,}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\Span}{\operatorname{span}}
\newcommand{\gen}{\operatorname{gen}}
\newcommand{\ra}{\longrightarrow}
\newcommand{\Fr}{\operatorname{Fr}}
\newcommand{\der}{\operatorname{der}}
\newcommand{\grade}{\operatorname{grade}}
\newcommand{\half}{\textstyle{\frac{1}{2}}}
\newcommand{\g}{\mathfrak{g}}
\newcommand{\bt}{\begin{tabular}{c}}
\newcommand{\et}{\end{tabular}}
\def\bf{\begin{flushright}}
\def\ef{\end{flushright}}

\begin{document}\title{Nijenhuis-Richardson algebra and Fr\"olicher-Nijenhuis
Lie module\thanks{Submitted to: Larissa Sbitneva, \fbox{Lev Sabinin}
and Ivan P. Shestakov, Editors, Non-Associative Algebra and Its
Applications, Marcel Dekker, INC., New York
2004.}\;\thanks{Supported by el Consejo Nacional de Ciencia y
Tecnolog\'{\i}a (CONACyT de M\'exico), Grant \# U 41214 F. Supported
by Programa de Apoyo a Proyectos de Investigaci\'on e Innovaci\'on
Tecnol\'ogica, UNAM, Grant \# IN 105402.}}
\author{Jos\'e de Jes\'us Cruz Guzm\'an and Zbigniew Oziewicz\thanks{Zbigniew
Oziewicz is a member of Sistema Nacional de Investigadores in
M\'exico, Expediente \# 15337}\\Universidad Nacional Aut\'onoma de
M\'exico\\Facultad de Estudios Superiores Cuautitl\'an\\Apartado
Postal \# 25, C.P. 54714 Cuautitl\'an Izcalli\\Estado de M\'exico\\
cruz@servidor.unam.mx, oziewicz@servidor.unam.mx}
\date{May 30, 2004}\maketitle
%\thanks{This paper is in final form and no version of it will be
%submitted for publication elsewhere}

\hyphenation{ca-te-go-ry}

%\baselineskip=24pt  \renewcommand{\baselinestretch}{2}

\begin{abstract} Not associative Nijenhuis-Richardson graded algebra on
universal module over Gra{\ss}mann algebra of differential forms
allows a novel/algo\-rith\-mic definition of the
Fr\"olicher-Nijenhuis Lie $\R$-algebra. Some consequences are
derived. The signature of the five-dimensional Frobenius subalgebra
of the Nijenhuis-Richardson algebra is calculated.
%Leibniz algebra, Gerstenhaber algebra, Poisson algebra, Frobenius
%bialgebra
\end{abstract}
\textbf{2000 Mathematics Subject Classification.} Primary 16W25
Derivation; Secondary 17A32 Leibniz algebra, 16W30 Coalgebra,
bialgebra.

\noindent\textbf{Keywords:} universal Gra{\ss}mann module,
Nijenhuis-Richardson algebra, Fr\"olicher-Nijenhuis Lie module,
Leibniz algebra, Frobenius algebra
%, Gerstenhaber algebra, Poisson algebra

\tableofcontents

\hyphenation{o-pe-ra-tio-nal pro-duct}
\pagestyle{myheadings}\markboth{\quad\hrulefill\quad Jos\'e de
Jes\'us Cruz Guzm\'an and Zbigniew Oziewicz\quad}{\quad
Fr\"olicher-Nijenhuis Lie module\quad\hrulefill\quad}

\section{Introduction}%\normalsize
Fr\"olicher and Nijenhuis in 1956 discovered Lie $\R$-algebra
implicit structure on a Gra{\ss}mann module of vector valued
differential forms. More on this was presented in Nijenhuis
contribution to Edinburg Congress in 1958. Peter Michor since 1985
together with collaborators published many papers and a monograph
[Kol\'ar, Michor, Slov\'ak 1993] deeply investigating all aspects of
Fr\"olicher and Nijenhuis Lie bracket. Dubois-Violette and Michor in
1995 found a common generalization of the Fr\"olicher-Nijenhuis
bracket and the Schouten bracket for the symmetric algebra of
multi-vector fields.

The Fr\"olicher and Nijenhuis Lie module and Lie $\R$-operation
found very important applications/interpretations in differential
geometry of connections (and in particular the Nijenhuis tensor that
describe the curvature of an almost product structure) [Gray 1967,
Gancarzewicz 1987, Kocik 1997, Krasil'shchik and Verbovetsky 1998,
Wagemann 1998], in algebraic geometry, in cohomology of Lie algebras
[Wagemann 1999], in special relativity theory, in Maxwell's theory
of electromagnetic field [Fecko 1997, Kocik 1997, Cruz and Oziewicz
2003], in Einstein's gravity theory [Minguzzi 2003], in classical
mechanics for symplectic structure [Gruhn and Oziewicz 1983, Gozzi
and Mauro 2000, Chavchanidze 2003].

From the point of view of applications there is a need, among other,
for the explicit/algorithmic definition/expression for the
Fr\"olicher and Nijenhuis Lie operation, such that can be
implemented for symbolic program.

In the present note we remaind the basic concepts, and we are
proposing a novel/algorithmic explicit definition of the Fr\"olicher
and Nijenhuis Lie $\R$-operation in terms of the primary
non-associative (Lie-admissible) $\F$-algebra structure on universal
Gra{\ss}mann module of vector-valued differential forms, that was
introduced by Nijenhuis and Richardson in a year 1967.

The non-associative Nijenhuis-Richardson primary algebra, that we
need in order to define Fr\"olicher and Nijenhuis Lie operation, is
a natural extension of the associative algebra of endomorphisms,
trace-class $(1,1)$-fields, to algebra of $(\text{any},1)$-fields
with generalized Gra{\ss}mann-valued `trace'.

The main objective of this note is rethink the basic concepts,
introduce a novel/algorithmic definition of the differential
Fr\"olicher and Nijenhuis Lie Gra{\ss}mann-module, presentation some
consequences of this definition, and provide a detailed proofs of
some statements that otherwise it is hard to find in available
literature.

The Nijenhuis-Richardson not associative algebra possess the
associative subalgebra, that is the Frobenius algebra. For Frobenius
algebra we refer to [Frobenius 1903, Curtis \& Reiner 1962, Kauffman
1994, Voronov 1994, Beidar et al. 1997, Kadison 1999, Baez 2001,
Caenepeel et al. 2002]. In the last Sections we briefly define the
Frobenius algebra, and initiate study of the five-dimensional
Frobenius associative subalgebra of the Nijenhuis-Richardson not
associative algebra.

References include all known to us publications related to the
subject of the present paper, even so we do not made comments about
some of them.

\begin{center}\begin{tabular}{c|l}\multicolumn{2}{c}{\textbf{Some
Notation}}\\\hline $\F$&- denotes the associative, unital and
commutative ring,\\& e.g. $\R$-algebra.\\\hline $\der_\R\F$&-denotes
the Lie $\F$-module of the derivations,\\$\equiv\der_\R(\F,\F)$&Lie
$\F$-modul of the vector fields.\\\hline $M={_\F M}$&- denotes the
projective $\F$-module of the differential\\&$1$-forms (the Pfaffian
forms), $\dim_\F M<\infty,$ with\\& a derivation
$d\in\der_\R(\F,M).$\\
&Then $M=(\der_\R\F)^*\equiv\Mod_\F(\der_\R\F,\F).$\\\hline
$M^*$&- denotes `dual of dual' $\F$-modul of the vector fields,\\
&$M^*\equiv\Mod_\F(M,\F)=(\der_\R\F)^{**}\simeq\der_\R\F.$\\
$(-)^{AB}$&- is an abbreviation for $(-1)^{(\grade A)(\grade
B)}.$\\\hline\end{tabular}\end{center}%\smallskip

\section{Universal Gra{\ss}mann module} In the sequel the
Gra{\ss}mann $\F$-factor-algebra of differential multi-forms is
denoted by $M^\wedge\equiv M^\otimes/I,$ where $I<M^\otimes$ is an
ideal in a free tensor $\F$-algebra, generated by
$\alpha\otimes\alpha$ $\forall\;\alpha\in M.$ A left
$M^\wedge$-module $M^\wedge\otimes_\F M^*\simeq\Mod_\F(M,M^\wedge)$
is said to be an $M^*$-universal Gra{\ss}mann-module, known
variously as the module of `vector-valued differential forms' or
module of `vector-forms'.

An $\R$-linear or $\F$-linear homogeneous endomorphism
$D\in\End(M^\wedge)$ with $\grade D\in\Z,$ is said to be a
$\Z_2$-graded derivation (skew derivation, anti-derivation),
$D\in\der(M^\wedge),$ if the graded Leibniz axiom holds.
%\begin{gather*}\{D,e^{l/r}_\alpha\}=e^{l/r}_{D\alpha}\quad\text{or}\quad
%[D,e^{r/l}_\beta]=e^{r/l}_{D\beta}\cdot(-1)^{D\cdot\grade}.\end{gather*}
Derivation is said to be \textit{algebraic} if $D|\F=0.$
%Then $\{j^{l/r}_\alpha,D^*\}=j^{l/r}_{D\alpha}.$

A $\Z$-graded Lie $\F$-algebra of $\F$-derivations of the
Gra{\ss}mann $\F$-algebra, $\der_\F(M^\wedge),$ is a left
$M^\wedge$-module. We are going to describe an $M^\wedge$-module
isomorphism that Nijenhuis and Richardson in 1967 extended to
isomorphism of graded commutators (actually this is an isomorphism
of Gerstenhaber algebras, see Lemma \ref{NR2}, etc),
\begin{gather}i\in\Mod_{M^\wedge}(M^\wedge\otimes_\F
M^*,\der_\F(M^\wedge)).\label{i}\end{gather}

Every derivation of a Gra{\ss}mann algebra $D\in\der(M^\wedge\equiv
M^\otimes/I)$ is uniquely determined by values of $D$ on generating
$\R$-algebra $\F$ and on $\F$-module $M$:
$D|\F\in\der_\R(\F,M^\wedge)$ and $D|M\in\der_\R(M,M^\wedge),$ if
and only if $(D|M)^\otimes I\subset I.$ Therefore a $\Z$-homogeneous
derivation $D$ with a $\grade D\leq-2$ must be the trivial zero
derivation.

A $\F$-module map $p\in\Mod_\F(M,M^\wedge)$ lifts to the unique
$\Z_2$-graded $\F$-derivation $i_p$ with $\grade(i)=0,$ such that $
i_p|\F=0$ and $ i_p|M=p,$
\begin{gather}\Mod_\F(M,M^\wedge)\simeq M^\wedge\otimes_\F M^*\ni
p\quad\stackrel{i}{\longmapsto}\quad
i_p\in\der_\F(M^\wedge),\label{i1}\end{gather}

Let $\alpha,\beta\in M^\wedge,$ $X\in M^*\simeq\der_\R\F$ and
$p\equiv\alpha\otimes_\F X\in(M^\wedge\otimes_\F M^*).$ We
abbreviate $\beta\wedge p=\beta p.$ Then [e.g Dubois-Violette and
Michor 1995]
\begin{gather}i(\alpha\otimes_\F X)\equiv e_\alpha\circ
i_X,\quad i_X\in\der_\F(M^\wedge),\quad i_{\alpha p}=\alpha\,
i_p,\label{i2}\\\grade(e_\alpha\circ
i_T)=-1+\grade\alpha.\notag\end{gather}

If $p\in\der_\F(M^\wedge),$ then $(i\circ |M)p=p.$ Therefore the
restriction `$|M$' is the inverse of
\eqref{i}-\eqref{i1}-\eqref{i2}, $i^{-1}=|M,$ and there is a
bijection,
\begin{gather}\begin{CD}\der_\F(M^\wedge)@>{|M=i^{-1}}>>
M^\wedge\otimes_\F M^*.\end{CD}\label{inv}\end{gather}

\noindent\textbf{Example.} A vector field $T\in
M^*\simeq\Mod_\F(M,\F)$ lifts to an algebraic derivation $T\mapsto
 i_T\in\der_\F(M^\wedge)$ with $\grade(i_T)=-1.$

\subsection{Nijenhuis-Richardson algebra}
Consider $p,q\in M^\wedge\otimes_\F M^*\simeq\Mod_\F(M,M^\wedge).$
Under this identification Nijenhuis and Richardson in 1967 defined
not associative $\F$-algebra as follows.
\begin{NR}[Nijenhuis-Richardson algebra]\label{NR} Let $\alpha,\beta\in M^\wedge.$
\begin{gather*}\{\Mod_\F(M,M^\wedge)\}\otimes_\F\{\Mod_\F(M,M^\wedge)\}
\ra\{\Mod_\F(M,M^\wedge)\},\\p\otimes_\F q\longmapsto
pq\equiv(ip)\circ q\in\{\Mod_\F(M,M^\wedge)\}.\\
\text{If}\quad p=\alpha\otimes_\F P\quad\text{and}\quad
q=\beta\otimes_\F Q,\quad\text{then}\quad
pq=(\alpha\wedge(i_P\beta))\otimes_\F Q.\end{gather*}\end{NR}

Clearly $(\alpha p)q=\alpha(pq).$ However for $\alpha\in M^\wedge,$
$p\alpha\equiv(ip)\alpha$ and every vector valued differential form
is his own $M^\wedge$-module derivation, e.g. [Dubois-Violette and
Michor 1994],
\begin{gather}p(\alpha q)=(p\alpha)q+(-)^{p\alpha}\alpha(pq),\\
\{p\otimes_\F(\alpha q)\}=(i_p\alpha)q+(-)^{p\alpha}\{p\otimes_\F
q\}.\end{gather}

The Nijenhuis-Richardson $\Z$-graded $\F$-algebra is not
associative, not unital, and not commutative,
\begin{gather}(pq)r\equiv i(i_p\circ q)\circ r\quad\neq\quad i_p\circ
(i_q\circ r)\equiv p(qr),\\i_{pq}=i_p\circ i_q+i_{p\wedge
q}\quad\in\der(M^\wedge).\\\text{If $\grade\,q=-1$}\quad\text{then
$\forall\;p,$}\quad pq=0.\end{gather}

\section{Leibniz/Loday and Gerstenhaber algebra} Let $\F$ be a ring and
$A$ be $\F$-bimodule. A category of $\F$-bimodules is a monoidal
abelian category.

\begin{Leibniz}[Leibniz/Loday algebra, Loday 1993]
%A Leibniz algebra/object in a (strict) monoidal abelian category is
%an object with a pair of binary morphisms ... subject the following
%condition

%\begin{figure}[h]\caption[Poisson algebra]{Poisson algebra}
%\input{poisson.pic}\end{figure}

%\begin{figure}\caption[Poisson algebra]{Poisson algebra}\begin{center}
%\epsfig{file=poisson.ps}\end{center}\end{figure}

%\begin{figure}\caption[Poisson algebra]{Poisson algebra}\begin{center}
%\includegraphics{poisson.ps}\end{center}\end{figure}

%\begin{gather}\xymatrix{\ar@{-}[dr]&&\ar@{-}[dl]\\&\ar@{-}[d]&\\&&}
%\end{gather}

A pair of binary operations/morphisms, $\cap$ and $[\cdot,\cdot],$
is said to be the Leibniz/Loday algebra if
\begin{gather}[\cdot,\cdot]\in\der\cap,\qquad\begin{CD}\text{carrier}
@>{[\cdot,\cdot]}>>\der\cap.\end{CD}\end{gather} A graded Leibniz
algebra is a pair of homogeneous binary operations $\cap$ and
$[\cdot,\cdot]$ on a $\Z$-graded object/carrier such that
$\forall\;a,b\in\text{carrier},$ $[a\equiv[a,\cdot]\in\der\cap,$
\begin{gather}([a)\circ\cap_b=\cap_{[a,b]}+(-1)^{(a+[\cdot,\cdot])(b+\cap)}
\cdot\cap_b\circ([a)\quad\in\End\,A.\end{gather}\end{Leibniz}

\begin{Gerst}[Gerstenhaber algebra]\label{Gerst} The $\Z$-graded Leibniz
algebra $(\cap,[\cdot,\cdot])$ is said to be the graded Poisson
algebra or the graded Gerstenhaber algebra if
\begin{gather*}\grade[\cdot,\cdot]+\grade\cap=\begin{cases}\text{even}&-
\text{the Poisson algebra,}\\\text{odd}&- \text{the Gerstenhaber
algebra.}\end{cases}\end{gather*}\end{Gerst} Definition \ref{Gerst}
[Oziewicz and Paal 1995] generalize the Gerstenhaber [1963]
structure carried by the Hochschild cohomology of an associative
algebra $\cap.$ In this definition both binary operations need not
to be graded commutative, $\cap$ need not to be associative, and
$[\cdot,\cdot]$ need not to be Lie-admissible. However a crossing
$2\mapsto 2$ needs to be the Artin braid [Oziewicz, R\'o\.za\'nski
and Paal 1995]. A concept of the Lie-Cartan pair introduced by
Jadczyk and Kastler [1987, 1991] is a generalization of Leibniz
algebra to pair of objects, it is a two-sorted Leibniz/Loday
algebra.

\begin{commutator}Let $A,B,C$ be $\R$- or $\F$-
linear $\Z$-homogeneous graded endomorphisms
$A,B,C\in\End(M^\wedge).$ We abbreviate $(-1)^{(\grade A)(\grade
B)}$ to $(-)^{AB}.$ The graded commutator (bracket) needs the Koszul
rule of signs
\begin{gather}\{A\otimes_{\R/\F}B\}\equiv A\circ B-(-)^{AB}B\circ A,\\
\grade\{A\otimes B\}=\grade\{\cdot,\cdot\}+\grade A+\grade
B.\notag\end{gather} Thanks associativity of a composition this is
an example of the $\Z$-graded Poisson algebra
\begin{gather}\{A\otimes(B\circ C)\}=\{A\otimes B\}\circ C+(-)^{AB}\cdot
B\circ\{A\otimes C\}.\label{der}\end{gather} An associative
$\Z_2$-graded $\R$- and $\F$-algebra $\End(M^\wedge)$ with the above
commutator is a $\Z$-graded Poisson $\F$-algebra and a Lie ring. The
Jacobi identity is a consequence of \eqref{der},
\begin{gather*}\{A\otimes\{B\otimes C\}\}=\{\{A\otimes B\}\otimes C\}
+(-)^{AB}\{B\otimes\{A\otimes C\}\}.
%,\\\{e_A,e_B\}=0=\{i_A,i_B\}.
\end{gather*}\end{commutator}
\begin{lem1}[Lie super algebra of derivations]\label{lem1} Let $A,B\in
\der(M^\wedge).$ Then $\{A\otimes B\}\in\der(M^\wedge).$\end{lem1}

\begin{proof} Every commutator (graded or `not graded' with trivial grading)
is an inner derivation in the Lie admissible ring of an
($\Z$-graded) abelian group-endo\-mor\-phisms. This implies that the
commutator of derivations (of a ring) is again the derivation.

Therefore the space of derivations is a $\Z_2$-graded Lie algebra
(\ie a super-algebra), a sub-algebra of $\End(M^\wedge)$ with
$\{\cdot\otimes\cdot\}\equiv\{,\}.$

Independently one can check Lemma \ref{lem1} by direct computation.
In particular $\{D,D\}=(1-(-)^D)\cdot D^2,$ therefore for a
derivation $D,$ a map $D^2$ is again a nontrivial derivation if
$\grade D=\text{odd}.$\end{proof}

\begin{NR2}[Nijenhuis and Richardson 1967]\label{NR2} Let $p,q\in M^\wedge
\otimes_\F M^*.$ The $\F$-module isomorphism \eqref{i}-\eqref{i2} is
a graded Lie $\F$-algebra map:
\begin{gather}\{i_p\otimes_\F i_q\}=i\{p\otimes_\F q\}\quad\in\der_\F(M^\wedge).
\end{gather}\end{NR2}\begin{proof} An equality of algebraic derivations must be
verified on restriction $i^{-1}\equiv|M.$\end{proof}

\section{Fr\"olicher and Nijenhuis decomposition}
\subsection{Universal property of derivation} The derivation $d\in\der_\R(\F,M)$
has the universal property: for $D\in\der_\R(\F,M^\wedge),$ there is
the unique $\F$-module map, $j_D\in\Mod_\F(M,M^\wedge),$ such that
$D=j_D\circ d,$ $\grade j=-1,$
\begin{gather}\begin{CD}\F@>{d}>>M\\@|@VV{j_D}V\\\F@>{D}>>M^\wedge\end{CD}
\notag\\\begin{CD}\der_\R(\F,M^\wedge)@>{j}>>\Mod_\F(M,M^\wedge)@>{i}
>>\der_\F(M^\wedge).\end{CD}\end{gather}\label{u1}

In particular $d=j_d\circ d\;\Rightarrow\;j_d=\id_M.$  The grade
operator is a derivation,
\begin{gather}\End_\F M=\Mod_\F(M,M)\ni\id_M\overset{i}{\longmapsto}
\grade\equiv i_{\id}\in\der_\F(M^\wedge),\\\{(i\circ
j)\,d,d\}=d.\end{gather} From the universal property of
$d\in\der_\R(\F,M)$ it follows the $\F$-module isomorphism of the
vector fields, $\der_\R\F\equiv \der_\R(\F,\F)$ with the $\F$-dual
$\F$-module, $M^*\equiv\Mod_\F(M,\F)\equiv\F^M.$ Let $T\in\der\F,$
then
\begin{gather*}\forall\;f\in\F,\quad Tf\equiv(df)T\equiv j_Tdf\in\F,\\
\begin{CD}\der_\R(\F,\F)@>{j}>>M^*\\\der_\R(\F,\F)@<{d^*}<<M^*.
\end{CD}\end{gather*}
Therefore $\der_\R\F\ni T=j_T\circ d=d^*(j_T)=(d^*\circ j)T.$

\subsection{Lie-\'Slebodzi\'nski derivation} The
Gra{\ss}mann-Hopf $\F$-algebra $M^\wedge$ with the unique lifted
graded differential, $d\in\der_\R(M^\wedge),\quad\grade d=+1,\quad
d^2=0\in\der_\R(M^\wedge),$ is said to be the differential
$\N$-graded algebra (DGA), de Rham complex. The following
$\R$-derivation with $\grade\La=+1$ is said to be the (right/left)
Lie-\'Slebodzi\'nski derivation of the endomorphism algebra $\End,$
\begin{gather}\La^{r/l}\in\der^{r/l}_\R(\End_\R(M^\wedge))\equiv\der^{r/l}_R(\circ),\notag\\
\begin{CD}\End_\R(M^\wedge)\ni\,A@>{\La^r}>>\La^r_A\equiv\{A,d\}\in
\End_\R(M^\wedge),\\\der_\R(M^\wedge)\ni\,p@>{\La^r}>>\La^r_p\equiv\{p,d\}\in
\der_\R(M^\wedge),\end{CD}\\d^2=0\quad\Longrightarrow\quad\La^2=0.\end{gather}
The last implication follows from graded Jacobi identity
$\La^2A=\{A,d^2\}.$

Let $A\in\End(M^\wedge)$ be a $\Z$-graded $\F$- or $\R$-map, and
$f\in\F.$ Then
\begin{gather}\La^r_A\equiv\{A,d\}\equiv\,A\circ d-(-)^A\cdot d\circ\,A
\equiv(-)^{1+A}\La^l_A\quad\in\End_\R(M^\wedge),\\\La_f\equiv\{f,d\}
=-e_{df}\equiv-(df)\wedge\ldots,\\
\La^r_{A\circ B}=(-1)^B\La^r_A\circ B+A\circ\La^r_B.\end{gather}

For a multivector fields $X,Y\in M^{*\wedge},$ $i_{X\wedge
Y}=i_Y\circ i_X\in\End(M^\wedge)$ (for $\grade X\geq 2,$
$i_X\not\in\der(M^\wedge)$), and
$\La_X\equiv\{i_X,d\}\in\End_\R(M^\wedge)$ [Tulczyjew 1974].

For a $1$-vector field, $X\in\der_\R\F\equiv\der_\R(\F,\F)\simeq
M^*\ni jX,$ lifted to $\F$-derivation of the Gra{\ss}mann algebra
$(i\circ j)X\in\der_\F(M^\wedge),$ the $0$-grade directional
$\R$-derivation along a 1-vector field $X\in\der\F,$
$\La_X\equiv\{(i\circ j)X,d\}\in\der_\R(M^\wedge),$ was invented by
\'Slebodzi\'nski [1931]. For $X\in\der\F,$ and for $f\in\F,$ we have
\begin{gather}\La_X\equiv\La_{(i\circ j)X},\quad(\La^2)X=\{\La_X,d\}=0,\\
\La_X f=(i\circ j)_Xdf=j_Xdf=Xf.\end{gather}

The name `Lie derivation' along the vector field $X\in\der\F,$ was
introduced by D. van Dantzig (collaborator of Schouten). The
Lie-\'Slebodzi\'nski derivation is implicit in [Cartan 1922].

The Lie-\'Slebodzi\'nski $M^\wedge$-module graded right/left
derivations $$\La^l_A\equiv\{d,A\}=(-)^{1+A}\La^r_A,$$ possess the
following Leibniz expressions for $\alpha\in M^\wedge$ and $q\in
M^\wedge\otimes_\F M^*,$
\begin{align}\La^r_{i(\alpha q)}&=(-)^{1+\alpha+q}(d\alpha)\wedge
q+\qquad\alpha\wedge\La^r_{iq},\label{Dubois3}\\\La^l_{i(\alpha
q)}&=\hspace{1.6cm}(d\alpha)\wedge
q+(-)^\alpha\alpha\wedge\La^l_{iq}.\notag\end{align}

\subsection{Fr\"olicher and Nijenhuis decomposition}
In the sequel we use the universal property \eqref{u1}, and to
simplify notation we write $j$ instead of the composition
$j\circ(|M).$ In this convention \eqref{u1} reads
\begin{gather}\begin{CD}\der_\R(M^\wedge)\quad@>{i\;\circ\;
j\;\circ\;(|\F)}>>\quad\der_\F(M^\wedge).\end{CD}\end{gather}

\noindent\textbf{Theorem 4.3.1 (Fr\"olicher and Nijenhuis 1956).}
%\begin{FN1}[Fr\"olicher and Nijenhuis 1956]
Any $\R$-derivation $D\in\der_\R(M^\wedge)$ possess the following
unique decomposition
\begin{gather}D=(\La\circ i\circ j+i\circ j\circ\La)D=\{i_{jD},d\}+i_{j\{D,d\}}.
\label{FN1}\end{gather}%\end{FN1}
\begin{proof} First we need remaind
the definitions of `vector-forms' \eqref{u1}, $$jD,\quad
j\La_D\quad\in M^\wedge\otimes_F M^*.$$ For $D\in\der_\R(M^\wedge),$
$D|\F\in\der_\R(\F,M^\wedge).$ Universality of $d\in\der_\R(\F,M)$
gives
\begin{gather}D|\F\equiv(jD)\circ d,\quad jD=0\Longleftrightarrow D|\F=0,\\
\La_D|\F\equiv(j\La_D)\circ d.\end{gather} The Fr\"olicher and
Nijenhuis decomposition \eqref{FN1} is an equality of derivations,
$D=0$ iff $D|\F=0$ and $D|d\F=0.$ We must check that the F-N
decomposition \eqref{FN1} is an identity on a ring $\F$ and on exact
differential one-forms $d\F<M.$\end{proof}

\section{Main definition} Let $A,B\in\End_\F(M^\wedge),$ \ie $Af=0.$ Then
\begin{align*}\La^r_{(A\circ B)}|\F&=(A\circ
B)|d\F,\quad\La^l_{A\circ
B}|\F=\ldots\\\{A\otimes_\R\La^r_B\}|\F&=(A\circ B)|d\F.\end{align*}
Set an $M^\wedge$-module map \eqref{inv},
$i^{-1}\equiv|M:\der_\F(M^\wedge)\ra M^\wedge\otimes_\F M^*.$ Let
$p,q,pq,qp\in M^\wedge\otimes_\F M^*,$ where $pq$ is the
Nijenhuis-Richardson nonassociative product. We have
\begin{align}\La_{i{(pq)}}|\F&=(pq)d|\F,\\
\{i_p\otimes_\R\La_{iq}\}|\F&=i_p\circ\La_{iq}|\F=i_p\circ i_q\circ
d|\F=(i_p\circ q)d|\F.\end{align} This proves that
\begin{gather}\La^{r/l}_{i(pq)}-\{i_p\otimes_\R\La^{r/l}_{iq}\}\in\der_\F(M^\wedge).
\end{gather}

The Fr\"olicher-Nijenhuis differential binary operation on the
$\R$-Lie $M^\wedge$-module $\der_\F(M^\wedge)\simeq
M^\wedge\otimes_\F M^*,$ is denoted by $[\cdot\otimes_\R\cdot],$
with $\grade[\cdot\otimes_\R\cdot]=+1,$
%is an example of the Gerstenhaber $\R$-algebra.
\begin{gather*}\begin{CD}(M^\wedge\otimes_\F M^*)\otimes_R(M^\wedge\otimes_\F
M^*)@>{[\cdot\otimes_\R\cdot]}>>(M^\wedge\otimes_\F
M^*).\end{CD}\end{gather*}

\begin{FNalg} We define the following algorithmic/explicit form of the
Fr\"olicher-Nijenhuis $\R$-bracket,
\begin{gather}(-)^q\,i[p\otimes_\R q]\equiv\La^r_{i{(pq)}}-
\{i_p\otimes_\R\La^r_{iq}\}\qquad\in\der_\F(M^\wedge),\label{FN4}\\
\boxed{(-)^q[p\otimes_\R q]\equiv i^{-1}\left(\La^r_{i(pq)}
-\{i_p\otimes_\R\La^r_{iq}\}\right)}\quad\in M^\wedge\otimes_\F
M^*.\label{FN2}\end{gather} In particular if $p$ is an idempotent
(with respect to Nijenhuis-Richardson product), $p^2=p\in
M^\wedge\otimes_\F M^*,$ then $\grade p=0$ and
\begin{gather}i[p\otimes_\R p]=\La^r_{ip}-\{i_p\otimes_\R\La^r_{ip}\}=2i_pdi_p.
\label{FN3}\end{gather}\end{FNalg}

\begin{lem2} The binary $\R$-operation \eqref{FN2} is
graded commutative
\begin{gather}[p\otimes_\R q]=(-1)^{p+q+pq}\cdot[q\otimes_\R
p].\end{gather}\end{lem2}
\begin{proof}For $p,q\in M^\wedge\otimes_\F M^*,$ and for
$A,B\in\der_\F(M^\wedge),$ we have
\begin{gather}i\{p\otimes_\F q\}=\{i_p\otimes_\F
i_q\}\quad\Longleftrightarrow
\quad i_{pq}-(-)^{pq}\,i_{qp}=\{i_p\otimes_\F i_q\},\\
\La_{A\otimes_\F B}=\{A\otimes_\R\,\La_B\}+(-)^B\,\{\La_A\otimes_\R\,B\},\\
\La_{i(pq)}=(-)^{pq}\,\La_{i(qp)}+\{i_p\otimes_\R\,\La_{iq}\}+
(-)^q\,\{\La_{ip}\otimes_\R\,i_q\}.\end{gather} All this implies
that
\begin{align}(-)^q[p\otimes_\R\,q]&=\La_{i(pq)}-\{i_p\otimes_\R\,\La_{iq}\}\\
&=(-)^{pq}\,\La_{i(qp)}+(-)^q\,\{\La_{ip}\otimes_\R\,i_q\}\\
&=(-)^{p+pq}\,[q\otimes_\R\,p].\qed\end{align}\renewcommand{\qed}{}\end{proof}

In order to relate Definition \eqref{FN4}-\eqref{FN2} with the
original \textit{implicit} Definition by Fr\"olicher and Nijenhuis
[1956], we need to calculate the Lie-\'Slebodzi\'nski map on
\eqref{FN4},
\begin{gather}\La_{i[p\otimes_\R q]}=(-)^{1+q}\,
\{\{i_p\otimes_\R\La_{iq}\}\otimes_\R
d\}=\{\La_{ip}\otimes_\R\La_{iq}\}.\end{gather}

The original, implicit Definition by Fr\"olicher-Nijenhuis is as
follows. By the Jacobi identity we have,
\begin{gather}\La\circ\La=0\quad\Longrightarrow\notag\\
\{\La_A\otimes_\R\,d\}=0\quad\&\quad
\{\{\La_A\otimes_\R\,\La_B\}\otimes_\R\,d\}=0.\end{gather} The
Fr\"olicher and Nijenhuis decomposition [1956] \eqref{FN1} implies
that for $A,B\in\der_\F(M^\wedge)$ a derivation $[A\otimes_\R
B]\in\der_\F(M^\wedge)$ exists (in an implicit way) such that
\begin{gather}\La_{[A\otimes_\R B]}\equiv\{\La_A\otimes_\R\La_B\}
\in\der_\R(M^\wedge),\label{FN}\\
[A\otimes_\R B]=(-1)^{A+B+AB}\cdot[B\otimes_\R A].\end{gather}

\begin{exp1} If $\grade q=-1$ we set $q=X\in M^*.$ Then $\forall\;p\in
M^\wedge\otimes_\F M^*,$ $pq=0\in M^\wedge\otimes_\F M^*.$ In this
case the Definition \eqref{FN4}-\eqref{FN2} is simplified
\begin{gather}i[p\otimes_\R X]=\{i_p\otimes_\R\La_{iX}\}.\end{gather}
Evaluating above brackets on exact 1-form $df\in M,$ is showing that
the Fr\"olicher and Nijenhuis Lie $M^\wedge$-module generalize Lie
$\F$-module of the vector fields
\begin{gather}[p\otimes_\R
X]df=i_pd(Xf)-(\La_{iX})pdf.\end{gather}\end{exp1}

\begin{com1} Vinogradov in 1990, in an attempt of
unification of the Schouten Lie module of multi-vector fields
[Schouten 1940, Nijenhuis 1955], with the Fr\"olicher and Nijenhuis
Lie-operation, introduced new $\R$-bracket as the sum of double
graded commutator of derivations. The value of the Vinogradov binary
bracket do not vanish on a ring of the scalars and therefore is not
given by the tensor field. Vinogradov proposed the following
explicit $\R$-bracket for $A,B\in\End_\F(M^\wedge)$
\begin{gather}2[A\otimes_\R\,B]_V\equiv\{\La_A\otimes_\R\,B\}-
(-)^B\,\{A\otimes_\R\,\La_B\}.\end{gather} An evaluation of the
Lie-\'Slebodzi\'nski map gives
\begin{gather}\La_{[A\otimes_\R\,B]_V}=\{\La_A\otimes_\R\,\La_B\}.\end{gather}
Contrary to our Definition \eqref{FN2} where $[p\otimes_\R\,q]\in
M^\wedge\otimes_\F M^*,$ the Vinogradov bracket do not define a
tensor field, $[A\otimes_\R\,B]_V|\F\neq 0.$\end{com1}

\subsection{Consequence: modul derivation} The notion of the Leibniz/Loday
algebra can be weakened by relaxing the condition of an algebra
derivation to a module derivation. De Rham complex $M^\wedge$ with
$d\in\der_\R(M^\wedge)$ is a DGA. Then an $M^\wedge$-module with a
binary operation $[\cdot\otimes_\R\cdot]$ is said to be
Leibniz/Loday $\R$-algebra if $[\cdot\otimes_\R\cdot]$ is
$M^\wedge$-module derivation.
\begin{thmoz}[e.g. Dubois-Violette and Michor 1994] Let
$p,q\in M^\wedge\otimes_\F M^*$ and $\alpha\in M^\wedge.$ We
abbreviate $\alpha\wedge q$ to $\alpha q.$ The following Leibniz
formula for the $M^\wedge$-module graded derivation holds
\begin{gather*}[p\otimes_\R(\alpha q)]=(\La_{ip}\alpha)q-
(-)^{p(\alpha+q+1)}\,(d\alpha)(qp)+(-)^{\alpha(p+1)}\,\alpha
[p\otimes_\R q].\end{gather*}\end{thmoz}

The above clue $M^\wedge$-module graded derivation is rather known,
however frequently presented without proof. We claim that the proof
is a trivial consequence of Definition \eqref{FN4}-\eqref{FN2}.
Straightforward calculations using \eqref{Dubois3} proves the above
theorem.

Another important easy consequence of Definition
\eqref{FN4}-\eqref{FN2} is the graded Jacobi relation that is an
example of the graded Leibniz derivation. With this respect it is
instructive to compare with Kanatchikov [1996], where the graded
Jacobi relation was derived for `semi-bracket'
$\{i_p\otimes_\R\La^r_{iq}\},$ that do not coincide with the
Fr\"olicher-Nijenhuis bracket \eqref{FN4}-\eqref{FN2}.

\section{Bianchi identity} In this section $p\equiv\tau\otimes_\F P\in M\otimes_\F
M^*$ with $\tau P=1\in\F.$
\begin{Zero} The composition $i_p=e_\tau\circ i_P\in\der_\F(M^\wedge)$
implies $i_P\circ e_\tau|\F=\id_\F\cdot\tau P,$ and $(i_p)^2=i_p,$
\begin{gather}\begin{CD}M@>{P}>>\F\\@V{p}VV ||\\
M@<{e_\tau}<<\F\end{CD}\hspace{1.5cm}
\begin{CD}\F@<{P}<<M\\@V{\id}VV||\\
\F@>{e_\tau}>>M\end{CD}\hspace{1.5cm}
\begin{CD}M^\wedge@>{i_P}>>M^\wedge\\@V{p}VV ||\\
M^\wedge@<{e_\tau}<<M^\wedge\end{CD}\label{split}\end{gather}
However $i_P\circ e_\tau$ does not split on $M^\wedge,$ $i_P\circ
e_\tau=(\tau P)\id-e_\tau\circ i_P\quad\neq\id.$\end{Zero}

\begin{Angular}\label{angular} Let $p^2=p\in M^\wedge\otimes_\F M^*,$ then
$\grade\,p=0.$ The angular rotation tensor $\omega$ of the
$(1,1)$-tensor field $p,$ is defined as follows
\begin{gather*}i_\omega\equiv(\id-i_p)\circ d\circ\,i_p\quad=-\La_{ip}\circ\,i_p.
\end{gather*}\end{Angular} We will show that $i_\omega\in\der_\F(M^\wedge)$
and therefore $\omega$ is a $(2,1)$-tensor field. The name `angular
rotation of idempotent' is motivated in the proof of the next Lemma.
\begin{thm2}[Anholonomy]\label{lem1a} Let $p^2=p,$
$i_p\equiv e_\tau\circ i_P\in\der_\F(M^\wedge).$ Then
\begin{align*}i.\quad&\omega=\half\cdot[p\otimes_\R p].\\
ii.\quad& \omega=(\omega\tau)\otimes_\F P=(i_P(\tau\wedge
d\tau))\otimes_\F P.\\
iii.\quad&i_\omega=\{(d-\tau\wedge\La_{iP})\otimes_\R i_p\},\quad
\{(d-\tau\wedge\La_{iP})\otimes_\R i_P\}=0.\\
iv.\quad&(d-\tau\wedge\La_{iP})^2=(\omega\tau)\wedge\La_{iP}\simeq
\operatorname{curvature}.\end{align*}\end{thm2}
\begin{proof} The proof of (i)-(ii) is straightforward, by direct
inspection. The equalities (iii) and (iv) of derivations are a
little more involved. The identity (iii), tells that the tensor
field $\omega$ is `the spatial divergence' of the connection $p,$ is
even more convincing, than adopted Definition \ref{angular}, to
interpret $\omega$ as the angular rotation tensor field. A two-form
$d\tau$ sometimes is called the vortex form of the connection $p\in
M\otimes_\F M^*$ [Cattaneo].\end{proof}

The differential operator, $(e_{\tau/(\tau P)}\circ\La_{iP})\circ
(\id-i_p),$ is invariant with respect to the dilation
\begin{gather}P\mapsto fP,\quad e_{\tau/(\tau
fP)}\circ\La_{fiP}=e_\tau\circ\La_{iP}-e_{df/f}\circ
i_p.\end{gather}

\noindent\textbf{Bianchi identity.} Luigi Bianchi introduced his
identity in Lezioni di geometria $\dots,$ three Volumes published in
[1902--1909]. We refer also to [Kol\'ar, Michor and Slov\'ak 1993].
The Bianchi identity for a connection $p\in M\otimes_\F M^*$ tells
that
\begin{gather} \half[[p\otimes_\R p]\otimes_\R p]=[\omega\otimes_\R p]
=\{\omega\otimes_\R(d-\La_{ip})\}=0.\end{gather}

\section{Frobenius algebra} Let $\F$ denotes an associative and
commutative unital ring. Let $A$ be $\F$-module ($\F-\F$-module) and
$A^*\equiv\Mod_\F(A,\F)$ be a dual $\F$-module, together with the
right and the left evaluations and co-evaluations, also known as the
closed/pivotal structures which axioms are given by the Reidemeister
zero moves,
\begin{gather}\begin{CD}
A^*\otimes_\F A@>{\text{left evaluation}}>>\F\\
A\otimes_\F A^*@>{\text{right evaluation}}>>\F\\
A^*\otimes_\F A@<{\text{left co-evaluation}}<<\F\\
A\otimes_\F A^*@<{\text{right co-evaluation}}<<\F
\end{CD}\end{gather}

An $\F$-algebra $m=\guy$ with a Frobenius covector (a co-unit)
$\varepsilon$ is said to be co-unit-class $\F$-algebra,
\begin{gather}\guy\in(2\mapsto 1)\equiv\Mod_\F(A\otimes_\F A,A),\notag\\
\varepsilon\in(1\mapsto 0)\equiv\Mod_\F(A,\F)\equiv A^*.\end{gather}
The composition $(\text{co-unit}\,\circ\,\guy)$ is a binary form
equivalent to unary left/right $\F$-module map
$h^{l/r}\in\Mod_\F(A,A^*),$
\begin{gather}\begin{CD}A\otimes A@>{\varepsilon\circ m=
h^l\circ(\ev^l\otimes\id)=(\id\otimes\ev^r)\circ h^r}>>\F\\
A@>{h^l,h^r}>>A^*\end{CD}\end{gather} If a form $h^l$ or/and $h^r$
is non-degenerate, $\ker(h)=0\in A,$ then $\{m,\varepsilon)\}$ is
said to be Frobenius $\F$-algebra [Ferdinand Georg Frobenius
(1849-1917), 1903].

An $\F$-co-algebra $\triangle=\guc$ with unit $\eta$ is said to be
unit-class co-algebra,
\begin{gather}\guc\in(1\mapsto 2)\equiv\Mod_\F(A,A\otimes_\F A),\notag\\
\eta/1\in(0\mapsto 1)\equiv\Mod_\F(\F,A)\simeq A.\end{gather} The
composition $\guc\circ\eta$ is a co-binary form that is equivalent
to left/right unary $\F$-module map $f^{l/r}\in\Mod_\F(A^*,A),$
\begin{gather}\begin{CD}A\otimes_\F A@<{\triangle\circ\eta=(f^l\otimes
\coev^l)\circ\coev^l=\coev^r\circ(\id\otimes f^r)}<<\F\\
A@<{f^l,f^r}<<A^*\end{CD}\end{gather} If this co-binary form
$\triangle\circ\eta$ is non-degenerate, $\ker(f^l/f^r)=0\in A^*,$
then $\{\triangle,(\text{unit}=\eta)\}$ is said to be Frobenius
$\F$-co-algebra.

A Frobenius $\F$-algebra is both Frobenius algebra and Frobenius
co-algebra subject two Frobenius axioms [Frobenius 1903, Curtis and
Reiner 1962, Kauffman 1994, Voronov 1994, Kadison 1999, Caenepeel et
al. 2002, Baez 2001], $$\gud\sim\guw\sim\gue.$$ The Frobenius axioms
do not imply uniqueness of $\guc$ for a given model of $\guy,$ and
vice-versa. The Frobemius axioms can be rephrased as
\begin{gather*}\guy\in\text{bicomod}(||,|),\qquad\guc\in\text{bimod}(|,||).
\end{gather*}

A Clifford algebra is a particular example of a Frobenius algebra
where unary `handle' $\guy\circ\guc=\guo\in(1\mapsto 1)$ is diagonal
[Oziewicz 2003, Figure 10]. Such Frobenius algebra is also said to
be `canonical'. A Frobenius algebra is antipode-less [Oziewicz 1997,
1998].

\begin{trace}A trace on $\F$-algebra $A$ is an $\F$-module map, a
covector $\tr\in\Mod_\F(A,F)\equiv A^*,$ \ie a co-unit/(Frobenius
covector), such that $\forall\;u,v\in A,$ $\tr(uv)=\tr(vu).$ An
$\F$-algebra $A$ with a trace is said to be trace-class
$\F$-algebra.

A unit $\eta\in A\simeq\Mod_\F(\F,A)$ is said to be co-trace if
$\triangle\circ\eta=\triangle^{op}\circ\eta.$ An $\F$-co-algebra
with co-trace, $\cotr=\tr^*,$ is said to be co-trace-class
co-algebra,
\begin{gather}\triangle\circ\cotr=\triangle^{op}\circ\cotr.\end{gather}

The composition $(\tr\circ m)$ is a symmetric binary form, and
$(\triangle\circ\cotr)$ is a symmetric co-binary form.\end{trace}

The Nijenhuis-Richardson $\Z$-graded $\F$-algebra restricted to zero
grade endomorphisms $M\otimes M^*$ is associative and unital
trace-clase algebra,
\begin{gather}\begin{CD}M\otimes_\F M^*@>{\text{trace = counit}}>>\F,\end{CD}
\qquad\tr(pq)=\tr(qp).\end{gather} One can extend $\F$-valued trace
to $M^\wedge$-valued counit=`super-trace' over the
Nijenhuis-Richardson nonassociative graded $\F$-algebra
\begin{gather}\begin{CD}M^\wedge\otimes_\F
M^*@>\text{`trace'}>>M^\wedge\end{CD},\quad \tr(\alpha\otimes_\F
P)\equiv i_P\alpha\in M^\wedge.\end{gather}

\section{Frobenius subalgebra of Nijenhuis-Richardson algebra}
\begin{atom}[Atomic idempotent] An idempotent $p^2=p\in A$ in
an algebra $A$ is said to be an \textit{atom} if $p\wedge(pAp)=0\in
A^{\wedge 2}$ [Jones, Statistical Mechanics, 1989].\end{atom}

The Nijenhuis-Richardson nonassociative $\F$-algebra possess
important associative subalgebra of endomorphisms $\End_\F
M\equiv\Mod_\F(M,M)$ (the endomorphism algebra with trivial center
is said to be the von Neumann factor). The endomorphism subalgebra
is not stable under Fr\"olicher-Nijenhuis Lie differential
$\R$-operation, if $p\in\End_\F M$ then $[p\otimes_\R
p]\not\in\End_\F M.$

We consider unital subalgebra of endomorphism algebra, generated by
finite set of \textit{primitive} idempotents (an idempotent $p^2=p$
is said to be primitive if $p=a+b$ for idempotents $a$ and $b$ with
$ab=ba=0$ imply that $a=0$ or $b=0$). It appears that in the generic
case such subalgebra `of idempotents' is Frobenius.

A set $n\in\N$ of primitives idempotents $\{p_1,\ldots,p_n\},$
$\tr(p_i)=1\in\F,$ and unit $u,$ with a finite trace $\tr u=d\in\N,$
generate not commutative trace-class Frobenius $\F$-algebra $\Fr_n$
(relations are given below) with symmetric form $h\equiv\tr\circ
m\in\Mod_\F((\Fr_n)^{\otimes 2},\F).$ This particular bi-associative
and bi-unital/bi-trace Frobenius $\F$-algebra $\Fr_n$ is a
sub-algebra of Nijenhuis-Richardson algebra, Definition \ref{NR}.

A Frobenius $\F$-algebra ${\Fr}_n$ of atomic/simple idempotents is
subject of the following relations,
\begin{gather}(p_i)^2=p_i,\quad i=1,\ldots,n,\\
\forall\;w\in{\Fr}_n,\qquad
p_iwp_j\,\tr(p_ip_j)=p_ip_j\,\tr(p_iwp_j).\end{gather}

Every pair of atomic idempotents $p$ and $q$ with $\tr p=\tr
q=1\in\F,$ satisfy the Galois connection (name introduced by Ore), a
property that is also called a generalized
inverse\begin{gather}pqp=\tr(pq)\,p\quad\text{and}\quad
qpq=\tr(pq)\,q.\label{Galois}\end{gather} This remains the relations
of the Jones algebra and of the von Neumann finite dimensional
algebra generated by atoms $p$ and $q$ [Jones 1983, \S 3].

From this it follows that a length of every word in Frobenius
$\F$-algebra $\Fr_n$ must be $\leq 2,$ and the $\F$-dimensions are
\begin{gather*}\dim_\F(\Fr_n)=1+n^2\quad=1,2,5,10,17,26,\ldots\end{gather*}

\begin{thm4}[Laplace expansion] The Frobenius covector is given by a trace
$tr\in(\Fr)^*.$ The following Laplace expansion holds, also called
`\textit{weak} coalgebra' condition. In the Sweedler notation for
three words $a,b,c\in\Fr_n),$
\begin{gather}\tr(abc)=\Sigma\,\tr(a_1c)\tr(a_2b).\label{Laplace}\end{gather}
\end{thm4}

In particular for $a=b=c=u\equiv\eta,$
\begin{gather}\N\ni d\equiv\tr\circ\cotr\equiv\tr(u)=\Sigma\,\tr(u_1)
\tr(u_2),\\\guc u\neq u\otimes
u\stackrel{\guy}{\longmapsto}u\stackrel{\tr}{\longmapsto}d.\end{gather}

\begin{thm5}[Frobenius coalgebra] Let $\{e_i\in\Fr_n\}$ be a basis
diagonalizing $h=\tr\circ\guy,$ \ie $h(e_i\otimes
e_j)\equiv\tr(e_ie_j)=h_i\delta_{ij}.$ Then
$$\guc\,e_i=\tr(e_ie_ke_l)\frac{e_l}{h_l}\otimes\frac{e_k}{h_k}.$$
\end{thm5}

The Frobenius algebra of atomic idempotents is antipode-less.

\section{Frobenius algebra of two idempotents} The bilinear form on
2-dimensional $\F$-algebra $\Fr_1=\Span_\F\{u,p\}$ for $1<d$ is
positive definite $(++).$ To see this, let choose the volume form as
$z_1\equiv u\wedge p\in(\Fr_1)^{\wedge 2}.$ Then
$\det_zh\equiv(h^\wedge z)z=d-1.$ The form $h\equiv\tr\circ\guy$ in
the basis $\{u,p\}$ and in the basis $\{u-p,p\}$ (after Gram-Schmidt
orthogonalization) possess the following basis-dependent matrix
presentations
\begin{gather}h\begin{pmatrix}u\\p\end{pmatrix}=\begin{pmatrix}
d&1\\1&1\end{pmatrix}\begin{pmatrix}u^*\\p^*\end{pmatrix},\quad
h\begin{pmatrix}u-p\\p\end{pmatrix}=\begin{pmatrix}
d-1&0\\0&1\end{pmatrix}\begin{pmatrix}u^*\\u^*+p^*\end{pmatrix}.\end{gather}

A coalgebra $\Fr_1$ is group-like (no no-zero primitives). The
co-unital co-multiplication is not unital
\begin{gather}\guc(u-p)=\frac{(u-p)\otimes(u-p)}{\tr(u-p)},\quad
\guc\,p=p\otimes p.\end{gather}

The Lagrange/Sylvester theorem and Gram-Schmidt orthogonalization
allows to calculate the signature for Frobenius $\F$-algebra $\Fr_n$
for any $n\in\N.$ Here we wish to report signature for
five-dimensional Frobenius $\F$-algebra $\Fr_2=\gen\{p,q\}$
generated by two atomic idempotents.

\begin{thm3}[Signature] Let $t\equiv\tr(pq)\neq\{-1,0,+1\}.$ The signature
of the bilinear form $h\simeq\tr\circ m:(\Fr_2)^{\otimes 2}\ra\F$
for five-dimensional $\F$-algebra $\Fr_2,$ $\dim_\F(\Fr_2)=5,$
depends on $d\equiv\tr(u)\in\N$ only.
\begin{gather*}\operatorname{Signature}\; \text{of $h$}\;=\begin{cases}
-++++&\text{if}\quad d>2,\\-+++0&\text{if}\quad d=2,\\
-+++-&\text{if}\quad d<2.\end{cases}\end{gather*}\end{thm3}
\begin{proof} Let $p$ and $q\in\Fr_2$ be generating atomic idempotents.
The center $Z\Fr_2$ of Frobenius $\F$-algebra $\Fr_2$ is
two-dimensional,
\begin{gather}\dim_\F(Z\Fr_2)=2,\qquad u,(p-q)^2\in Z\Fr_2,\\
(pq+qp)^2=t(p+q)^2,\quad(p-q)^4=-(t-1)(p-q)^2.\end{gather}

Let a volume form for a $\F$-module $\Fr_2$ be $z_2\equiv u\wedge
p\wedge q\wedge pq\wedge qp\,\in(\Fr_2)^{\wedge 5}.$ Then
$\det_z(\tr\circ\, m)= -(d-2)(t-1)^4t^2.$ In the basis
$\{u,p,q,pq,qp\}$ the bilinear form $h\equiv\tr\circ\,m$ has the
following basis-dependent-matrix
\begin{gather}h\begin{pmatrix}u\\p\\q\\pq\\qp\end{pmatrix}=
\begin{pmatrix}
d&1&1&t&t\\1&1&t&t&t\\1&t&1&t&t\\t&t&t&t^2&t\\t&t&t&t&t^2\end{pmatrix}
\begin{pmatrix}u^*\\p^*\\q^*\\(pq)^*\\(qp)^*\end{pmatrix}
\end{gather}
For $t\neq\{-1,0,+1\},$ the particular basis of $\Fr_2$
diagonalizing the form $h=\tr\circ\cotr$ is
\begin{gather}u+\frac{(p-q)^2}{t-1},\quad qp,\quad p+tq-(pq+qp),\quad
q-\frac{pq+qp}{t+1},\quad pq-\frac{qp}{t}.\end{gather} In this basis
the matrix of the scalar product $h$ is diagonal,
\begin{gather}h\simeq\text{diag}\left(d-2,t^2,(t-1)^2,-\frac{t-1}{t+1},
t^2-1\right).\qed\end{gather}\renewcommand{\qed}{}\end{proof}

\section{Conclusion} The Fr\"olicher and Nijenhuis Lie $\R$-algebra
structure on universal Gra{\ss}mann-module of differential
multi-forms found increasing number of important
applications/interpretations both in pure algebra and in
differential geometry of Ehresmann connections [Kocik 1997, Wagemann
1998], as well as in many branches of mathematical physics, in the
special and in the general theory of relativity [Minguzzi 2003], in
Maxwell's theory of electromagnetic field [Fecko 1997, Kocik 1997,
Cruz and Oziewicz 2003], in Hamilton-Jacobi theory in classical
mechanics [Gruhn and Oziewicz 1983], in symplectic geometry of the
Lagrangian and Hamiltonian mechanics [Chavchanidze 2003], etc.

From the point of view of these numerous fundamental applications
there is a need for the algorithmic computational programming
methods to deals with many structural aspects of this non trivial
Lie $\R$-algebra. The present paper was motivated by this need of
explicit/algorithmic easy to handle definition of the Fr\"olicher
and Nijenhuis Lie operation. We are proposing here such definition
of the Fr\"olicher and Nijenhuis Lie operation
\eqref{FN4}-\eqref{FN2}. This definition has a clear advantage that
can be implemented for computational symbolic program in computer
algebra.

Many identities that hold in Fr\"olicher and Nijenhuis Lie
Gra{\ss}mann-module follows much easily from proposed definition.

It is important that the Definition \eqref{FN4}-\eqref{FN2} of Lie
$\R$-algebra needs nonassociative Fr\"olicher-Richardson
$\F$-operation on universal Gra{\ss}mann-module. The
Fr\"olicher-Richardson nonassociative $\F$-algebra deserve future
studies in many respects. The Fr\"olicher-Richardson algebra include
associative endomorphism subalgebra. Of special interests, from
fundamental physical theories, quantum mechanics and relativity
theory, are endomorphism subalgebras generated by atomic
idempotents. Such generic subalgebras are Frobenius algebras, they
possess non-degenerate scalar product that gives antipode-less
algebra structure. In the last Sections the Frobenius algebra is
illustrated on example of the five-dimensional algebra generated by
two atomic idempotents. We believe that the correct environment for
these particular Frobenius associative algebras must be
nonassociative Fr\"olicher-Richardson algebra, because the
Fr\"olicher and Nijenhuis differential Lie operation do not preserve
associative endomorphism algebra. If $p\in M\otimes_\F M^*$ is an
endomorphism, then the Fr\"olicher and Nijenhuis differential Lie
operation \eqref{FN4}-\eqref{FN2} gives $[p\otimes_\R p]\not\in
M\otimes_\F M^*,$ but $[p\otimes_\R p]$ is inside the
Fr\"olicher-Richardson algebra. We conjecture that the Frobenius
associative algebra could be related/identified with the kinematics
and the Fr\"olicher-Richardson not associative algebra with
dynamics,
\begin{center}\begin{tabular}{cc}\hline\\
Kinematics&Dynamics\\\hline Special relativity&\bt General relativity\\Gravity\et\\
Frobenius algebra&\bt Fr\"olicher-Richardson\\algebra\et\\\hline
\end{tabular}\end{center}

\begin{thebibliography}{99}  %bez numeracji\item[Bondi]{}
%\bibitem{} Abrams Lowell (1996). Two-dimensional topological quantum field
%theories and Frobenius algebras, Journal of Knot Theory and
%Ramifications \textbf{5} 569--587.
%\bibitem{} Abrams Lowell (1999) Modules, comodules and cotensor products
%over Frobenius algebras, Journal of Algebra \textbf{219} 201-213.
\bibitem{} Baez John (2001). Frobenius algebra,\\
http://math.ucr.edu/home/baez/week174.html,
http://w4.lns.cornell.edu/spr/2001-12/msg0037326.html.
\bibitem{} Cabras A., Vinogradov A. M., (1992). Extensions of the
Poisson bracket to differential forms and multi-vector fields,
Journal of Geometry and Physics \textbf{9} 75--100.
\bibitem{} Beidar K. I., Fong Y. and Stolin A., (1997). On
Frobenius algebras and the quantum Yang-Baxter equation,
Transactions of the American Mathematical Society \textbf{349} (9)
3823--3836.
\bibitem{} Caenepeel Stefaan, Militaru Gigel, Zhu Shenglin, (2002).
Frobenius and Separable Functors for Generalized Module Categories
and Nonlinear Equations, Lecture Notes in Mathematics \# 1787,
Springer, Berlin.
\bibitem{} Cap A., Kriegl A., Michor Peter W., Van{\u z}ura Jiri,
(1991). The Fr\"olicher--Nijenhuis bracket in non commutative
differential geometry, Universit\"at Wien, preprint.
\bibitem{} Cartan \'Elie Joseph, (1922). Le{\c c}ons sur les invariants
int\'egraux, Paris 1922.
\bibitem{} Chavchanidze George, (2003). Non-Noether symmetries and their
influence on phase space geometry, Journal of Geometry and Physics
\textbf{48} 190--202; math-ph/0211014.
\bibitem{} Coquereaux R., Jadczyk Arkadiusz, (1991). Differential
and integral geometry of Grassmann algebras, Review Mathematical
Physics \textbf{3} 63.
\bibitem{} Cruz Guzm\'an Jos\'e de Jes\'us, Oziewicz Zbigniew,
(2003). Fr\"olicher - Nijenhuis algebra and four Maxwell equations
for not inertial observer, Bulletin de la Soci\'et\'e des Sciences
et des Lettres de {\L}\'od\'z, Volume \textbf{LIII}, S\'erie:
Recherches sur les D\'eformations Volume \textbf{XXXIX} 107--140, PL
ISSN 0459-6854.
\bibitem{} Curtis Charles W., Reiner Irving, (1962). Representation
Theory of Finite Groups and Associative Algebras, Interscience
Publishers a division of John Wiley \& Sons, New York; Nauka, Moskva
1969.
\bibitem{} Dubois-Violette Michel, Michor Peter W., (1995). A
common generalization of the Fr\"olicher-Nijenhuis bracket and the
Schouten bracket for symmetry multi vector fields, Indagationes
Math. New Series, \textbf{6} 51--66; alg-geom/9401006 (version
August 2003).
\bibitem{} Fecko Mari\'an, (1997). On $3+1$ decomposition with respect
to an observer field via differential forms, Journal of Mathematical
Physics \textbf{38} (9) 4542--4560.
\bibitem{} Frobenius Ferdinand Georg (1903). Theorie der hyperkomplexen
Gr\"o{\ss}en, Sitzungsber. Press. Akad. Wissen., Leipzig,
\textbf{24} 504--537, 634--645.
\bibitem{} Fr\"olicher A., Nijenhuis A., (1956). Theory of vector valued
differential forms, Part I, Indagationes Math. \textbf{18} 338--359.
%\bibitem{} Fr\"olicher A., 1956, P KON NED AKAD WET A, \textbf{59} 338
\bibitem{} Gancarzewicz Jacek, (1987). Geometria R\'o\.zniczkowa,
PWN Warszawa 1987.
\bibitem{} Gozzi E., Mauro D., (2000). A new look at the Schouten-Nijenhuis,
Fr\"olicher-Nijenhuis and Nijenhuis-Richardson brackets for
symplectic spaces, Journal of Mathematical Physics \textbf{41}
1916--1933; hep-th/9907065.
\bibitem{} Gray P., (1967). Pseudo-Riemannian almost product manifolds and
submersions, Journal of Mathematics and Mechanics \textbf{16}
715--737.
\bibitem{} Gruhn Wojciech, Oziewicz Zbigniew, (1983). On Jacobi's theorem
from a year 1838 in classical mechanics, Hadronic Journal \textbf{6}
(6) 1579--1605, MR\#85f:70027.
\bibitem{} Jadczyk Arkadiusz, and Daniel Kastler Daniel, (1987). Graded
Lie-Cartan pairs II, Annals of Physics \textbf{179} 169.
\bibitem{} Jones Vaughan F. R., (1983). Index for subfactors, Inventiones
mathematicae \textbf{72} 1--25.
\bibitem{} Kadison L., (1999). New examples of Frobenius extensions,
University Lecture Series \# 14, American Mathematical Society.
\bibitem{} Kanatchikov Igor V., (1996). From the Poincar\'e-Cartan
form to a Gerstenhaber algebra of Poisson brackets in field theory,
in: Antoine J. P., Ali S. T., `Quantization, Coherent States and
Complex Structures', Plenum Press, New York.
\bibitem{} Kanatchikov Igor V., (1996). On non-commutative
Gerstenhaber algebra. Graded Leibniz algebra and non-commutative
Gerstenhaber algebra in field theory. Not published.
\bibitem{} Kauffman Louis H., (1994). Ribbon Hopf algebras and
invariants of 3-manifolds. In: Keller Jaime and Oziewicz Zbigniew,
Editors, `Differential Geometric Methods in Theoretical Physics',
Universidad Nacional Aut\'onoma de M\'exico Press.
\bibitem{} Kocik Jerzy, (1997). Relativistic observer and Maxwell's
equations: an example of a non-principal Ehresmann connection,
http://www.math.siu.edu/Kocik/.
\bibitem{} Kolar I., Michor Peter W., (1987). Determination of all
natural bilinear operators of the type of the Fr\"olicher--Nijenhuis
bracket, Supplement Rendiconti Circolo Matematico Palermo, Serie II
\textbf{16} 101--108.
\bibitem{} Kolar I., Michor Peter W., Slov\'ak J., (1993). Natural
Operations in Differential Geometry, Springer-Verlag, New York.
\bibitem{} Loday Jean-L., (1993). Une version non commutative des
alg\'ebres de Lie. Les alg\'ebres de Leibniz. L'Enseign. Math.
\textbf{39} 269.
\bibitem{} Michor Peter W., (1987). Remarks on the Fr\"olicher--Nijenhuis
bracket. In: Demeter Krupka and A. {\v S}vec, Editors, Differential
Geometry and its Applications, D. Reidel, pp. 197--220.
\bibitem{} Krasil'shchik Joseph S., Verbovetsky Aleksander M,
(1998). Homological methods in equations of mathematical physics:
Fr\"olicher--Nijenhuis brackets and algebras with flat connections,
Lectures in Slovakia, unpublished.
\bibitem{} Minguzzi E., (2003). Simultaneity and generalized connections
in general relativity, gr-qc/0204063.
\bibitem{} Minguzzi E., (2003). The gauge nature of simultaneity, Classical
and Quantum Gravity \textbf{20} 2443--2456.
%\bibitem{} M\"uger Michael, (2002). From subfactors to categories and
%topology I. Frobenius algebras in and Morita equivalence of tensor
%categories, math.CT/0111204 v2, 17 January 2002.
\bibitem{} Nijenhuis A., (1955). Jacobi-type identities for bilinear
differential concomitants of certain tensor fields, Part I,
Indagationes Math. \textbf{17} 390--403.
%\bibitem{} Nijenhuis A., 1955, P KON NED AK WET A A, V58, 390
\bibitem{} Nijenhuis A., Richardson R., (1967). Deformation of Lie
algebra structures, Journal of Mathematics and Mechanics \textbf{17}
89--105.
\bibitem{} Zbigniew Oziewicz, (1992). Fr\"olicher and Nijenhuis operations
for non commutative rings, in: Sultan Catto and Alvany Rocha,
Editors, Differential Geometric Methods in Theoretical Physics,
World Scientific, Singapore, pp. 584--593.
%\bibitem{}  OZIEWICZ Z, 1990, 19 INT C DIFF GEOM M
\bibitem{} Oziewicz Zbigniew, Paal Eugen, R\'o\.za\'nski
Jerzy, (1994). Coalgebra and cohomology. In: Santos Gonz\'alez,
Editor, Non-Associative Algebra and Its Applications, Kluwer
Academic Publishers, Dordrecht 1994, ISBN 0-7923-3117-6 [In Series:
Mathematics and Its Applications, Volume \textbf{303}], pp.
314--322. hep-th/9312023.
\bibitem{} Oziewicz Zbigniew, Paal Eugen, R\'o\.za\'nski Jerzy
(1995). Derivations in braided geometry, Acta Physica Polonica
\textbf{B 26} (7) 245--278.
\bibitem{} Oziewicz Zbigniew, Paal Eugen, (1995). On Gersthenhaber
algebra, unpublished.
\bibitem{} Oziewicz Zbigniew, (1997). Clifford algebra and universal
Hopf algebra, Czechoslovak Journal of Physics \textbf{47} (12)
1267--1274, http://www.arxiv.org/abs/q-alg/9709016.
\bibitem{} Oziewicz Zbigniew, (1998). The Dirac operator as graph, and the
Clifford algebra, Pitman Research Notes in Mathematics Series \#
394, Addison Weseley Longman, Harlow-London 1998, ISBN
0-582-35681-4, pp. 210--224.
\bibitem{} Oziewicz Zbigniew, (2003). Operad of graphs, convolution and quasi
Hopf algebra, Contemporary Mathematics \textbf{318} (2003) 175--197.
\bibitem{} Soroka D V, Soroka, V A. (2003). Exterior differentials in
superspace and Poisson brackets, Journal of High Energy Physics (3)
AR 001, 11.
\bibitem{} \'Slebodzi\'nski W{\l}adys{\l}aw, (1931). Sur les \'equations
de Hamilton, Bulletin de l'Acad\'emie Royale Belgique \textbf{17}
(5), 864--870.
\bibitem{} Tulczyjew W{\l}odzimierz M., (1974). The graded Lie algebra
of multivector fields and the generalized Lie derivative of forms,
Bulletin de l'Acad\'emie Polonaise des Sciences, S\'erie des
Sciences Mathematiques \textbf{XXII} (9) (1974), 931--935.
\bibitem{} Tulczyjew W{\l}odzimierz M., (1974). The graded Lie
algebra of multivector fields, Bulletin de L'Acad\'emie Polonaise
des Sciences, S\'erie des Sciences Mathematiques \textbf{XXII} (9)
937-942.
\bibitem{} Voronov Alexander A., (1994). Topological field theories,
string backgrounds and homotopy algebras. In: Keller Jaime and
Oziewicz Zbigniew, Editors, `Differntial Geometric Methods in
Theoretical Physics', Universidad Nacional Aut\'onoma de M\'exico
Press.
\bibitem{} Vinogradov A. M., (1990). Unification of Schouten and Nijenhuis
brackets, cohomology and superdifferential operators, Soviet
Matematiczeskije Zametki \textbf{47} (6) 138--140.
\bibitem{} Wagemann Friedrich, (1998). Differential graded cohomology
and Lie algebras of holomorphic vector fields, math-ph/9806015.
\end{thebibliography}
\end{document}
\nwfilename{SandBoxNijenhuis.pamphlet}
