\documentclass[12pt,twoside]{article} %Marcel Dekker 2004 \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=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^*}@<{\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{[\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}@<{e_\tau}>>M\end{CD}\hspace{1.5cm} \begin{CD}M^\wedge@>{i_P}>>M^\wedge\\@V{p}VV ||\\ M^\wedge@<{e_\tau}@<{\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}@<{\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 $12,\\-+++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. 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