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\section*{multiplicative unitary}

[[!redirects pentagon relation]] [[!redirects multiplicative unitaries]] [[!redirects fundamental operator]]

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

While the following idea is originally in operator setup and with an involution, consider the following. Let $H$ be a finite-dimensional vector space. Consider the invertible operator $W : H\otimes H \to H\otimes H$ satisfying the pentagon identity

\begin{displaymath}
W_{1 2} W_{1 3} W_{2 3} = W_{2 3} W_{1 2}
\end{displaymath}
in the space of linear endomorphisms of $H\otimes H\otimes H$. Then the formula

\begin{displaymath}
\Delta(h) = W (h\otimes 1) W^{-1}
\end{displaymath}
define a coassociative coproduct on $H$. Usually we replace the structure of the coproduct with knowing $W$, which can be easier to define in infinite-dimensional analogues when the coproduct needs to take values in some hard to manage completions.

for all $h\in H$. For finite-dimensional Hopf algebras $W(g\otimes h) = g_{(1)}\otimes g_{(2)} h$ and $W^{-1}(g\otimes h) = g_{(1)}\otimes (S g_{(2)}) h$ and then we can reproduce the antipode via the formula

\begin{displaymath}
S h = (\epsilon\otimes id)\circ W^{-1}(h\otimes - )
\end{displaymath}
We can also make a discussion in terms of the dual space $H^*$. Then the coproduct on $H^*$ which is dual to the product on $H$ is also obtained from $W$ by the formula

\begin{displaymath}
\Delta_{H^*}(\psi) = W^{-1} (1\otimes\psi) W
\end{displaymath}
\hypertarget{literature_and_further_directions}{}\subsubsection*{{Literature and further directions}}\label{literature_and_further_directions}

In the setup of operator algebras, the multiplicative unitaries were introduced as so called Kac–Takesaki operator. Following some ideas on noncommutative extensions of Pontrjagin duality (in Tannaka-Krein spirit) by George's Kac and also M. Takesaki, Lecture Notes in Mathematics. 247, Berlin: Springer; 1972. pp. 665–785. The followup work of Baaj and Skandalis introduced two more fundamental axioms, regularity and irreducibility, important in $C^*$-algebraic setup.

\begin{itemize}%
\item Saad Baaj, Georges Skandalis, \emph{Unitaires multiplicatifs et dualité pour les produits croisés de $C^*$-algèbre}, Annales scientifiques de l'École Normale Supérieure \textbf{26}:4 (1993) 425-488 \href{https://doi.org/10.24033/asens.1677}{numdam}; \emph{Transformations pentagonales (Pentagonal transformations)}, Comptes Rendus de l'Académie des Sciences, I - Mathematics \textbf{327}:7 (1998) 623-628, \href{a: href="https://doi.org/10.1016/S0764-4442(99)80090-1"}{a\char58\char32href\char61\char34https\char58\char47\char47doi\char46org\char47\char49\char48\char46\char49\char48\char49\char54\char47S\char48\char55\char54\char52\char45\char52\char52\char52\char50\char40\char57\char57\char41\char56\char48\char48\char57\char48\char45\char49\char34}doi{\tt \symbol{60}}/a{\tt \symbol{62}}


\item Saad Baaj, Étienne Blanchard, Georges Skandalis, \emph{Unitaires multiplicatifs en dimension finieet leurs sous-objets}, Annales de l’institut Fourier, tome 49, no4 (1999), p. 1305-1344 \href{http://www.numdam.org/item?id=AIF_1999__49_4_1305_0}{numdam}



\end{itemize}
The monograph on Kac algebras is

\begin{itemize}%
\item M. Enock, J. M. Schwartz, \emph{Kac algebras and duality of locally compact groups}, Springer-Verlag, 1992, , x+257 pp. \href{http://books.google.com/books/about/Kac_algebras_and_duality_of_locally_comp.html?id=U6e6aD1gj3oC}{gBooks}, \href{http://www.ams.org/mathscinet-getitem?mr=1215933}{MR94e:46001}

\end{itemize}
and a more recent view of duality between Hopf algebra approach and an approach to quantum groups via multiplicative unitaries is in the book

\begin{itemize}%
\item Thomas Timmermann, \emph{An invitation to quantum groups and duality: From Hopf algebras to multiplicative unitaries and beyond}, Europ. Math. Soc. 2008.

\end{itemize}
Introduction to Ch. 7 says in Timmerman's book says

\begin{quote}%
Multiplicative unitaries are fundamental to the theory of quantum groups in the setting of $C^*$-algebras and [[von Neumann algebra]]s, and to generalizations of [[Pontrjagin duality]]. Roughly, a multiplicative unitary is one single map that encodes all structure maps of a quantum group and of its generalized Pontrjagin dual simultaneously.


\end{quote}
Woronowicz has introduced managaeble multiplicative unitaries

\begin{itemize}%
\item [[S. L. Woronowicz]], \emph{From multiplicative unitaries to quantum groups}, Internat. J. Math. \textbf{7} (1996), 127–149.

\end{itemize}
It is useful to look at the survey

\begin{itemize}%
\item Johan Kustermans, Stefaan Vaes, \emph{The operator algebra approach to quantum groups}, Proc Natl Acad Sci USA 97(2): 547–552 (2000) \href{https://doi.org/10.1073/pnas.97.2.547}{doi}


\item A. Van Daele, S. Van Keer, \emph{The Yang-Baxter and pentagon equation}, Compositio Mathematica \textbf{91}:2 (1994) 201-221 \href{http://www.numdam.org/item/CM_1994__91_2_201_0}{numdam}



\end{itemize}
The categorical background of the pentagon equation has been studied in

\begin{itemize}%
\item [[Ross Street]], \emph{Fusion operators and cocycloids in monoidal categories}, Applied Categorical Structures 6: 177–191 (1998) \href{https://doi.org/10.1023/A:1008655911796}{doi}

\end{itemize}
A finite dimensional version is reformulated in section 3 of

\begin{itemize}%
\item S. Majid, \emph{Quantum random walks and time reversal} \href{https://doi.org/10.1142/S0217751X93001818}{doi}

\end{itemize}
and reprinted in Majid's, \emph{Foundations of quantum group theory}, 1995, as Theorem 1.7.4. Majid has stated in this finite-dimensional case, ideas about [[quantum group Fourier transform]] (see there and Majid's book). This has been used in

\begin{itemize}%
\item Laurent Freidel, [[nlab:Shahn Majid]], \emph{Noncommutative harmonic analysis, sampling theory and the Duflo map in $2+1$ quantum gravity}, Classical Quantum Gravity \textbf{25} (2008), no. 4, 045006 \href{http://www.ams.org/mathscinet-getitem?mr=2388191}{MR2009f:83058}, \href{http://dx.doi.org/10.1088/0264-9381/25/4/045006}{doi}

\end{itemize}
More categorical treatment and relation to [[Hopf-Galois extension]]s is in

\begin{itemize}%
\item A. A. Davydov, \emph{Pentagon equation and matrix bialgebras}, Commun. Alg. \textbf{29}(6), 2627–2650 (2001) \href{https://doi.org/10.1081/agb-100002412}{doi}

\end{itemize}
In the language of finite-dimensional Heisenberg doubles see also the treatment of fundamental operator in

\begin{itemize}%
\item G. Militaru, \emph{Heisenberg double, pentagon equation, structure and classification of finite-dimensional Hopf algebras}, J. London Math. Soc. (2) 69 (2004) 44--64 (\href{http://dx.doi.org/10.1112/S0024610703004897}{doi}).

\end{itemize}
Kashaev has explained the pentagon relations for quantum [[dilogarithm]] as coming from the pentagon for the canonical element in the double.

\begin{itemize}%
\item R.M. Kashaev, \emph{Heisenberg double and the pentagon relation}, St. Petersburg Math. J. 8 (1997) 585-- 592 \href{http://arxiv.org/abs/q-alg/9503005}{q-alg/9503005}.

\end{itemize}
The multiplicative unitary representing quantum dilogarithm has been studied analytically, in the disguise of a quantum exponential, in relation to the construction of ``noncompact quantum ax+b group'' in

\begin{itemize}%
\item [[S. L. Woronowicz]], \emph{Quantum exponential function}, Reviews in Mathematical Physics \textbf{12}:06, 873-920 (2000) \href{https://doi.org/10.1142/S0129055X00000344}{doi}

\end{itemize}
The formalism is also in

\begin{itemize}%
\item S. L. Woronowicz, From multiplicative unitaries to quantum groups, Internat. J. Math. 7(1) (1996) 127–149 \href{https://doi.org//10.1142/S0129167X96000086}{doi}, MR 1369908
\item P. M. Sołtan, S. L. Woronowicz, \emph{From multiplicative unitaries to quantum groups. II}, J. Funct. Anal 252(1), 42–67 (2007) \href{https://doi.org/10.1016/j.jfa.2007.07.006}{doi}
\item [[Ralf Meyer]], Sutano Roy, [[S. L. Woronowicz]], \emph{Quantum group-twisted tensor products of $C^*$-algebras}, Int. J. \textbf{25}:02, 1450019 (2014) \href{https://doi.org/10.1142/S0129167X14500190}{doi}; \emph{Semidirect products of $C^*$-quantum groups: multiplicative unitaries approach}, Commun. Math. Phys. \textbf{351}, 249–282 (2017) \href{https://doi.org/10.1007/s00220-016-2727-3}{doi}
\item Ralf Meyer, Sutanu Roy, \emph{Braided multiplicative unitaries as regular objects}, Adv. Stud. Pure Math., Operator Algebras and Mathematical Physics, M. Izumi, Y. Kawahigashi, M. Kotani, H. Matui, N. Ozawa, eds. (Tokyo: Mathematical Society of Japan, 2019) 153 - 178 \href{https://doi.org/10.2969/aspm/08010153}{doi}

\end{itemize}
The role of quantum torus is here quite clear; later treatments of more general quantum [[dilogarithm]]s influenced by Kontsevich-Soibelman work on [[wall crossing]] and related Goncharov's cluster varieties quantization also witness the appearance of the similar quantum torus.



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