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
in the space of linear endomorphisms of $H\otimes H\otimes H$. Then the formula
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
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
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.
Saad Baaj, Georges Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de $C^*$-algèbre, Annales scientifiques de l’École Normale Supérieure 26:4 (1993) 425-488 numdam; Transformations pentagonales (Pentagonal transformations), Comptes Rendus de l’Académie des Sciences, I - Mathematics 327:7 (1998) 623-628, a: href="https://doi.org/10.1016/S0764-4442(99)80090-1"doi</a>
Saad Baaj, Étienne Blanchard, Georges Skandalis, Unitaires multiplicatifs en dimension finieet leurs sous-objets, Annales de l’institut Fourier, tome 49, no4 (1999), p. 1305-1344 numdam
The monograph on Kac algebras is
and a more recent view of duality between Hopf algebra approach and an approach to quantum groups via multiplicative unitaries is in the book
Introduction to Ch. 7 says in Timmerman’s book says
Multiplicative unitaries are fundamental to the theory of quantum groups in the setting of $C^*$-algebras and von Neumann algebras, 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.
Woronowicz has introduced managaeble multiplicative unitaries
It is useful to look at the survey
Johan Kustermans, Stefaan Vaes, The operator algebra approach to quantum groups, Proc Natl Acad Sci USA 97(2): 547–552 (2000) doi
A. Van Daele, S. Van Keer, The Yang-Baxter and pentagon equation, Compositio Mathematica 91:2 (1994) 201-221 numdam
The categorical background of the pentagon equation has been studied in
monoidal categories_, Applied Categorical Structures 6: 177–191 (1998) doi
A finite dimensional version is reformulated in section 3 of
and reprinted in Majid’s, 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
More categorical treatment and relation to Hopf-Galois extensions is in
In the language of finite-dimensional Heisenberg doubles see also the treatment of fundamental operator in
Kashaev has explained the pentagon relations for quantum dilogarithm as coming from the pentagon for the canonical element in the double.
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
The formalism is also in
The role of quantum torus is here quite clear; later treatments of more general quantum dilogarithms 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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