nLab multiplicative unitary

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

$$W_{1 2} W_{1 3} W_{2 3} = W_{2 3} W_{1 2}$$

in the space of linear endomorphisms of $H\otimes H\otimes H$. Then the formula

$$\Delta(h) = W (h\otimes 1) W^{-1}$$

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

$$S h = (\epsilon\otimes id)\circ W^{-1}(h\otimes - )$$

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

$$\Delta_{H^*}(\psi) = W^{-1} (1\otimes\psi) W$$

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.

• 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

• M. Enock, J. M. Schwartz, Kac algebras and duality of locally compact groups, Springer-Verlag, 1992, , x+257 pp. gBooks, MR94e:46001

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

• Thomas Timmermann, An invitation to quantum groups and duality: From Hopf algebras to multiplicative unitaries and beyond, Europ. Math. Soc. 2008.

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

• S. L. Woronowicz, From multiplicative unitaries to quantum groups, Internat. J. Math. 7 (1996), 127–149.

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

• Ross Street, Fusion operators and cocycloids in

  monoidal categories_, Applied Categorical Structures 6: 177–191 (1998) doi

A finite dimensional version is reformulated in section 3 of

• Shahn Majid, Quantum random walks and time reversal doi

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

• Laurent Freidel, Shahn Majid, Noncommutative harmonic analysis, sampling theory and the Duflo map in $2+1$ quantum gravity, Classical Quantum Gravity 25 (2008) no. 4, 045006 MR2009f:83058, doi

More categorical treatment and relation to Hopf-Galois extensions is in

• A. A. Davydov, Pentagon equation and matrix bialgebras, Commun. Alg. 29(6), 2627–2650 (2001) doi

In the language of finite-dimensional Heisenberg doubles see also the treatment of fundamental operator in

• Gigel Militaru, Heisenberg double, pentagon equation, structure and classification of finite-dimensional Hopf algebras, J. London Math. Soc. (2) 69 (2004) 44–64 doi

Kashaev has explained the pentagon relations for quantum dilogarithm as coming from the pentagon for the canonical element in the double.

• R.M. Kashaev, Heisenberg double and the pentagon relation, St. Petersburg Math. J. 8 (1997) 585– 592 q-alg/9503005.

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

• S. L. Woronowicz, Quantum exponential function, Reviews in Mathematical Physics 12:06, 873-920 (2000) doi

The formalism is also in

• S. L. Woronowicz, From multiplicative unitaries to quantum groups, Internat. J. Math. 7(1) (1996) 127–149 doi, MR 1369908
• P. M. Sołtan, S. L. Woronowicz, From multiplicative unitaries to quantum groups. II, J. Funct. Anal 252(1), 42–67 (2007) doi
• Ralf Meyer, Sutano Roy, S. L. Woronowicz, Quantum group-twisted tensor products of $C^*$-algebras, Int. J. 25:02, 1450019 (2014) doi; Semidirect products of $C^*$-quantum groups: multiplicative unitaries approach, Commun. Math. Phys. 351, 249–282 (2017) doi
• Ralf Meyer, Sutanu Roy, 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 doi

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.