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multiplicative unitary

Idea

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

W 12W 13W 23=W 23W 12 W_{1 2} W_{1 3} W_{2 3} = W_{2 3} W_{1 2}

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

Δ(h)=W(h1)W 1 \Delta(h) = W (h\otimes 1) W^{-1}

define a coassociative coproduct on HH. Usually we replace the structure of the coproduct with knowing WW, 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 hHh\in H. For finite-dimensional Hopf algebras W(gh)=g (1)g (2)hW(g\otimes h) = g_{(1)}\otimes g_{(2)} h and W 1(gh)=g (1)(Sg (2))hW^{-1}(g\otimes h) = g_{(1)}\otimes (S g_{(2)}) h and then we can reproduce the antipode via the formula

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

We can also make a discussion in terms of the dual space H *H^*. Then the coproduct on H *H^* which is dual to the product on HH is also obtained from WW by the formula

Δ H *(ψ)=W 1(1ψ)W \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 *C^*-algebraic setup.

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 *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

The categorical background of the pentagon equation has been studied in

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.