The discrete quantum group $su_q(2)$ and its dual

Abstract: Discrete quantum groups were introduced as duals of compact quantum groups by Podleś and Woronowicz in 1990. They have been studied intrinsically by Effros and Ruan (1994) and by the author (1996). In a more recent note (2025), we have given a slightly updated treatment, viewing the duality between discrete and compact quantum groups as a special case of the more general duality of algebraic quantum groups. Along these lines, we start in this paper with the discrete quantum group $su_q(2)$, not constructed as the dual of the compact quantum group $SU_q(2)$ but rather from the Hopf algebra deformation of the enveloping algebra of the Lie algebra of $SU(2)$, as given by Jimbo (1985). The passage to the discrete quantum group as studied in earlier papers is not completely trivial as we will see. This is a known phenomenon. We consider the dual of this discrete quantum group in the sense of duality of algebraic quantum groups and see that this indeed is the compact quantum group $SU_q(2)$.

• The paper presents an explicit operator-algebraic construction of the discrete quantum group su_q(2), defining its coproduct, counit, and antipode.
• It develops a detailed representation theory for su_q(2), including finite-dimensional *-representations and explicit Clebsch-Gordan decompositions.
• The paper rigorously establishes the duality between su_q(2) and the compact quantum group SU_q(2), providing explicit formulas for integrals, cointegrals, and modular elements.

Detailed Summary of "The discrete quantum group $su_q(2)$ and its dual" (2603.29701)

Introduction and Motivation

This paper rigorously constructs the discrete quantum group $su_q(2)$, starting from the Hopf $^*$-algebra deformation of the enveloping algebra of the Lie algebra $\mathfrak{su}(2)$ introduced by Jimbo, and develops its dual. In contrast to the standard approach where discrete quantum groups arise as duals of compact quantum groups (notably $SU_q(2)$), the methodology here treats $su_q(2)$ directly within the operator algebraic framework of quantum groups.

A key focus is to establish the full discrete quantum group structure on $su_q(2)$, concretely constructing the coproduct, counit, and antipode at the operator algebra level, and then to realize its dual algebraically as the compact quantum group $SU_q(2)$ of Woronowicz.

Discrete Quantum Groups and Their Algebraic Setting

Discrete quantum groups are formulated as algebraic quantum groups, specifically as multiplier Hopf $^*$-algebras $A$ which are direct sums of full matrix algebras, equipped with a coassociative coproduct $su_q(2)$0, a counit $su_q(2)$1, and an antipode $su_q(2)$2. The structural regularity conditions (non-degeneracy and compatibility of algebraic operations with *-structure) are verified in detail.

Important foundational results are established, such as the existence and uniqueness of cointegrals and integrals, their explicit formulation, and the interplay of modular structures (modular automorphism, scaling group, modular element) necessary for further duality theory.

Construction of $su_q(2)$3: Hopf $su_q(2)$4-Algebra Level

The quantum group $su_q(2)$5 is constructed from the Jimbo deformation $su_q(2)$6 of $su_q(2)$7 with generators $su_q(2)$8, and commutation relations dependent on the deformation parameter ($su_q(2)$9, $^*$0). The deformed coproduct is

$^*$1

with a well-defined $^*$2-structure and antipode.

The paper provides a thorough analysis of the representation theory of $^*$3: irreducible finite-dimensional $^*$4-representations are constructed and indexed by $^*$5, in exact analogy with classical $^*$6 theory. The combinatorics of Clebsch-Gordan decomposition on tensor products is explicitly realized.

A comprehensive discussion of the Casimir element in $^*$7 is given, including deformed centrality relations and its spectral action on irreducible representations.

Operator Algebra Framework: Discrete Quantum Group Structure

$^*$8 as a discrete quantum group is realized via the direct sum $^*$9 where each $\mathfrak{su}(2)$0 is the algebra of endomorphisms on the $\mathfrak{su}(2)$1-dimensional Hilbert space associated to the irreducible representation $\mathfrak{su}(2)$2. The multiplier algebra $\mathfrak{su}(2)$3 is constructed as the direct product.

The coproduct on $\mathfrak{su}(2)$4 is reconstructed from matrix-level data using explicit representation theory and a careful analysis (\emph{e.g.}, injectivity of the direct sum representation, support of morphisms, compatibility of the representation-level coproduct with the algebraic one). The paper gives explicit formulas for the extension of $\mathfrak{su}(2)$5, counit, and antipode to $\mathfrak{su}(2)$6 and its multiplier algebra, utilizing the underlying representation ring data.

A major technical point is the construction and analysis of the scaling group, modular automorphisms, and unitary antipode. The scaling group $\mathfrak{su}(2)$7 and its polar decomposition intertwine with the action of the conjugate-linear involution $\mathfrak{su}(2)$8 on the Hilbert spaces, and the computation of $\mathfrak{su}(2)$9 on generators is matched with module automorphisms.

Integrals, Cointegrals, and Modular Data

The structure of (left and right) integrals, cointegrals, and the modular element is computed explicitly. The paper gives direct expressions for the cointegral $SU_q(2)$0 and the integrals $SU_q(2)$1 in terms of traces over matrices weighted by deformation parameters: $SU_q(2)$2 where $SU_q(2)$3 are matrix units and $SU_q(2)$4 is a normalization constant. The modular automorphism, modular element, and squared antipode action are all expressed in the explicit matrix framework.

Duality and the Appearance of $SU_q(2)$5

The dual of $SU_q(2)$6 in the algebraic quantum group sense is concretely identified. The dual algebra $SU_q(2)$7 is formed as the direct sum of dual spaces to the matrix blocks $SU_q(2)$8, and the identification proceeds via evaluation of functionals on products of irreducible representations.

The pairs $SU_q(2)$9 corresponding to the fundamental $su_q(2)$0 matrix elements are shown to satisfy the defining relations for the compact quantum group $su_q(2)$1; coproducts, counits, and antipodes are matched, and a $su_q(2)$2-isomorphism with the $su_q(2)$3 algebra is constructed explicitly. The dual Haar weight is computed, and its modular automorphism group is described at the level of $su_q(2)$4.

A surjectivity result is established for the canonical map from the operator algebraic presentation to the Woronowicz $su_q(2)$5 algebra, with injectivity conjectured but left open.

Strong Claims and Technical Results

• The paper provides an explicit, operator-algebraic construction of the discrete quantum group $su_q(2)$6 from the representation theory of the deformed enveloping algebra, filling a gap in the literature where most treatments exist only at the Hopf algebra or purely algebraic level.
• It is shown that the dual, in the precise algebraic quantum group sense, is \textbf{exactly} the compact quantum group $su_q(2)$7 as in Woronowicz's theory, not just as a formal object but via an explicit $su_q(2)$8-homomorphic realization.
• The fully rigorous operator-algebraic extension of quantum group data (coproduct, antipode, integrals, cointegrals) from the algebraic representations is constructed in detail, clarifying technical issues often glossed over in the existing physics and quantum algebra literature.

Implications and Future Directions

The explicit operator algebraic realization of $su_q(2)$9 as a discrete quantum group and its duality with $su_q(2)$0 provides a template for analogous constructions for other quantum groups and their duals, with rigorous operator-theoretic control over all quantum group data.

Potential directions include:

• Extension to non-semisimple quantum groups or further classes in the multiplier Hopf algebra framework.
• Analytic completion to $su_q(2)$1- or von Neumann algebraic quantum groups, enabling applications in quantum probability, noncommutative geometry, and mathematical physics.
• Deeper exploration of the role of the modular structures, scaling group, and polar decomposition of the antipode in harmonic analysis on quantum groups.
• Investigation into the structure and properties of possible quantum subgroups or quantum subfactors, and consequences for representation theory and tensor categories.

The explicit representation-theoretic approach underlines the need for careful analytic and operator-algebraic foundations in quantum group theory, particularly relevant for applications beyond the purely algebraic context.

Conclusion

This paper establishes a comprehensive and explicit construction of the discrete quantum group $su_q(2)$2 from its Hopf $su_q(2)$3-algebra deformation, together with a rigorous derivation of its dual as the compact quantum group $su_q(2)$4. The operator algebraic framework and thorough representation-theoretic analysis provide a firm foundation for duality theory in the context of discrete and compact quantum groups. The results clarify technical subtleties in analytic extensions, antipode and integral structures, and lay groundwork for further investigations into more general and physically relevant quantum group settings.