Quantum Analogs of Tensor Product Representations of su(1,1)

Abstract: We study representations of $U_q(su(1,1))$ that can be considered as quantum analogs of tensor products of irreducible *-representations of the Lie algebra $su(1,1)$. We determine the decomposition of these representations into irreducible *-representations of $U_q(su(1,1))$ by diagonalizing the action of the Casimir operator on suitable subspaces of the representation spaces. This leads to an interpretation of the big $q$-Jacobi polynomials and big $q$-Jacobi functions as quantum analogs of Clebsch-Gordan coefficients.

• The paper introduces quantum analogs of tensor product representations of su(1,1) by employing the Casimir operator to guide the decomposition.
• It utilizes orthogonal polynomial theory to analyze both continuous and discrete spectra in representations, highlighting unique series such as the principal and strange series.
• The study shows that modifying the tensor product through big q-Jacobi functions yields a complete quantum analog that mirrors classical decomposition methods.

Quantum Analogs of Tensor Product Representations of $su(1,1)$

The paper explores the concept of quantum analogs for tensor product representations associated with the Lie algebra $su(1,1)$. It investigates the representations of the quantum algebra $_q(su(1,1))$, identifying their decompositions through the action of the Casimir operator and interpreting mathematical entities such as big $q$-Jacobi polynomials as quantum analogs of classical Clebsch–Gordan coefficients. This study expands the understanding of quantum algebra representations and their relation to orthogonal polynomials.

Quantum Algebra $_q(su(1,1))$

The quantized universal enveloping algebra $_q(su(1,1))$ is central to the paper and consists of generators $K$, $K^{-1}$, $E$, and $F$, with defining relations. It supports five irreducible $*$-representations series: positive discrete $\pi^+_k$, negative discrete $\pi^-_k$, principal unitary $\pi^{\mathrm{P}}_{\rho,\rho'}$, complementary $\pi^{\mathrm{C}}_{\lambda,\lambda'}$, and strange $\pi^{\mathrm{S}}_{\sigma,\sigma'}$, each having unique spectral properties tied to the Casimir operator $\Omega$.

The structure and decomposition of these representations are pivotal as they relate to how quantum analogs of tensor products behave.

Decomposition of Representations

Negative and Positive Discrete Series

The decomposition of quantum analog representations, specifically $\mathcal{T} = (\pi^- \otimes \pi^+)\Delta$, involves orthogonal polynomial theory, concluding that big $q$-Jacobi functions serve as quantum analogs for Clebsch–Gordan coefficients. These provide the 'building blocks' for expanding irreducible representation decomposition, given by:

1. Continuous spectrum characterized by the principal unitary series.
2. Discrete spectrum including the strange series and potentially complementary series, depending upon the representation parameters such as $k_1$ and $k_2$.

Tensor Product Completion

Additional representation $\mathcal{T}'$ modifies $\mathcal{T}$ to be a complete tensor product. These are interpreted using big $q$-Jacobi polynomials, confirming their role in quantum representations without classical analogs. This aligns more closely with tensor product behaviors featuring unique $q$-dependent coefficients.

Principal Unitary Series

Considering two principal unitary series, this paper extends the decomposition into vector-valued big $q$-Jacobi functions. Quantum analog representations use these functions to map closely to their classical counterparts through decomposition showing both continuous spectra (principal series) and discrete cases (strange series), highlighting how orthogonality relations transfer between classical and quantum paradigms.

Conclusion

The quantum analogs explored in $_q(su(1,1))$ manifest in complex mathematical structures like the big $q$-Jacobi polynomials and functions, significantly influencing decompositions and the understanding of tensor products in quantum algebra. These insights demonstrate the adaptability of classical mathematical methods to quantum contexts, laying the groundwork for deeper inquiries into categories of quantum group theory and representations.