[Paper Review] q-deformed algebras $U_q(so_n)$ and their representations

A. M. Gavrilik, N. Z. Iorgov|ArXiv.org|Sep 24, 1997

Algebraic structures and combinatorial models7 references3 citations

TL;DR

This paper constructs finite-dimensional irreducible representations of the nonstandard q-deformed algebra $U_q(so_n)$ using a q-analogue of the Gel'fand-Tsetlin basis, proving that representation operators satisfy the defining trilinear relations for $q$ not a root of unity. The key contribution is the explicit realization of q-deformed representations with highest weights identical to those of the classical $so(n)$ algebra, demonstrating the formalism's suitability for nonstandard q-deformations.

ABSTRACT

For the nonstandard $q$-deformed algebras $U_q(so_n)$, defined recently in terms of trilinear relations for generating elements, most general finite dimensional irreducible representations directly corresponding to those of nondeformed algebras $so(n)$ (i.e., characterized by the same sets of only integers or only half-integers as in highest weights of the latter) are given explicitly in a $q$-analogue of Gel'fand-Tsetlin basis. Detailed proof, for $q$ not equal to a root of unity, that representation operators indeed satisfy relevant (trilinear) relations and define finite dimensional irreducible representations is presented. The results show perfect suitability of the Gel'fand-Tsetlin formalism concerning (nonstandard) $q$-deformation of $so(n)$.

Motivation & Objective

• To construct finite-dimensional irreducible representations of the nonstandard q-deformed algebra $U_q(so_n)$.
• To establish a q-analogue of the Gel'fand-Tsetlin basis for $U_q(so_n)$, mirroring the structure of classical $so(n)$ representations.
• To prove that the constructed representation operators satisfy the defining trilinear relations of $U_q(so_n)$ when $q$ is not a root of unity.
• To demonstrate that the highest weights of these representations are characterized by the same sets of integers or half-integers as in the nondeformed $so(n)$ case.
• To validate the suitability of the Gel'fand-Tsetlin formalism for nonstandard q-deformations of $so(n)$.

Proposed method

• The authors define $U_q(so_n)$ via trilinear relations among its generating elements, following a nonstandard deformation scheme.
• They construct representations using a q-analogue of the Gel'fand-Tsetlin basis, extending the classical basis to the quantum setting.
• Representation operators are explicitly defined in terms of q-deformed raising and lowering operators acting on basis vectors.
• The proof of closure under the trilinear relations is carried out for $q$ not equal to a root of unity, ensuring non-degeneracy.
• The weight systems of the representations are shown to match those of the classical $so(n)$ algebra, with highest weights specified by integers or half-integers.
• The construction is verified to yield finite-dimensional irreducible representations by checking the defining algebraic relations and weight space structure.

Experimental results

Research questions

• RQ1 Can finite-dimensional irreducible representations of the nonstandard q-deformed algebra $U_q(so_n)$ be explicitly constructed using a q-analogue of the Gel'fand-Tsetlin basis?
• RQ2 Do the highest weights of these q-deformed representations correspond exactly to the same integer or half-integer labels as in the classical $so(n)$ algebra?
• RQ3 Do the representation operators of $U_q(so_n)$ satisfy the defining trilinear relations of the algebra for generic $q$?
• RQ4 Is the Gel'fand-Tsetlin formalism applicable and suitable for nonstandard q-deformations of $so(n)$?
• RQ5 What is the precise structure of the weight spaces and the action of generators in the q-deformed setting?

Key findings

• The paper explicitly constructs finite-dimensional irreducible representations of $U_q(so_n)$ using a q-analogue of the Gel'fand-Tsetlin basis.
• The highest weights of these representations are characterized by the same sets of integers or half-integers as in the classical $so(n)$ algebra.
• Representation operators satisfy the defining trilinear relations of $U_q(so_n)$ for $q$ not equal to a root of unity.
• The construction confirms the formal suitability of the Gel'fand-Tsetlin formalism for nonstandard q-deformations of $so(n)$.
• The weight space decomposition and action of generators are fully determined and consistent with the classical case in the $q \to 1$ limit.
• The results are valid for generic $q$, with the proof relying on algebraic verification of relations in the representation space.