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Finite dimensional representations of the quantum group $GL_{p,q}(2)$ using the exponential map from $U_{p,q}(gl(2))$
High Energy Physics - Theory · Submitted 28 Nov 1994
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra $U_{p,q}(gl(2))$ to the quantum group $GL_{p,q}(2)$, we show how the $(2j+1)$-dimensional representations of $GL_{p,q}(2)$ can be obtained by `exponentiating' the well-known $(2j+1)$-dimensional representations of $U_{p,q}(gl(2))$ for $j$ $=$ $1,{3/2},...
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Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra $U_{p,q}(gl(2))$ to the quantum group $GL_{p,q}(2)$, we show how the $(2j+1)$-dimensional representations of $GL_{p,q}(2)$ can be obtained by `exponentiating' the well-known $(2j+1)$-dimensional representations of $U_{p,q}(gl(2))$ for $j$ $=$ $1,{3/2},... $; $j$ $=$ 1/2 corresponds to the defining 2-dimensional $T$-matrix. The earlier results on the finite-dimensional representations of $GL_q(2)$ and $SL_q(2)$ (or $SU_q(2)$) are obtained when $p$ $=$ $q$. Representations of $U_{\bar{q},q}(2)$ $(q$ $\in$ $\C \backslash \R$ and $U_q(2)$ $(q$ $\in$ $\R \backslash \{0\})$ are also considered. The structure of the Clebsch-Gordan matrix for $U_{p,q}(gl(2))$ is studied. The same Clebsch-Gordan coefficients are applicable in the reduction of the direct product representations of the quantum group $GL_{p,q}(2)$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra $U_{p,q}(gl(2))$ to the quantum group $GL_{p,q}(2)$, we show how the $(2j+1)$-dimensional representations of $GL_{p,q}(2)$ can be obtained by `exponentiating' the well-known $(2j+1)$-dimensional representations of $U_{p,q}(gl(2))$ for $j$ $=$ $1,{3/2},... | contribution 0.81 | 0.93 | 0.84 | asserted 1.00 |
| s1 | $; $j$ $=$ 1/2 corresponds to the defining 2-dimensional $T$-matrix. | result 0.78 | 0.54 | 0.08 | asserted 1.00 |
| s2 | The earlier results on the finite-dimensional representations of $GL_q(2)$ and $SL_q(2)$ (or $SU_q(2)$) are obtained when $p$ $=$ $q$. | interpretation 0.47 | 0.26 | 0.08 | asserted 1.00 |
| s3 | Representations of $U_{\bar{q},q}(2)$ $(q$ $\in$ $\C \backslash \R$ and $U_q(2)$ $(q$ $\in$ $\R \backslash \{0\})$ are also considered. | contribution 0.92 | 0.75 | 0.04 | asserted 0.98 |
| s4 | The structure of the Clebsch-Gordan matrix for $U_{p,q}(gl(2))$ is studied. | contribution 0.99 | 0.84 | 0.07 | asserted 0.94 |
| s5 | The same Clebsch-Gordan coefficients are applicable in the reduction of the direct product representations of the quantum group $GL_{p,q}(2)$. | result 0.92 | 0.75 | 0.15 | asserted 1.00 |
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Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra $U_{p,q}(gl(2))$ to the quantum group $GL_{p,q}(2)$, we show how the $(2j+1)$-dimensional representations of $GL_{p,q}(2)$ can be obtained by `exponentiating' the well-known $(2j+1)$-dimensional representations of $U_{p,q}(gl(2))$ for $j$ $=$ $1,{3/2},... $; $j$ $=$ 1/2 corresponds to the defining 2-dimensional $T$-matrix. The earlier results on the finite-dimensional representations of $GL_q(2)$ and $SL_q(2)$ (or $SU_q(2)$) are obtained when $p$ $=$ $q$. Representations of $U_{\bar{q},q}(2)$ $(q$ $\in$ $\C \backslash \R$ and $U_q(2)$ $(q$ $\in$ $\R \backslash \{0\})$ are also considered. The structure of the Clebsch-Gordan matrix for $U_{p,q}(gl(2))$ is studied. The same Clebsch-Gordan coefficients are applicable in the reduction of the direct product representations of the quantum group $GL_{p,q}(2)$.
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3 arXiv papers cite this in 3 sentences
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It can be regarded as the canonical object that condenses the so-called quantum duality principle [ref] and, since then, the universal $T$-matrix has been constructed for a number of quantum algebras and groups, thereby establishing a comprehensive and explicit framework for Hopf algebra duality, with applications in the construction of $q$-coherent states, the derivation of quantum universal $R$-matrices, as well as in quantum gravity and representation theory (see [ref] and references therein).
Universal $T$-matrices for quantum Poincar\'e groups: contractions and quantum reference frames · Introduction · 0.96
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