dorsal/arxiv
View SchemaDuals of coloured quantum universal enveloping algebras and coloured universal $\cal T$-matrices
| Authors | C. Quesne |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9711012 |
| URL | https://arxiv.org/abs/q-alg/9711012 |
| DOI | 10.1063/1.532378 |
| Journal | J. Math. Phys. 39 (1998) 1199-1222 |
Abstract
We extend the notion of dually conjugate Hopf (super)algebras to the coloured Hopf (super)algebras ${\cal H}^c$ that we recently introduced. We show that if the standard Hopf (super)algebras ${\cal H}_q$ that are the building blocks of ${\cal H}^c$ have Hopf duals ${\cal H}_q^*$, then the latter may be used to construct coloured Hopf duals ${\cal H}^{c*}$, endowed with coloured algebra and antipode maps, but with a standard coalgebraic structure. Next, we review the case where the ${\cal H}_q$'s are quantum universal enveloping algebras of Lie (super)algebras $U_q(g)$, so that the corresponding ${\cal H}_q^*$'s are quantum (super)groups $G_q$. We extend the Fronsdal and Galindo universal ${\cal T}$-matrix formalism to the coloured pairs $(U^c(g), G^c)$ by defining coloured universal ${\cal T}$-matrices. We then show that together with the coloured universal $\cal R$-matrices previously introduced, the latter provide an algebraic formulation of the coloured RTT-relations, proposed by Basu-Mallick. This establishes a link between the coloured extensions of Drinfeld-Jimbo and Faddeev-Reshetikhin-Takhtajan pictures of quantum groups and quantum algebras. Finally, we illustrate the construction of coloured pairs by giving some explicit results for the two-parameter deformations of $\bigl(U(gl(2)), Gl(2)\bigr)$, and $\bigl(U(gl(1/1)), Gl(1/1)\bigr)$.
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"abstract": "We extend the notion of dually conjugate Hopf (super)algebras to the coloured\nHopf (super)algebras ${\\cal H}^c$ that we recently introduced. We show that if\nthe standard Hopf (super)algebras ${\\cal H}_q$ that are the building blocks of\n${\\cal H}^c$ have Hopf duals ${\\cal H}_q^*$, then the latter may be used to\nconstruct coloured Hopf duals ${\\cal H}^{c*}$, endowed with coloured algebra\nand antipode maps, but with a standard coalgebraic structure. Next, we review\nthe case where the ${\\cal H}_q$\u0027s are quantum universal enveloping algebras of\nLie (super)algebras $U_q(g)$, so that the corresponding ${\\cal H}_q^*$\u0027s are\nquantum (super)groups $G_q$. We extend the Fronsdal and Galindo universal\n${\\cal T}$-matrix formalism to the coloured pairs $(U^c(g), G^c)$ by defining\ncoloured universal ${\\cal T}$-matrices. We then show that together with the\ncoloured universal $\\cal R$-matrices previously introduced, the latter provide\nan algebraic formulation of the coloured RTT-relations, proposed by\nBasu-Mallick. This establishes a link between the coloured extensions of\nDrinfeld-Jimbo and Faddeev-Reshetikhin-Takhtajan pictures of quantum groups and\nquantum algebras. Finally, we illustrate the construction of coloured pairs by\ngiving some explicit results for the two-parameter deformations of\n$\\bigl(U(gl(2)), Gl(2)\\bigr)$, and $\\bigl(U(gl(1/1)), Gl(1/1)\\bigr)$.",
"arxiv_id": "q-alg/9711012",
"authors": [
"C. Quesne"
],
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"q-alg",
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"doi": "10.1063/1.532378",
"journal_ref": "J. Math. Phys. 39 (1998) 1199-1222",
"title": "Duals of coloured quantum universal enveloping algebras and coloured universal $\\cal T$-matrices",
"url": "https://arxiv.org/abs/q-alg/9711012"
},
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