{"id":{"repo_id":"rgu","oai_identifier":"oai:rgu-repository.worktribe.com:2807515"},"canonical_url":"https://search.dev.ndltd.org/etd/rgu/oai:rgu-repository.worktribe.com:2807515","repository":{"repo_id":"rgu","name":"Robert Gordon University","base_url":"https://rgu-repository.worktribe.com/oaiprovider"},"display":{"title":"Multiparameter quantum groups: contractions and coloured generalisations.","abstract":"This thesis is devoted to various algebraic as well as differential geometric aspects of multi- parameter quantum groups. In particular, we study some two-parameter quantum groups and associated structure of Hopf algebras. We focus our attention on a new quantum group, Gr,s, depending on two deformation parameters and five generators. The first four generators of this Hopf algebra form a Hopf subalgebra which coincides exactly with the single-parameter dependent GLq(2)quantum group when q = r-1. However, the two-parameter dependent GLp,q(2) can also be realised through the generators of the Gr,s Hopf algebra provided the sets of deformation parameters p,q and r,s are related to each other in a particular fashion. This new algebra can, therefore, be used to realise both GLq(2) and GLp,q(2) quantum groups. Alternatively, the Gr,s structure can be considered as a two-parameter quantisation of the classical GL(2) x GL(l) group. As a starting point for further investigation of the Gr,s structure, we give an explicit description of its dual algebra and show that it is isomorphic to a single-parameter quantum deformation of U(gl(2)) x U(u(1)). We also formulate a differential calculus on Gr,s which provides a realisation of the calculus on GLp,q(2). We then look at the contraction procedure, which is used to obtain Jordanian deformations from the well-known quantum deformations. The quantum group GLp,q(2) is known to be related to the Jordanian GLh,h'(2) via the contraction procedure. We contract the Gr,s quantum group to obtain its Jordanian analogue Gm,k, which provides a realisation of GLh,h'(2) in a manner similar to the q-deformed case. The contraction procedure is also employed to obtain the inhomogeneous Jordanian multiparameter quantum groups from their 1-deformed counterparts. The scheme is then set in the wider context of the coloured extensions of these deformations, namely, the so-called ‘coloured’ quantum groups. The contraction procedure is generalised to the coloured quantum group GL lambda, mu over r(2) to yield a new Jordanian quantum group GL lambda, mu over m(2). Both Gr,s and Gm,k are then extended to their coloured versions which in turn provide realisations of GL lambda, mu over r(2) and GL lambda, mu over m(2). Finally, the duality between coloured quantum groups and coloured quantum algebras is explored. The algebra dual to the coloured quantum group GL lambda, mu over r(2) is derived explicitly. We also present a differential calculus on GL lambda, mu over r(2) by giving a coloured generalisation of the R-matrix approach. The results obtained in the thesis are believed to be of significance in the further study of coloured quantum groups as well as in various mathematical and physical aspects of quantum groups.","abstract_html":"This thesis is devoted to various algebraic as well as differential geometric aspects of multi- parameter quantum groups. In particular, we study some two-parameter quantum groups and associated structure of Hopf algebras. We focus our attention on a new quantum group, Gr,s, depending on two deformation parameters and five generators. The first four generators of this Hopf algebra form a Hopf subalgebra which coincides exactly with the single-parameter dependent GLq(2)quantum group when q = r-1. However, the two-parameter dependent GLp,q(2) can also be realised through the generators of the Gr,s Hopf algebra provided the sets of deformation parameters p,q and r,s are related to each other in a particular fashion. This new algebra can, therefore, be used to realise both GLq(2) and GLp,q(2) quantum groups. Alternatively, the Gr,s structure can be considered as a two-parameter quantisation of the classical GL(2) x GL(l) group. As a starting point for further investigation of the Gr,s structure, we give an explicit description of its dual algebra and show that it is isomorphic to a single-parameter quantum deformation of U(gl(2)) x U(u(1)). We also formulate a differential calculus on Gr,s which provides a realisation of the calculus on GLp,q(2). We then look at the contraction procedure, which is used to obtain Jordanian deformations from the well-known quantum deformations. The quantum group GLp,q(2) is known to be related to the Jordanian GLh,h'(2) via the contraction procedure. We contract the Gr,s quantum group to obtain its Jordanian analogue Gm,k, which provides a realisation of GLh,h'(2) in a manner similar to the q-deformed case. The contraction procedure is also employed to obtain the inhomogeneous Jordanian multiparameter quantum groups from their 1-deformed counterparts. The scheme is then set in the wider context of the coloured extensions of these deformations, namely, the so-called ‘coloured’ quantum groups. The contraction procedure is generalised to the coloured quantum group GL lambda, mu over r(2) to yield a new Jordanian quantum group GL lambda, mu over m(2). Both Gr,s and Gm,k are then extended to their coloured versions which in turn provide realisations of GL lambda, mu over r(2) and GL lambda, mu over m(2). Finally, the duality between coloured quantum groups and coloured quantum algebras is explored. The algebra dual to the coloured quantum group GL lambda, mu over r(2) is derived explicitly. We also present a differential calculus on GL lambda, mu over r(2) by giving a coloured generalisation of the R-matrix approach. The results obtained in the thesis are believed to be of significance in the further study of coloured quantum groups as well as in various mathematical and physical aspects of quantum groups.","abstract_has_math":false,"creators":["Parashar, Deepak"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["A. Solomon"],"committee_chairs":[],"committee_members":[],"year":2000,"date_issued":"2000","date_published":"2000","updated_at":"2026-07-24T04:10:12Z","subjects":["Multi-parameter quantum groups","Hopf algebras","Quantum deformations","Differential calculus on quantum groups","Jordanian deformations","Coloured quantum groups"],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["oai:rgu-repository.worktribe.com:2807515","https://doi.org/10.48526/rgu-wt-2807515"],"render_values":[{"text":"oai:rgu-repository.worktribe.com:2807515","href":null,"code":true},{"text":"https://doi.org/10.48526/rgu-wt-2807515","href":"https://doi.org/10.48526/rgu-wt-2807515","code":true}]}]},"links":{"outbound_url":"https://rgu-repository.worktribe.com/2807515/1/PARASHAR%202000%20Multiparameter%20quantum%20groups","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["A. 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In particular, we study some two-parameter quantum groups and associated structure of Hopf algebras. We focus our attention on a new quantum group, Gr,s, depending on two deformation parameters and five generators. The first four generators of this Hopf algebra form a Hopf subalgebra which coincides exactly with the single-parameter dependent GLq(2)quantum group when q = r-1. However, the two-parameter dependent GLp,q(2) can also be realised through the generators of the Gr,s Hopf algebra provided the sets of deformation parameters p,q and r,s are related to each other in a particular fashion. This new algebra can, therefore, be used to realise both GLq(2) and GLp,q(2) quantum groups. Alternatively, the Gr,s structure can be considered as a two-parameter quantisation of the classical GL(2) x GL(l) group. As a starting point for further investigation of the Gr,s structure, we give an explicit description of its dual algebra and show that it is isomorphic to a single-parameter quantum deformation of U(gl(2)) x U(u(1)). We also formulate a differential calculus on Gr,s which provides a realisation of the calculus on GLp,q(2). We then look at the contraction procedure, which is used to obtain Jordanian deformations from the well-known quantum deformations. The quantum group GLp,q(2) is known to be related to the Jordanian GLh,h'(2) via the contraction procedure. We contract the Gr,s quantum group to obtain its Jordanian analogue Gm,k, which provides a realisation of GLh,h'(2) in a manner similar to the q-deformed case. The contraction procedure is also employed to obtain the inhomogeneous Jordanian multiparameter quantum groups from their 1-deformed counterparts. The scheme is then set in the wider context of the coloured extensions of these deformations, namely, the so-called ‘coloured’ quantum groups. The contraction procedure is generalised to the coloured quantum group GL lambda, mu over r(2) to yield a new Jordanian quantum group GL lambda, mu over m(2). Both Gr,s and Gm,k are then extended to their coloured versions which in turn provide realisations of GL lambda, mu over r(2) and GL lambda, mu over m(2). Finally, the duality between coloured quantum groups and coloured quantum algebras is explored. The algebra dual to the coloured quantum group GL lambda, mu over r(2) is derived explicitly. We also present a differential calculus on GL lambda, mu over r(2) by giving a coloured generalisation of the R-matrix approach. The results obtained in the thesis are believed to be of significance in the further study of coloured quantum groups as well as in various mathematical and physical aspects of quantum groups."]},{"key":"dc:title","label":"Title","values":["Multiparameter quantum groups: contractions and coloured generalisations."]}]}],"canonical_facts":{"dc:contributor.advisor":["A. Solomon"],"dc:contributor.sponsor":["RGU Internal Funding","Association of Commonwealth Universities","Japan Society for the Promotion of Science"],"dc:creator":["Parashar, Deepak"],"dc:date":["2000-07-31"],"dc:date.issued":["2000"],"dc:description.abstract":["This thesis is devoted to various algebraic as well as differential geometric aspects of multi- parameter quantum groups. In particular, we study some two-parameter quantum groups and associated structure of Hopf algebras. We focus our attention on a new quantum group, Gr,s, depending on two deformation parameters and five generators. The first four generators of this Hopf algebra form a Hopf subalgebra which coincides exactly with the single-parameter dependent GLq(2)quantum group when q = r-1. However, the two-parameter dependent GLp,q(2) can also be realised through the generators of the Gr,s Hopf algebra provided the sets of deformation parameters p,q and r,s are related to each other in a particular fashion. This new algebra can, therefore, be used to realise both GLq(2) and GLp,q(2) quantum groups. Alternatively, the Gr,s structure can be considered as a two-parameter quantisation of the classical GL(2) x GL(l) group. As a starting point for further investigation of the Gr,s structure, we give an explicit description of its dual algebra and show that it is isomorphic to a single-parameter quantum deformation of U(gl(2)) x U(u(1)). We also formulate a differential calculus on Gr,s which provides a realisation of the calculus on GLp,q(2). We then look at the contraction procedure, which is used to obtain Jordanian deformations from the well-known quantum deformations. The quantum group GLp,q(2) is known to be related to the Jordanian GLh,h'(2) via the contraction procedure. We contract the Gr,s quantum group to obtain its Jordanian analogue Gm,k, which provides a realisation of GLh,h'(2) in a manner similar to the q-deformed case. The contraction procedure is also employed to obtain the inhomogeneous Jordanian multiparameter quantum groups from their 1-deformed counterparts. The scheme is then set in the wider context of the coloured extensions of these deformations, namely, the so-called ‘coloured’ quantum groups. The contraction procedure is generalised to the coloured quantum group GL lambda, mu over r(2) to yield a new Jordanian quantum group GL lambda, mu over m(2). Both Gr,s and Gm,k are then extended to their coloured versions which in turn provide realisations of GL lambda, mu over r(2) and GL lambda, mu over m(2). Finally, the duality between coloured quantum groups and coloured quantum algebras is explored. The algebra dual to the coloured quantum group GL lambda, mu over r(2) is derived explicitly. We also present a differential calculus on GL lambda, mu over r(2) by giving a coloured generalisation of the R-matrix approach. The results obtained in the thesis are believed to be of significance in the further study of coloured quantum groups as well as in various mathematical and physical aspects of quantum groups."],"dc:identifier":["oai:rgu-repository.worktribe.com:2807515","https://doi.org/10.48526/rgu-wt-2807515"],"dc:identifier.uri":["https://rgu-repository.worktribe.com/2807515/1/PARASHAR%202000%20Multiparameter%20quantum%20groups"],"dc:language":["en"],"dc:relation.isreferencedby":["https://rgu-repository.worktribe.com/output/2807515"],"dc:subject":["Multi-parameter quantum groups","Hopf algebras","Quantum deformations","Differential calculus on quantum groups","Jordanian deformations","Coloured quantum groups"],"dc:title":["Multiparameter quantum groups: contractions and coloured generalisations."],"dc:type":["Thesis"]},"updated_at":"2026-07-24T04:10:12Z"}