On combined standard–nonstandard or hybrid (q,h)-deformations

Aneva, Boyka; Arnaudon, D.; Dobrev, V. K.; Mihov, Stoyan

doi:10.1063/1.1343881

Cited by 16 publications

(24 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The procedure is generalizable to any other Lie algebra with one or other deformation, but also to quantum algebras with hybrid deformations, i.e, quantum algebras combining both kinds of deformations [9,10].…”

Section: Discussionmentioning

confidence: 99%

Approximate symmetries and quantum algebras

Celeghini

Olmo

2003

Europhys. Lett.

View full text Add to dashboard Cite

In Quantum Mechanics operators must be hermitian and, in a direct product space, symmetric. These properties are saved by Lie algebra operators but not by those of quantum algebras. A possible correspondence between observables and quantum algebra operators is suggested by extending the definition of matrix elements of a physical observable, including the eventual projection on the appropriate symmetric space. This allows to build in the Lie space of representations one-parameter families of operators belonging to the enveloping Lie algebra that satisfy an approximate symmetry and have the properties required by physics.MSC: 81R50, 81R40, 17B37

show abstract

Section: Discussionmentioning

confidence: 99%

Approximate symmetries and quantum algebras

Celeghini

Olmo

2003

Europhys. Lett.

View full text Add to dashboard Cite

show abstract

“…Studying the classification of [8] we noticed altogether five nonsingular such R-matrices. The triangular ones were introduced in [7] and their duals were found and studied in detail in [9]. In the latter paper we called these bialgebras exotic.…”

Section: Introductionmentioning

confidence: 99%

“…For the supergroup GL(1|1) there are also two: the standard GL pq (1|1) [3][4][5] and the hybrid (standard-nonstandard) GL qh (1|1) [6] two-parameter deformations. Recently, in [7] it was shown that there are no more deformations of GL (2) or GL(1|1). In particular, it was shown that these four deformations match the distinct triangular 4 × 4 R-matrices from the classification of [8] which are deformations of the trivial R-matrix (corresponding to undeformed GL(2)).…”

Section: Introductionmentioning

confidence: 99%

Duality and representations for new exotic bialgebras

Arnaudon

Dobrev

Mihov

2002

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

We find the exotic matrix bialgebras which correspond to the two nontriangular nonsingular 4 × 4 R-matrices in the classification of Hietarinta, namely, R S0,3 and R S1,4 . We find two new exotic bialgebras S03 and S14 which are not deformations of the of the classical algebras of functions on GL(2) or GL(1|1). With this we finalize the classification of the matrix bialgebras which unital associative algebras generated by four elements. We also find the corresponding dual bialgebras of these new exotic bialgebras and study their representation theory in detail. We also discuss in detail a special case of R S1,4 in which the corresponding algebra turns out to be a special case of the two-parameter quantum group deformation GL p,q (2).

show abstract

“…In the classification of [8] there are altogether five nonsingular such R-matrices. The three triangular ones were introduced in [7] and their duals were found and studied in detail in [9]. The study of the two non-triangular cases was started in [10].…”

Section: Introductionmentioning

confidence: 99%

“…The list, of course, includes the four cases which are deformations of classical ones: two two-parameter deformations of each of GL (2) and GL(1|1), namely, the standard GL pq (2) [1], nonstandard (Jordanian) GL gh (2) [2], the standard GL pq (1|1) [3][4][5] and the hybrid (standardnonstandard) GL qh (1|1) [6]. (Later, in [7] it was shown that there are no more deformations of GL (2) or GL(1|1).) The list includes also five exotic cases which are not deformations of the classical algebra of functions over the group GL(2) or the supergroup GL(1|1).…”

Section: Introductionmentioning

confidence: 99%

Spectral Decomposition and Baxterization of Exotic Bialgebras and Associated Noncommutative Geometries

Arnaudon

Chakrabarti

Dobrev

et al. 2003

Int. J. Mod. Phys. A

Self Cite

View full text Add to dashboard Cite

We study the geometric aspects of two exotic bialgebras S03 and S14 introduced in math.QA/0206053. These bialgebras are obtained by the FaddeevReshetikhin-Takhtajan RTT prescription with non-triangular R-matrices which are denoted R 03 and R 14 in the classification of Hietarinta, and they are not deformations of either GL(2) or GL(1/1). We give the spectral decomposition which involves two, resp., three, projectors. These projectors are then used to provide the Baxterisation procedure with one, resp., two, parameters. Further, the projectors are used to construct the noncommutative planes together with the corresponding differentials following the Wess-Zumino prescription. In all these constructions there appear non-standard features which are noted. Such features show the importance of systematic study of all bialgebras of four generators.

show abstract

On combined standard–nonstandard or hybrid (q,h)-deformations

Cited by 16 publications

References 20 publications

Approximate symmetries and quantum algebras

Approximate symmetries and quantum algebras

Duality and representations for new exotic bialgebras

Spectral Decomposition and Baxterization of Exotic Bialgebras and Associated Noncommutative Geometries

Contact Info

Product

Resources

About