Spectral Decomposition and Baxterisation of Exotic Bialgebras and Associated Noncommutative Geometries

We study the geometric aspects of two exotic bialgebras S03 and S14 introduced in math.QA/0206053. These bialgebras are obtained by the Faddeev-Reshetikhin-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.

Note:

14 pages, LaTeX2e, using packages: cite, amsfonts, amsmath, subeqn/ v2: small improvements, coincides with journal version

References(22)

Figures(0)

[1]

Yu.I. Manin, E.E Mukhin and D.V. Zhdanovich, Non-standard quantum deformations of GL(n) and constant solutions of the Yang-Baxter equation, Progress of Theor. Phys. Suppl. 102 (1990) 203-218

E.E. Demidov

[2]

The two parametric extension of h deformation of GL(2) and the differential calculus on its quantum plane

Amir Aghamohammadi
(
- IPM, Tehran and
- Azzahra U.
)

- Mod.Phys.Lett.A 8 (1993) 2607-2613
•
e-Print:
- hep-th/9306071
•
DOI:
- 10.1142/S0217732393002981

[3]

A Two parameter deformation of the SU(1/1) superalgebra and the XY quantum chain in a magnetic field

Haye Hinrichsen
(
- Bonn U.
)
,
Vladimir Rittenberg
(
- CERN
)

- Phys.Lett.B 275 (1992) 350-354
•
e-Print:
- hep-th/9110074
•
DOI:
- 10.1016/0370-2693(92)91601-5

[4]

Two parameter quantum deformation of GL(1|1)

L. Dabrowski
,
Lu- Yu Wang
(
- ICTP, Trieste
)

- Phys.Lett.B 266 (1991) 51-54
•
DOI:
- 10.1016/0370-2693(91)90742-9

[5]

The two parameter deformation of the supergroup GL (1|1), its differential calculus and its lie algebra

C. Burdik
(
- Charles U.
)
,
R. Tomasek
(
- Charles U.
)

- Lett.Math.Phys. 26 (1992) 97-104
•
DOI:
- 10.1007/BF00398806

[6]

Classification of the quantum deformation of the superalgebra gl(1|1)

- J.Phys.A 31 (1998) 4049-4072
•
e-Print:
- q-alg/9705024
•
DOI:
- 10.1088/0305-4470/31/17/014

[7]

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

B.L. Aneva
,
D. Arnaudon
,
A. Chakrabarti
,
V.K. Dobrev
,
S.G. Mihov

- J.Math.Phys. 42 (2001) 1236-1249

[7]

e-Print:
- math/0006206

[8]

Solving the two-dimensional constant quantum Yang-Baxter equation

Jarmo Hietarinta
(
- Turku U.
)

- J.Math.Phys. 34 (1993) 1725-1756
•
DOI:
- 10.1063/1.530185

[9]

Duality for exotic bialgebras

D. Arnaudon
(
- Annecy, LAPTH
)
,
A. Chakrabarti
(
- Ecole Polytechnique, CPHT
)
,
V.K. Dobrev
(
- Northumbria U. and
- Sofiya, Inst. Nucl. Res.
)
,
S.G. Mihov
(
- Sofiya, Inst. Nucl. Res.
)

- J.Phys.A 34 (2001) 4065-4082
•
e-Print:
- math/0101160
•
DOI:
- 10.1088/0305-4470/34/19/308

[10]

Duality and representations for new exotic bialgebras, J. Math Phys., to appear

D. Arnaudon
,
A. Chakrabarti
,
V.K. Dobrev
,
S.G. Mihov

[10]

Duality and Representations for New Exotic Bialgebras

D. Arnaudon
(
- Annecy, LAPTH
)
,
A. Chakrabarti
(
- Ecole Polytechnique, CPHT
)
,
V.K. Dobrev
(
- Sofiya, Inst. Nucl. Res. and
- ICTP, Trieste
)
,
S.G. Mihov
(
- Sofiya, Inst. Nucl. Res.
)

- J.Math.Phys. 43 (2002) 6238-6264
•
e-Print:
- math/0206053
•
DOI:
- 10.1063/1.1516845

[11]

Quantum groups and Yang-Baxter equations

A.P. Isaev

- Sov.J.Part.Nucl. 26 (1995) 501-526,
- Sov. J. Part. Nucl. 26 (1995) 501-526. (Fiz. Elem. Chastits At. Yadra 26 (1995) 1204-1263)

[12]

Modified braid equations, Baxterizations and noncommutative spaces for the quantum groups GLq(N), SOq(N) and Spq(N), J. Math Phys., to appear

A. Chakrabarti
,
R. Chakrabarti

[12]

e-Print:
- math/0206093

[13]

Covariant Differential Calculus on the Quantum Hyperplane

Julius Wess
(
- Karlsruhe U.
)
,
Bruno Zumino
(
- UC, Berkeley and
- CERN
)

- Nucl.Phys.B Proc.Suppl. 18 (1991) 302-312,
- In *Annecy 1990, Proceedings, Recent advances in field theory* 302-312. (Nucl. Phys. B, Proc. Suppl. 18B (1990) 302-312) and CERN Geneva - TH. 5697 (90,rec.May) 21 p,
- In *Li, M. (ed.) et al.: Physics in non-commutative world, vol. 1* 63-73
•
DOI:
- 10.1016/0920-5632(91)90143-3

[14]

Quantum groups

J. Schwenk

[15]

Modified braid equations for SOq(3) and noncommutative spaces

A. Chakrabarti

- J.Math.Phys. 43 (2002) 1569-1583

[15]

e-Print:
- math/0103070

[16]

Quantization of Lie Groups and Lie Algebras

- Leningrad Math.J. 1 (1990) 193-225,
- Alg.Anal. 1 (1989) 1, 178-206

[17]

RTT relations, a modified braid equation and noncommutative planes

A. Chakrabarti

- J.Math.Phys. 42 (2001) 2653-2666

[17]

e-Print:
- math/0009178