Geodesic
BRST Operator for Quantum Lie Algebras and Differential Calculus on Quantum Groups
Teoretičeskaâ i matematičeskaâ fizika, Tome 129 (2001) no. 2, pp. 298-316 Cet article a éte moissonné depuis la source Math-Net.Ru

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For a Hopf algebra $\mathcal A$, we define the structures of differential complexes on two dual exterior Hopf algebras: (1) an exterior extension of $\mathcal A$ and (2) an exterior extension of the dual algebra $\mathcal A^*$. The Heisenberg double of these two exterior Hopf algebras defines the differential algebra for the Cartan differential calculus on $\mathcal A$. The first differential complex is an analogue of the de Rham complex. When $\mathcal A^*$ is a universal enveloping algebra of a Lie (super)algebra, the second complex coincides with the standard complex. The differential is realized as an (anti)commutator with a BRST operator $Q$. We give a recursive relation that uniquely defines the operator $Q$. We construct the BRST and anti-BRST operators explicitly and formulate the Hodge decomposition theorem for the case of the quantum Lie algebra $U_{\mathrm q}(gl(N))$. 
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A. P. Isaev; O. V. Ogievetskii. BRST Operator for Quantum Lie Algebras and Differential Calculus on Quantum Groups. Teoretičeskaâ i matematičeskaâ fizika, Tome 129 (2001) no. 2, pp. 298-316. http://geodesic.mathdoc.fr/item/TMF_2001_129_2_a9/

[1] S. L. Woronowicz, Commun. Math. Phys., 122 (1989), 125 | DOI | MR | Zbl

[2] P. Aschieri, L. Castellani, Int. J. Mod. Phys. A, 8 (1993), 1667 | DOI | MR | Zbl

[3] A. Klimyk and K. Schmudgen, Quantum Groups and their Representations, Part IV, Texts and Monographs in Physics, Springer, Berlin, 1997 | MR

[4] A. P. Isaev, EChAYa, 28:3 (1997), 685 | MR

[5] A. Connes, Noncommutative Geometry, Academic Press, New York, 1994 | MR | Zbl

[6] Yu. I. Manin, TMF, 92 (1992), 425 ; G. Maltsiniotis, Commun. Math. Phys., 151 (1993), 275 ; C. R Acad. Sci. Sér. I, 311 (1990), 831 ; B. Tsygan, Sel. Math., 12 (1993), 75 | MR | Zbl | DOI | MR | Zbl | MR | Zbl | MR | Zbl

[7] B. Jurco, Lett. Math. Phys., 22 (1991), 177 | DOI | MR | Zbl

[8] N. Yu. Reshetikhin, L. A. Takhtadzhyan, L. D. Faddeev, Algebra i analiz, 1:1 (1989), 178 | MR

[9] A. Sudbery, Phys. Lett. B, 284 (1992), 61 ; A. P. Isaev and P. N. Pyatov, Phys. Lett. A, 179 (1993), 81 ; E-print hep-th/9211093 | DOI | MR | DOI | MR

[10] P. Schupp, P. Watts, and B. Zumino, Lett. Math. Phys., 25 (1992), 139 | DOI | MR | Zbl

[11] A. P. Isaev and P. N. Pyatov, J. Phys. A, 28 (1995), 2227 ; E-print hep-th/9311112 | DOI | MR | Zbl

[12] L. D. Faddeev and P. N. Pyatov, “The differential calculus on quantum linear groups”, Contemporary Mathematical Physics, eds. R. L. Dobrushin et al., AMS, Providence, RI, 1996, 35 ; ; Л. Д. Фаддеев, П. Н. Пятов, “Квантование дифференциального исчисления на линейных группах”, Проблемы современной теоретической физики, 96-212, ред. А. П. Исаев, ОИЯИ, Дубна, 1996, 19 E-print hep-th/9402070 | MR

[13] I. Ya. Aref'eva, G. E. Arutyunov, and P. B. Medvedev, J. Math. Phys., 35:12 (1994), 6658 | DOI | MR | Zbl

[14] G. E. Arutyunov, A. P. Isaev, and Z. Popowicz, J. Phys. A, 28 (1995), 4349 ; E-print q-alg/9502002 | DOI | MR | Zbl

[15] U. Carow-Watamura, M. Schlieker, S. Watamura, and W. Weich, Commun. Math. Phys., 142 (1991), 605 | DOI | MR | Zbl

[16] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Univ. Press, Princeton, 1992 | MR | Zbl

[17] J. W. van Holten, Nucl. Phys. B, 339 (1990), 158 | DOI | MR

[18] C. Chryssomalakos, J. A. de Azcarraga, A. J. Macfarlane, and J. C. Perez Bueno, J. Math. Phys., 40 (1999), 6009 | DOI | MR | Zbl

[19] D. Bernard, Prog. Theor. Phys. Suppl., 102 (1990), 49 ; Phys. Lett. B, 260 (1991), 389 | DOI | MR | Zbl | DOI | MR

[20] S. Watamura, Commun. Math. Phys., 158 (1993), 67 | DOI | MR | Zbl

[21] A. P. Isaev, J. Math. Phys., 35 (1994), 6784 ; ; Lett. Math. Phys., 34 (1995), 333 ; E-print hep-th/9402060E-print hep-th/9403154 | DOI | MR | Zbl | DOI | MR | Zbl

[22] S. L. Woronowicz, Publ. RIMS Kyoto, 23 (1987), 117 | DOI | MR | Zbl

[23] T. Brzezinski, Lett. Math. Phys., 27 (1993), 287 | DOI | MR | Zbl

[24] D. I. Gurevich, Algebra i analiz, 2:4 (1990), 119 | MR | Zbl

[25] O. V. Radko and A. A. Vladimirov, J. Math. Phys., 38 (1997), 5434 | DOI | MR | Zbl

[26] P. Schupp, Cartan calculus: differential geometry for quantum groups, E-print hep-th/9408170 | MR

[27] S. Baaj, G. Skandalis, Ann. Sci. \hboxÉc. Norm. Sup. 4 série, 26 (1993), 425 ; Р. М. Кашаев, Алгебра и анализ, 8:4 (1996), 63 ; E-print q-alg/9503005 | MR | Zbl | MR | Zbl

[28] C. Burdik, A. P. Isaev, and O. V. Ogievetsky, Standard complex for quantum Lie algebras, E-print math.QA/0010060 | MR