Geodesic
Parabolic twists for linear algebras $A_{n-1}$
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 20, Tome 347 (2007), pp. 187-213 Cet article a éte moissonné depuis la source Math-Net.Ru

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New solutions of twist equations for universal enveloping algebras $U(A_{n-1})$ are found. They can be presented as products of full chains of extended Jordanian twists $\mathcal F_{\widehat{ch}}$, Abelian factors (“rotations”) $\mathcal F^R$ and sets of quasi-Jordanian twists $\mathcal F^{\widehat J}$. The latter are the generalizations of Jordanian twists (with carrier $b^2$) for special deformed extensions of the Hopf algebra $U(b^2)$. The carrier subalgebra $g_{\mathcal P}$ for the composition $\mathcal F_{\mathcal P}=\mathcal F^{\widehat J}\mathcal F^R\mathcal F_{\widehat{ch}}$ is a nonminimal parabolic subalgebra in $A_{n-1}$, $g_{\mathcal P}\cap\mathbb N_g^-\ne\varnothing$. The parabolic twisting elements $\mathcal F_{\mathcal P}$ are obtained in the explicit form. The details of the construction are illustrated by considering the examples $n=4$ and $n=11$. 
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V. D. Lyakhovsky. Parabolic twists for linear algebras $A_{n-1}$. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 20, Tome 347 (2007), pp. 187-213. http://geodesic.mathdoc.fr/item/ZNSL_2007_347_a11/

[1] A. L. Onischik, Topology of transitive transformation groups, Phys. Mat. Lit., Moskow, 1995

[2] M. Chaichian, P. P. Kulish, K. Nishijima and A. Tureanu, On a Lorenz-invariant interpretation of noncommutative space-time and its implications in noncommutative QFT, , 2004 arXiv: /hep-th/0408069 | MR

[3] P. Achieri, M. Dimitrijevic, F. Meyer and J. Wess, Noncommutative Geometry and Gravity, , 2005 arXiv: /hep-th/0510059

[4] V. G. Drinfeld, Algebra and Analysis, 1:2 (1989), 30–46 | MR

[5] M. A. Semenov-Tian-Shansky, Teor. Mat. Fiz., 93 (1992), 302

[6] V. G. Drinfeld, Dokl. Akad. Nauk., 273:3 (1983), 531–535 | MR

[7] M. Gerstenhaber, A. Giaquinto, Lett. Math. Phys., 40:4 (1997), 337–353 | DOI | MR | Zbl

[8] P. Bonneau, M. Gerstenhaber, A. Giaquinto and D. Sternheimer, Journ. Math. Phys., 45:10 (2004), 3703–3741 | DOI | MR | Zbl

[9] N. Yu. Reshetikhin, “Multiparameter quantum groups and twisted quasitriangular Hopf algebras”, Lett. Math. Phys., 20 (1990), 331–335 | DOI | MR | Zbl

[10] O. V. Ogievetsky, “Hopf structures on the Borel subalgebra of $sl(2)$”, Proceedings of the winter school on geometry and physics, Zdikov, Czech Republic, January 1993, eds. J. Bures et al., Rend. Circ. Mat. Palermo, II. Ser., 37 (1994), Suppl. | MR | Zbl

[11] P. P. Kulish, V. D. Lyakhovsky and A. I. Mudrov, Journ. Math. Phys., 40 (1999), 4569–4586 | DOI | MR | Zbl

[12] P. P. Kulish, V. D. Lyakhovsky and M. A. del Olmo, Journ. Phys. A: Math. Gen., 32 (1999), 8671 ; arXiv: /math.QA/9908061 | DOI | MR | Zbl

[13] D. N. Ananikian, P. P. Kulish and V. D. Lyakhovsky, Algebra and Analysis, 14:3 (2002), 27–54

[14] V. D. Lyakhovsky, “Basic twisting factors and the factorization properties of twists”, Supersymmetries and Quantum symmetries, eds. E. Ivanov et al, 2002, 120–130

[15] P. P. Kulish, V. D. Lyakhovsky and A. A. Stolin, Czech. Journ. Phys., 50 (2000), 1291–11296 | DOI | MR

[16] M. Ilyin, V. Lyakhovsky, Czech. Journ. Phys., 56 (2006), 1191–1196 | DOI | MR | Zbl

[17] L. C. Kwek, V. D. Lyakhovsky, Czech. Journ. Phys., 51 (2001), 1374–1379 | DOI | MR | Zbl

[18] V. D. Lyakhovsky, M. A. del Olmo, Journ. Phys. A: Math. Gen., 35 (2002), 5731–5750 | DOI | MR | Zbl

[19] V. D. Lyakhovsky, Twist deformations in dual coordinates, arXiv: /math.QA/0312185

[20] V. D. Lyakhovsky, Zap. Nauchn. Sem. POMI, 317, 2004, 122–141 | Zbl

[21] V. D. Lyakhovsky, M. E. Samsonov, Journal of Algebra and its Applications, 1 (2002), 413–424 | DOI | MR | Zbl