By M. Chaichian, P. Prešnajder (auth.), R. Gielerak, J. Lukierski, Z. Popowicz (eds.)

In the seventies and eighties, clinical collaboration among the speculation component of the Physics division of Leipzig college and the Institute of Theoretical Physics of the college of Wroolaw was once demonstrated. This manifested itself, between different issues, within the association of standard, twice-yearly seminars positioned then again in Wrodaw and Leipzig. those Seminars in Theoretical Physics came about 27 instances, the final in the course of November 1990. that allows you to proceed the traditions of German-Polish contacts in theoretical physics, we made up our minds to begin a brand new sequence of Seminars in Theoretical Physics and identify them after the phenomenal German theoretical physicist Max Born who was once born in 1883 in Wrodaw. we are hoping that those seminars will proceed to give a contribution to raised clinical contacts and realizing among German and varnish theoretical physicists. the 1st Max Born Symposium was once held in Wojnowice fort, 20 km west of Wrodaw, 27 - 29 September 1991. Wojnowice fortress was once in-built the sixteenth century through the noble Boner kinfolk, within the Renaissance variety, and has been lately tailored as a small convention middle. the popular matters on the Symposium have been Quantum teams and Integrable types. The Symposium used to be prepared by way of medical professionals R. Gielerak and Z. Popowicz below the medical supervision of the undersigned.

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E. 14) These four types of involutions are implemented by their action on the Cartan-Weyl basis of Uq (Sp(4)) in the following way (for details see [9]) h: = ph. 15-16) follows that ef = e_l er = -e_2 er = -e_3 ef = -e_4 and the 0(3,2) generators (A,B = 0,1,2,3,4) = -l. 18) -i (e4 + e_4 - e2 - e_2) M14 = ! 19) where "lAB = diag( +, -, -, -, +). 18) can be found in [81. 16) we should assume that Iql = 1. 3. 1) with 1 = 1. 4) J. LUKIERSKI AND A. 7) The coproduct is f1Mi = Mi ® I + I ® Mi ~Po = Po ® I ~L_ +I .

We also use the standard notations Uq(K) and Uq(b±) for the Cartan and Borel subalgebras, generated by k;} and e±a;, k;;1 correspondingly. We put also Z2 zn zn ezpq(z):=I+z+(2)'+"'+-( ),+ ... = L : -)" ( q' n q' n;:::O n q' q4 _ 1 (a)q:= - - , q-l (19) (20) Now we proceed to a description of the Cartan-Weyl basis for the quantum (super)algebras Uq(g). 3 Cartan- Weyl basis for Uq(g) Let 6+ be the system of all positive roots with respect to II. We denote by 6+ the reduced root system which is obtained from 6+ by removing such roots a for which a/2 are roots.

Comm. Math. Phys. 134 (1990),421-431. S. Some application of quantum Weyl groups. The multiplicative formula for universal R-matrix for simple Lie algebras. Geom. and Phys 7:4 (1990), 1-14. D. Quantization of Lie groups and Lie algebras. Algebra and Analisis 1 (1989), 178-206. [13] Jimbo, M. Lett. Math. Phys. 11 (1986), 247-252. M. The universal R-matrix for quantum nontwisted affine Lie algebras. Funkz. Analiz i ego pril. 26;1 (1992),85-88. N. The Cartan-Weyl basis and the universal R-matrix for quantum Kac-Moody algebras and superalgebras.