Quantum Groups


  1. Last step in classification of irreducible representations of compact quantum groups
  2. Proof of the multiplicative formula for the universal R-matrix for quantized universal enveloping algebras of  simple Lie groups  (with Yan Soibel’man)
  3. Proof of the Poincare-Birkhoff-Witt theorem for Yangians
  4. Construction of Affine Yangians
  5. A short proof of the commutation relation for quantized enveloping algebras



  • S. Boyarchenko, S. Levendorskiĭ, On affine Yangians. Lett. Math. Phys. 32 (1994), no. 4, 269--274
  • S. Levendorskiĭ, Y. Soibelman, V. Stukopin, The quantum Weyl group and the universal quantum $R$-matrix for affine Lie algebra $A\sp {(1)}\sb 1$. Lett. Math. Phys. 27 (1993), no. 4, 253--264.
  • S. Levendorskiĭ,On generators and defining relations of Yangians. J. Geom. Phys. 12 (1993), no. 1, 1--11.
  •  S. Levendorskiĭ, On PBW bases for Yangians. Lett. Math. Phys. 27 (1993), no. 1, 37--42.
  • S. Levendorskiĭ, Twisted function algebras on a compact quantum group and their representations. (Russian) Algebra i Analiz 3 (1991), no. 2,180--198 translation in St. Petersburg Math. J. 3 (1992), no.2, 405--423
  • S. Levendorskiĭ, Y. Soibel'man, The quantum Weyl group and a multiplicative formula for the $R$-matrix of a simple Lie algebra. (Russian) Funktsional. Anal. i Prilozhen. 25 (1991), no. 2,73--76 translation in Functional Anal. Appl. 25 (1991), no. 2,
  • S. Levendorskiĭ, Y. Soibel'man, Algebras of functions on compact quantum groups, Schubert cells and quantum tori. Comm. Math. Phys. 139 (1991), no. 1, 141--170.
  • S. Levendorskiĭ, Y. Soibel'man, Quantum group $A\sb \infty$. Comm. Math. Phys. 140 (1991), no. 2, 399--414.
  •  S. Levendorskiĭ, Y. Soibel'man, Some applications of the quantum Weyl groups. J. Geom. Phys. 7 (1990), no. 2, 241--254.
  •  S. Levendorskiĭ, Twisted algebras of functions on a compact quantum group and their representations. (Russian) Funktsional. Anal. i Prilozhen. 24 (1990), no. 4,80--81 translation in Functional Anal. Appl. 24 (1990), no. 4, 330--332 (1991)


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