# zbMATH — the first resource for mathematics

A Poincaré-Birkhoff-Witt theorem for the quantum group of type $$A_ n$$. (English) Zbl 0677.17011
The author describes a linear basis for the $$q$$-analogue $$U_ q(G)$$ of the universal enveloping algebra in case $$G={\mathfrak sl}_{N+1}({\mathbb C})$$. It follows, that $$U_ q({\mathfrak sl}_{N+1}({\mathbb C}))$$ is a left (right) Noetherian ring without zero divisors provided $$q(q^ 8-1)\neq 0$$.

##### MSC:
 17B35 Universal enveloping (super)algebras 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 81Q99 General mathematical topics and methods in quantum theory
Full Text:
##### References:
 [1] V. G. Drinfeld: Hopf algebra and the quantum Yang-Baxter equation. Soviet Math. Dolk., 32, 254-258 (1985). · Zbl 0588.17015 [2] V. G. Drinfeld: Quantum groups. Proc. Int. Congr. Math., Berkeley, vol. 1 (1986) ; Amer. Math. Soc, pp. 798-820 (1988). [3] Y. Izumi: On a g-analogue of the universal enveloping algebra of the simple Lie algebra of type AN. Master thesis, Osaka University (1988) (in Japanese). [4] M. Jimbo: A ^-difference analogue of U(Q) and the Yang-Baxter equation. Lett. Math. Phys., 10, 63-69 (1985). · Zbl 0587.17004 · doi:10.1007/BF00704588 [5] V. G. Kac: Infinite Dimensional Lie Algebra. Progress in Mathematics, vol. 44, Birkhauser, Boston-Basel-Stuttgart (1983). · Zbl 0537.17001 [6] G. Lusztig: Quantum deformations of certain simple modules over enveloping algebra. Advances in Math., 70, 237-249 (1988). · Zbl 0651.17007 · doi:10.1016/0001-8708(88)90056-4 [7] M. Rosso: Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra. Commun. Math. Phys., 117, 581-593 (1988). · Zbl 0651.17008 · doi:10.1007/BF01218386
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.