On crystal bases of the $Q$-analogue of universal enveloping algebras

  title={On crystal bases of the \$Q\$-analogue of universal enveloping algebras},
  author={Masaki Kashiwara},
  journal={Duke Mathematical Journal},
  • M. Kashiwara
  • Published 1 July 1991
  • Mathematics
  • Duke Mathematical Journal
0. Introduction. The notion of the q-analogue of universal enveloping algebras is introduced independently by V. G. Drinfeld and M. Jimbo in 1985 in their study of exactly solvable models in the statistical mechanics. This algebra Uq(g) contains a parameter q, and, when q 1, this coincides with the universal enveloping algebra. In the context of exactly solvable models, the parameter q is that of temperature, and q 0 corresponds to the absolute temperature zero. For that reason, we can expect… 
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Canonical bases arising from quantized enveloping algebra, II, preprint
  • Canonical bases arising from quantized enveloping algebra, II, preprint
Thus we can define G: L()/qL() U(g) L() L(c)-by G(b) G(b) for b 3B() c B(). Then we have
  • Thus we can define G: L()/qL() U(g) L() L(c)-by G(b) G(b) for b 3B() c B(). Then we have
PiG(b) mod qL(oz) Hence we have (7.4.4) M c L/M c qL = @ Zz(j"b), bS and fi
  • R) N qLo). By (7.4.2) and Lemma 7.1.1 (i), we have ( (R) N) n Lo/(Q (R) N) n qLo (..+ Qb
Then the preceding lemma implies the desired result
  • Q.E.D
  • Dokl
  • 1985