Quantized enveloping algebra explained

ak{g} , the quantum enveloping algebra is typically denoted as U _{q(ak{g})} . The notation was introduced by Drinfeld and independently by Jimbo.

Among the applications, studying the

q \to 0 limit led to the discovery of crystal bases.

The case of

ak{sl} _{2}

Michio Jimbo considered the algebras with three generators related by the three commutators

[h,e] = 2 e, [h,f] = -2 f, [e,f] = \sinh( η h)/\sinh η . When η \to 0 , these reduce to the commutators that define the special linear Lie algebra ak{sl} _{2} . In contrast, for nonzero η , the algebra defined by these relations is not a Lie algebra but instead an associative algebra that can be regarded as a deformation of the universal enveloping algebra of ak{sl} _{2} .

See also

• Quantum group

References

• • Tjin . T. . An introduction to quantized Lie groups and algebras . International Journal of Modern Physics A . 10 October 1992 . 07 . 25 . 6175–6213 . 10.1142/S0217751X92002805 . hep-th/9111043 . 1992IJMPA...7.6175T . 119087306 . 0217-751X.

External links

• Quantized enveloping algebra at the nLab
• Quantized enveloping algebras at q = 1 at MathOverflow
• Does there exist any "quantum Lie algebra" imbedded into the quantum enveloping algebra U _{q(g)} ? at MathOverflow