TL;DRAbstract
After all the abstract algebra in the preceding two chapters, it is clearly high time for some substantial examples. This is the topic of the present chapter and the following one. In this chapter we describe quantum groups (quasitriangular Hopf algebras) that are deformations of the enveloping algebras of Lie algebras. The latter have already been introduced in Example 1.5.7 and are clearly cocommutative. But by deforming the comultiplication (and perhaps also the appearance of the multiplication), we can obtain examples that are not cocommutative and not commutative (if the Lie algebra is non-Abelian). The quasitriangular structure also becomes nontrivial. These examples can therefore be called quantum groups of enveloping algebra type. In principle, this deformation tends to be a systematic one related to the twisting construction given in Chapters 2.3–2.4, but this need not concern us as far as describing the resulting structures is concerned.
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After all the abstract algebra in the preceding two chapters, it is clearly high time for some substantial examples. This is the topic of the present chapter and the following one. In this chapter we describe quantum groups (quasitriangular Hopf algebras) that are deformations of the enveloping algebras of Lie algebras. The latter have already been introduced in Example 1.5.7 and are clearly cocommutative. But by deforming the comultiplication (and perhaps also the appearance of the multiplication), we can obtain examples that are not cocommutative and not commutative (if the Lie algebra is non-Abelian). The quasitriangular structure also becomes nontrivial. These examples can therefore be called quantum groups of enveloping algebra type. In principle, this deformation tends to be a systematic one related to the twisting construction given in Chapters 2.3–2.4, but this need not concern us as far as describing the resulting structures is concerned.
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