Two dual classes of bialgebras related to the concepts of “quantum group” and “quantum lie algebra”
Two dual classes of bialgebras related to the concepts of “quantum group” and “quantum lie algebra”
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DOI：
10.1080/00927879108824320
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发表时间：
1991
影响因子：
0.7
通讯作者：
R. Larson;T. Jacob
中科院分区：
文献类型：
--
作者：
R. Larson;T. Jacob
The concepts of" quantum groupn and" quantum universal enveloping algebra" seem to have arisen first in joint work by Soviet mathematicians [6, 12, 13, 15, 34, 56, 631. At present, there is no universally accepted general definition of these concepts. Rather, there exists a small family of distinguished Hopf algebras, whose representation theories, because they furnish, in a sense to be discussed below," deformations" of the classical representation theories of Lie groups and Lie algebras, permit remarkable applications to low-dimensional topology, to differential equations, to statistical mechanics, and (perhaps more conjecturally) to string theory. The purpose of the present paper is to define rigorously two classes of bialgebras,'braided bialgebrasn and" completed-triangular bialgebrasln which seem to include the small class of distinguished Hopf algebras which have proved useful for these applications.There seems to be a growing consensus that concepts related to the Yang-Baxter equations, and to braided monoidal categories (see [26]), are central in making possible the remarkable applications mentioned above. To appreciate this point of view, let us consider the category AM~ d of left modules over an associative c-algebra A. Consider in particular EBc F, where El Fe mod. In general, this no longer has an A-module structure. However, if A is a bialgebra, there is a natural A-module structure on E Bc F defined using the A A-module structure on E Bc F and the algebra homomorphism
