Generalized quantum current algebras

Two general families of new quantum deformed current algebras are proposed and identified both as infinite Hopf family of algebras, a structure which enable one to define “tensor products” of these algebras. The standard quantum affine algebras turn out to be a very special case of both algebra families, in which case the infinite Hopf family structure degenerates into standard Hopf algebras. The relationship between the two algebra families as well as their various special examples are discussed, and the free boson representation is also considered. Quantum groups, since proposed by Drinfeld [3, 4], have attracted the attentions of both mathematicians and theoretical physicists for more than ten years. And even after so many years’ extensive studies, the interest in quantum groups and their various extensions–quantum affine algebras [5, 19], Yangian doubles [17] and so on–is still not faded. Theoretical physicists are fascinated about quantum groups and their extensions because these algebraic objects are the right candidates for describing the dynamical symmetries in certain models in integrable quantum field theories and/or exactly solvable statistical physics. Mathematicians are interested in quantum groups and their extensions because these are the first known nontrivial Hopf algebras–non-commutative non-co-commutative associative algebras expected for quite some years but not realized until the discovery of Drinfeld. Recently, accompanying the search and investigation for elliptic quantum groups [7, 8, 9], it is realized that there are more general algebraic structures which are of interests to both mathematicians and physicists. One such example is the elliptic quantum groups proposed by Felder et al [7, 8, 9] which belong to the class of quasi-triangular quasi-Hopf algebras [6]– certain twists [15] of the standard Hopf algebra structures. From a pure mathematical point of view, the most important significance for the discovery of elliptic quantum groups might be that it reveals the possibility for the existence of non-co-associative and non-commutative non-co-commutative associative algebras. Such algebras also happen to describe the dynamical symmetry of some statistical models and hence greatly attracted the attention of theoretical physicists. Further investigations along this line showed that there are several kinds of elliptic quantum groups [7, 8, 9, 10, 11, 12, 14, 15, 18, 21], some are recognized as quasi-triangular quasi-Hopf algebras and some are still not. Due to the lack of co-algebraic structures for some of the elliptic E-mail: [email protected]