Quasigroup

 In mathematics and theoretical physics, the term quantum group denotes one amongst some different sorts of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups.   The term "quantum group" first appeared within the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a selected class of Hopf algebra. the identical term is additionally used for other Hopf algebras that deform or are near classical Lie groups or Lie algebras, like a "bicrossproduct" class of quantum groups introduced by Shahn Majid a bit after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras betting on an auxiliary parameter q or h, which become universal enveloping algebras of a particular Lie algebra, frequently semisimple or affine, when q = 1 or h = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group. 40. Quasigroup A set with a binary arithmetic operation (usually called multiplication) within which each of the equations ax=b and ya=b features a unique solution, for any elements a,b of the set. A quasi-group with a unit is termed a loop. A quasi-group may be a natural generalization of the concept of a gaggle. Quasi-groups arise in various areas of mathematics, as an example within the theory of projective planes, non-associative division rings, during a number of questions in combinatorial analysis, etc. The term "quasi-group" was introduced by R. Moufang; it absolutely was after her work on non-Desarguesian planes (1935), during which she elucidated the connection of such planes with quasi-groups, that the event of the idea of quasi-groups properly began. The mappings Ra:x→xa and La:x→ax are called right and left translations (or displacements) by the element a. during a quasi-group, translations are permutations of the underlying set (cf. Permutation of a set). The subgroup G of the group of permutations of the set Q generated by all translations of the quasi-group Q(⋅) is termed the group related to the quasi-group Q(⋅). there's an in depth relation between the structure of G which of Q(⋅). A homomorphic image of a quasi-group needn't, in general, be a quasi-group, but it's a groupoid with division. appreciate homomorphisms of a quasi-group onto a quasi-group are the so-called normal congruences. (A congruence (cf. Congruence (in algebra)) θ on Q(⋅) is normal if each of the relations acθbc and caθcb implies aθb.) In groups all congruences are normal. A sub-quasi-group H is termed normal if there exists a traditional congruence θ such H coincides with one in every of the congruence classes. There exist congruences on quasi-groups within which two or maybe all congruence classes with relevancy θ are sub-quasi-groups.