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Group Hopf Algebra
In mathematics, the group Hopf algebra of a given group is a certain construct related to the symmetries of group actions. Deformations of group Hopf algebras are foundational in the theory of quantum groups. Definition Let ''G'' be a group and ''k'' a field. The ''group Hopf algebra'' of ''G'' over ''k'', denoted ''kG'' (or ''k'' 'G'', is as a set (and a vector space) the free vector space on ''G'' over ''k''. As an algebra, its product is defined by linear extension of the group composition in ''G'', with multiplicative unit the identity in ''G''; this product is also known as convolution. Note that while the group algebra of a ''finite'' group can be identified with the space of functions on the group, for an infinite group these are different. The group algebra, consisting of ''finite'' sums, corresponds to functions on the group that vanish for cofinitely many points; topologically (using the discrete topology), these correspond to functions which are non-zero only on a fini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
