Let G be an arbitrary group and k a field. The group Hopf algebra of G over k, denoted kG (or k), is as a set (and 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 with compact support.
However, the group algebra k and the space of function kG := Hom(G,k) are dual: given an element of the group algebra and a function on the group these pair to give an element of k via which is a well-defined sum because it is finite.
Read more about this topic: Group Hopf Algebra
Other articles related to "definition, definitions":
... that resist human digestive enzymes, a definition that includes lignin and polysaccharides ... The definition was later changed to also include resistant starches, along with inulin and other oligosaccharides ... Official definition of dietary fiber differs a little among different institutions Organization (reference) Definition Institute of Medicine Dietary fiber consists ...
... the Brundtland Report, which included what is now one of the most widely recognised definitions "Sustainable development is development that meets the ... It offers an integrated vision and definition of strong sustainability ... It generates a more nuanced definition of sustainable development “the need to ensure a better quality of life for all, now and into the future, in a just and equitable manner, whilst living ...
... In medical dictionaries, definitions should to the greatest extent possible be simple and easy to understand, preferably even by the general public useful clinically or in ...
... ESPN launched its 720p high-definition simulcast, originally branded as ESPNHD, on March 20, 2001 ... along with most live events on ESPN, use high definition ... non-HD sources is presented in a standard definition, 43 format with stylized pillarboxes ...
... These three definitions may be described as the ethno-linguistic definition, the religious-cultural definition, and the patrilineal definition, respectively ...
Famous quotes containing the word definition:
“Im beginning to think that the proper definition of Man is an animal that writes letters.”
—Lewis Carroll [Charles Lutwidge Dodgson] (18321898)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than thisdevoted and obedient. This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (18201910)
“Its a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was mine.”
—Jane Adams (20th century)