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Primitive Elements (Hopf Algebra)
In algebra, a primitive element of a co-algebra ''C'' (over an element ''g'') is an element ''x'' that satisfies :\mu(x) = x \otimes g + g \otimes x where \mu is the co-multiplication and ''g'' is an element of ''C'' that maps to the multiplicative identity 1 of the base field under the co-unit (''g'' is called ''group-like''). If ''C'' is a bi-algebra, i.e., a co-algebra that is also an algebra (with certain compatibility conditions satisfied), then one usually takes ''g'' to be 1, the multiplicative identity of ''C''. The bi-algebra ''C'' is said to be primitively generated if it is generated by primitive elements (as an algebra). If ''C'' is a bi-algebra, then the set of primitive elements form a Lie algebra with the usual commutator bracket , y= xy - yx (graded commutator if ''C'' is graded). If ''A'' is a connected graded cocommutative Hopf algebra over a field of characteristic zero, then the Milnor–Moore theorem states the universal enveloping algebra In mathematics, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Co-algebra
In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras. Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions ( see below). Coalgebras occur naturally in a number of contexts (for example, representation theory, universal enveloping algebras and group schemes). There are also F-coalgebras, with important applications in computer science. Informal discussion One frequently recurring example of coalgebras occurs in representation theory, and in particular, in the representation theory of the rotation group. A primary task, of practical use in physics, is to obtain combinations of systems with different states of angular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
