Representation theory Lecture
نویسنده
چکیده
An algebra is a vector space (over C) with a multiplication such that A is a ring with identity, i.e. there is a map A × A → A, (a, b) 7→ ab, which is bilinear and satisfies the associative and distributive laws. The following are examples of algebras: (1) The group algebra of a group G is the vector space CG with basis G and with multiplication forced by the multiplication in G (and the bilinearity). (2) If M is a vector space (over C) then the space End(M) of C-linear transformations of M is an algebra under the multiplication given by composition of endomorphisms. (3) Given a basis B = {b1, . . . , bd} of the vector space M the algebra End(M) can be idenitified with the algebra Md(C) of d× d matrices T = (Tij)1≤i,j,≤d with entries in C via
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Representation Theory , Lecture 0
The aim of this lecture is to recall some standard basic things about the representation theory of finite dimensional algebras and finite groups. First, we recall restriction, induction and coinduction functors. Then we recall the Schur lemma and deduce consequences about the action of the center and the structure of completely reducible representations. Then we explain the structure and repres...
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