1
Basic notions of representation theory
1.1
What is representation theory?
In technical terms, representation theory studies representations of associative algebras. Its general
content can b e very
2
General results of representation theory
2.1
Subrepresentations in semisimple representations
Let A b e an algebra.
Denition 2.1. A semisimple (or completely reducible) representation of A is a dire
3
Representations of nite groups: basic results
Recall that a representation of a group G over a eld k is a k -vector space V together with a
group homomorphism : G GL(V ). As we have explained above,
4
Representations of nite groups: further results
4.1
Frobenius-Schur indicator
Suppose that G is a nite group and V is an irreducible representation of G over C. We say that
V is
- of complex type, i
5
Quiver Representations
5.1
Problems
Problem 5.1. Field embeddings. Recall that k (y 1 , ., ym ) denotes the eld of rational functions
of y1 , ., ym over a eld k . Let f : k [x1 , ., xn ] k (y1 , .,
1. 18.712 takehome assignment
1. Let Q be a quiver, i.e. a nite oriented graph. Let A(Q) be the path
algebra of Q over a eld k, i.e. the algebra whose basis is formed by paths
in Q (compatible with or