By Sjoerd Beentjes

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**Example text**

Note that a morphism of representations u : M → N is an isomorphism if and only if all constituent maps ui : Vi → Wi are k-linear isomorphisms. 4. From the above picture it is clear that composition of representations is well-defined and associative. We denote by Rep k (Q) the category of k-linear representation of Q. 1. Indeed, the unique zero object is the representation consisting of zero vector spaces and zero maps. Furthermore, kernels, cokernels, and direct sums are defined “vertex-wise” as one would expect.

S and h, h ∈ h. The last two equations are referred to as the Serre relations. Here ad : gA −→ gl(gA ) denotes the adjoint representation of gA , given by ad(x) = [x, −]. The Dynkin diagram of gA is Q, its root lattice is R = si=1 Z · αi , its Cartan matrix is again A. 6. 10) is indeed isomorphic to sl3 (C). 3 Intermezzo: quantum groups yielding a six-dimensional complex vector space V . 5. Recall that sln (C) is the Lie algebra of complex n × n-matrices of trace zero. Let eij denote the n × n-matrix which has zeroes everywhere except for a 1 in entry ij.

The parameter v is a formal variable, which we want to specialize at prime powers q = pn . But the Hopf algebra is defined over the field C(v), so a specialization at a value ν ∈ C∗ need not be well defined. Lusztig presented a solution which we will present shortly. 2. Giving the generators Ei , Fi , v h weights αi , −αi , 0 respectively equips Uv (g) with a grading by the root lattice R = Z · αi of g. Note that the Hall algebra associated to Q is graded by the Grothendieck group of A = Rep k (Q).

### An introduction to Hall algebras by Sjoerd Beentjes

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