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Let ∆ be a symmetric generalized Cartan matrix, and g = g(∆) the corresponding Kac–Moody Lie–algebra with triangular decomposition g = n−⊕h⊕n+ (see [K]. We denote by b+ = b+(∆) = h ⊕ n+ the Borel subalgebra. Let Uq(b+) be the quantization of the universal enveloping algebra of b+, it is defined by generators and relations as we will recall below. For ∆ of finite or affine type, we want to survey a construction of Uq(b+) using the representation theory of quivers, following [R2], [R3], [R4] and [R5].

From representations of quivers via Hall and Loewy algebras to quantum groups