# Lectures on Quantum Groups by Olivier Schiffman, Pavel Etingof

By Olivier Schiffman, Pavel Etingof

Revised moment version. The textual content covers the cloth offered for a graduate-level direction on quantum teams at Harvard college. The contents conceal: Poisoon algebras and quantization, Poisson-Lie teams, coboundary Lie bialgebras, Drinfelds double building, Belavin-Drinfeld class, countless dimensional Lie bialgebras, Hopf algebras, Quantized common enveloping algebras, formal teams and h-formal teams, endless dimensional quantum teams, the quantum double, tensor different types and quasi Hopf-algebras, braided tensor different types, KZ equations and the Drinfeld classification, Quasi-Hpf enveloping algebras, Lie associators, Fiber functors and Tannaka-Driein duality, Quantization of finite Lie bialgebras, common buildings, common quantization, Dequantization and the equivalence theorem, KZ associator and a number of zeta services, and Mondoromy of trigonometric KZ equations. Probems are given with each one topic and a solution key's incorporated. desk of contents Poisson algebras and quantization Poisson-Lie teams Coboundary Lie bialgebras Drinfeld's double development Belavin-Drinfeld class (I) limitless dimensional Lie bialgebras Belavin-Drinfeld type (II) Hopf algebras Quantized common enveloping algebras Formal teams and h-formal teams endless dimensional quantum teams The quantum double Tensor different types and quasi-Hopfalgebras Braided tensor different types KZ equations and the Drinfeld type Quasi-Hopf quantized enveloping algebras Lie associators Fiber functors and Tannaka-Krein duality Quantization of finite dimensional Lie bialgebras common buildings common quantization Dequantization and the Equalivalence

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Define a map θM : k≥2 M → − Z(G; M ) by θ(m) = k mk τk (one can k check that this converges, because δ(x ) = 0 + O(k) for all k). 19 that θM 41 is an isomorphism when IM = 0. 2, we deduce that θR/I k is iso for all k and thus that θR is iso. This implies that Z(G) = R{σp (F ), . . , σpn−1 (F )} ⊕ δ(C(G)) and thus that Ext(G, Ga ) = Z(G)/δ(C(G)) = R{σp , . . , σpn−1 }. 22. Given a formal group scheme H of dimension d over X, we define J = JH = {f ∈ OH | f (0) = 0} and ωH = J/J 2 and tH = HomOX (ωH , OX ).

If OX is torsion-free this implies easily that η itself is additive. In particular, as the Lazard ring is torsion-free we see that η is additive in the case of the universal FGL, and it follows by base change that it is additive for any formal group. 17. Let γ be any p-typical basic curve on G, inverse to a coordinate x. Let η be the canonical additive curve such that η ∗ d0 x = p d0 t. Then there is a unique series of elements uk ∈ OX (for k > 0) such that k γ(uk tp ). p γ(t) = η(t) + k>0 Proof.

As ξk = 0 for k < 3 and ξ3 = 1 it is immediate from the definitions that χ(x0 , x1 , x2 ) = x0 + x1 + x2 + O(2). 11. 8) then χ(a0 , a1 , a2 ) = 0 and thus x0 x1 x2 1 1 = 0. det 1 ξ(x0 ) ξ(x1 ) ξ(x2 ) Proof. We need to show that φ(a0 )φ(a1 )φ(a2 ) = 1 in Q(C), or equivalently that (1 − a0 /x)(1 − a1 /x)(1 − a2 /x) ∈ U A× . Consider the series h(x) = χ(a0 , a1 , x) ∈ A. As χ(x0 , x1 , x2 ) = x0 + x1 + x2 + O(2), we see that h is a W-series of degree one, and h(a2 ) = 0 so h(x) = v(x)(x − a2 ) for some v ∈ A× .