Degree of Parabolic Quantum Groups - Dipartimento di Matematica ...

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4. General Theory 40 There are two different candidates for UA the non restricted and the restricted integral form, which lead to different specializations (with markedly different representation theories) for certain values of ǫ. We are interested in the non restricted form. For more details one can see [CP95]. Introduce the elements with m ≥ 0, where qi = q di . [Ki; m] qi = Kiq m i − K−1 i q−m i qi − q −1 i ∈ U 0 Definition 4.2.2. The algebra UA is the A subalgebra of U generated by the elements Ei, Fi, K ±1 i and Li = [Ki; 0] , for i = 1, . . .,n. With the map qi ∆, S and η defined on the first set of generators as in 4.1.2 and with ∆(Li) = Li ⊗ Ki + K −1 i ⊗ Li (4.10a) S(Li) = −Li (4.10b) η(Li) = 0 (4.10c) Note. The defining relation of UA are as in 4.1.1 replacing 4.3 by and adding the relation EiFj − FjEi = δijLi (qi − q −1 i )Li = Ki − K −1 i Proposition 4.2.3. UA with the previous definition is a Hopf algebra. Moreover, UA is an integral form of U. Proof. See [CP95] or [DCP93] §12. Proposition 4.2.4. If ǫ 2di = 1 for all i, then (i) Uǫ is generated over C by the elements Ei, Fi, and K ±1 i with defining relations obtained from those in 4.1.1 by replacing q by ǫ (ii) The monomials are a C basis of Uǫ. E k1 β1 (iii) The L.S. relations holds in Uǫ. Proof. See [DCP93] §12. kN k1 · · ·EkN KαF · · ·F βN βN β1
4. General Theory 41 4.2.1 The center of Uǫ The center of Uǫ consists of “two parts”, one coming from the center at a “generic q”, the other from the fact that ǫ is a root of 1. We begin by describing the center at “q generic”. Let Z be the center of U. Any element z ∈ Z can be written as a linear combinations of the elements of the basis of U given by the P.B.W. theorem. Since z commutes with the Ki for all i, it follows that z = η∈Q + r,t∈Par(η) E r ϕr,tF t where ϕr,t ∈ U0 and Par(η) = (m1, . . .,mN) ∈ N N : miαi = η . Definition 4.2.5. The map h : Z ↦→ U 0 defined by h(z) = ϕ0,0 is called the Harish Chandra homomorphism Property. h is a homomorphism of algebras. Proof. See [DCP93], §18. h allows us to describe Z as a ring. Any element φ ∈ U0 may be regarded as a C(q)-valued function on the weight lattice Λ in an obvious way: if φ = n i=1 Kti i , where ti ∈ Z for all i, set ∀ λ ∈ Λ φ(λ) = qÈi ti(αi|λ) and extended to U 0 by linearity. Define an automorphism of C(q)-algebras γ : U 0 ↦→ U 0 by setting γ(Ki) = qiKi. Then Property. For all φ ∈ U 0 where ρ = 1 2 (α1 + . . . + αn). γ(φ)(λ) = φ(λ + ρ) Let Q ∗ 2 = {σ : Q ↦→ Z2}, it is easy to see that Q ∗ 2 automorphisms by i acts on U as a group of σ · Kβ = σ(β)Kβ (4.11) σ · Fα = σ(α)Fα (4.12) σ · Eα = Eα (4.13) for all α ∈ R + , β ∈ Q, and σ ∈ Q ∗ 2 . Note that W acts on Q∗ 2 : (ω · σ)(β) = σ(ω −1 (β));
Page 1 and 2: Università degli studi Roma Tre Do
Page 3 and 4: Contents ii Part II Quantum Groups
Page 5 and 6: Contents iv there exists a non empt
Page 7 and 8: Part I USEFUL ALGEBRAIC AND GEOMETR
Page 9 and 10: 1. Poisson algebraic Groups 3 Defin
Page 11 and 12: 1. Poisson algebraic Groups 5 1.2 L
Page 13 and 14: 1. Poisson algebraic Groups 7 1.2.2
Page 15 and 16: 1. Poisson algebraic Groups 9 Examp
Page 17 and 18: 1. Poisson algebraic Groups 11 1.3
Page 19 and 20: 1. Poisson algebraic Groups 13 Exam
Page 21 and 22: Proof. See [KS98], page 23. Proposi
Page 23 and 24: 1. Poisson algebraic Groups 17 Let
Page 25 and 26: 2. ALGEBRAS WITH TRACE In this chap
Page 27 and 28: 2. Algebras with trace 21 Example 2
Page 29 and 30: 3. TWISTED POLYNOMIAL ALGEBRAS In t
Page 31 and 32: 3. Twisted polynomial algebras 25 3
Page 33 and 34: 3. Twisted polynomial algebras 27 3
Page 35 and 36: 3. Twisted polynomial algebras 29 (
Page 37 and 38: 3. Twisted polynomial algebras 31 m
Page 39 and 40: 3. Twisted polynomial algebras 33 P
Page 41 and 42: 4. GENERAL THEORY We briefly recall
Page 43 and 44: 4. General Theory 37 Now we use the
Page 45: subject to the following relations:
Page 49 and 50: 4. General Theory 43 For α ∈ R +
Page 51 and 52: 4. General Theory 45 Let G be the c
Page 53 and 54: Theorem 4.4.1. Uǫ is a maximal ord
Page 55 and 56: 4. General Theory 49 Thus we introd
Page 57 and 58: 5. QUANTUM UNIVERSAL ENVELOPING ALG
Page 59 and 60: 5. Quantum universal enveloping alg
Page 79 and 80: BIBLIOGRAPHY [Abe80] Eiichi Abe. Ho
Page 81 and 82: Bibliography 75 [DCP97] C. De Conci
Page 83 and 84: Bibliography 77 [LS91a] S. Z. Leven
Page 85: A Algebra with trace ........... 19

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