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

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moreover, the action of Q ∗ 2 4. General Theory 42 can be obviously extended to an action of the semidirect product W ⋉ Q∗ 2 . Let W the subgroup generated by all conjugates σWσ −1 of W by elements σ ∈ Q∗ 2 . Theorem 4.2.6. The homomorphism γ ◦ h : Z ↦→ U 0 is injective and its image is precisely the set U 0W of fixed points of the action of W on U 0 . Proof. See [DCP93] or [CP95]. Example 4.2.7. Let g = sl2(C). Then W = W and U0W consist of the . Thus Laurent polynomials in K1 which are invariant under K1 ↦→ K −1 1 U 0W is generated as an algebra over C(q) by φ = K1 + K −1 −1 (q − q−1 . ) 2 It easy to check that the quantum Casimir element Ω = qK1 + q−1K −1 −1 (q − q−1 + EF ) 2 lies in Z and γ −1 (h(Ω)) = φ. It follows from theorem 4.2.6 that Ω generates Z as C(q)-algebra. We see now the quantum analogue of the Harish Chandra’s theorem on the central characters of the classical universal enveloping algebra. Let λ ∈ Λ and define a homomorphism λ : U 0 ↦→ C(q) by Ki ↦→ q (αi|λ) . Let χq,λ = λ ◦ γ −1 ◦ h : Z ↦→ C(q) Theorem 4.2.8. Let λ, µ ∈ Λ. Then χλ = χµ if and only if µ = ω(λ) for some ω ∈ W. Proof. See [CP95] Suppose now that ǫ ∈ C is a l th root of the unity such that l is odd and l > di for all i. Our aim is to describe the center Zǫ of Uǫ. We shall assume a different approach from that used in the generic case. Let Z1 = (Z ∩ A) /(q − ǫ), we call it the Harish Chandra part of the center. This is not the full center, in fact we have Proposition 4.2.9. The elements El α, F l α for α ∈ R + , and Kl i for i = 1, . . .,n lie in Zǫ. Proof. See [DCP93], §21.
4. General Theory 43 For α ∈ R + , β ∈ Q, set eα = El α, fα = F l α and kβ = Kl β ; we shall be the often write ei and fi for eαi and fαi . Let Z0 (resp. Z + 0 , Z− 0 , and Z0 0 subalgebra of Zǫ generated by the eα, fα and k ± i (resp.eα, fα and k ± i ). Proposition 4.2.10. (i) We have Z ± 0 ⊂ U ± ǫ . (ii) Multiplication defines an isomorphism of algebras Z − 0 ⊗ Z0 0 ⊗ Z + 0 ↦→ Z0. (iii) Z0 0 is the algebra of Laurent polynomial in the ki, and Z ± 0 is the polynomial algebra with generators the eα and fα respectively. (iv) We have Z ± 0 = U ± ǫ ∩ Zǫ (v) The subalgebra Z0 of Zǫ is preserved by the braid group algebra automorphism Ti. (vi) Uǫ is a free Z0 module with basis the set of monomial E k1 · · ·EkN β1 βN Ks1 1 . . .Ksn n F tN t1 · · ·F βN β1 for which 0 ≤ tj, si, kj < l, for i = 1, . . .,n and j = 1, . . .,N. Proof. See [DCP93], §21. Therefore, we can completely describe the center of Uǫ. Theorem 4.2.11. Zǫ is generated by Z1 and Z0. Proof. See [DCP93], §21. The preceding proposition shows that Uǫ is a finite Z0 module. It follows that Zǫ ⊂ Uǫ is finite over Z0, and hence integral over Z0. By the Hilbert basis theorem, Zǫ is a finitely generated algebra. Thus, the affine schemes Spec(Zǫ) and Spec(Z0), namely the sets of algebra homomorphism from Zǫ and Z0 to C, are algebraic varieties. In fact, it is obvious that Spec(Z0) is isomorphic to C 2N × (C ∗ ) n . Moreover the inclusion Z0 ֒→ Zǫ induces a projection τ : Spec(Zǫ) ↦→ Spec(Z0), and we have Proposition 4.2.12. Spec(Zǫ) is a normal affine variety and τ is a finite (surjective) map of degree l n . Proof. See [CP95] or [DCP93] §21. We conclude this section by discussing the relation between the center and the Hopf algebra structure of Uǫ.
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 and 46: subject to the following relations:
Page 47: 4. General Theory 41 4.2.1 The cent
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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