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

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2. Algebras with trace 22 Remark 2.2.8. (1) If R is an order in a central simple algebra of degree d then T equals the center of R, furthermore since the central simple algebra splits in a d dimensional matrix algebra which one may consider as a generic irreducible representation, it is easily seen that the open set ΩR is non empty. (2) If T is a finitely generated module over a subalgebra Z0, we can consider the finite surjective morphism τ : Spec(T) → Spec(Z0). Then by the properness of τ we get that the set Ω 0 R := a ∈ Spec(Z0) : τ −1 (a) ⊂ ΩR is a Zariski open dense subset of Spec(Z0). We will use this remark in the theory of quantum groups where there is a natural subalgebra Z0 which appears in the picture.
3. TWISTED POLYNOMIAL ALGEBRAS In this chapter we introduce the main notion of quasi polynomial algebras, or skew polynomial. Note that as the quantum enveloping algebras are the “quantum” version of the universal enveloping algebras of a Lie algebra, we can think that twisted polynomial algebras are the “quantum” version of the symmetric algebra of a Lie algebra. More details on twisted polynomial algebras can be found, for examples, in [DCP93] or [Man91]. 3.1 Useful notation and first properties Before giving the definition of twisted polynomial algebra, we want to introduce some notations, all will be useful in the sequel. Let fix an invertible element q ∈ C different from 1 and −1 so that the fraction 1 q−q −1 is well defined. For all n ∈ Z, set [n] = qn − q −n q − q −1 = qn−1 + q n−3 + · · · + q −n+3 + q −n+1 . We have the following relation: [−n] = − [n] [n + m] = q n [m] + q m [n] Observe that if q is not a root of unity then ∀ n ∈ Z, non zero, [n] = 0. If q is a primitive lth root of unity, with l > 2, define l if l is odd e = . if l is even. Now is easy to check that Property. If q is a primitive l root of unity then (i) [n] = 0 ⇔ n ≡ 0 mod e (ii) [n] l = [n]. l 2 We can now define the q analogue of the factorials and of the binomial coefficients
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: 2. Algebras with trace 21 Example 2
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 and 48: 4. General Theory 41 4.2.1 The cent
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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