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

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1. POISSON ALGEBRAIC GROUPS In this chapter we recall some basic facts about Poisson groups that will prove useful in the study of quantum groups. The interested reader can find more details in a vast variety of articles and monographs, for example in [CP95], [KS98] or [CG97]. 1.1 Poisson manifolds 1.1.1 Poisson algebras and Poisson manifolds Definition 1.1.1. A commutative associative algebra A over a field k is called a Poisson algebra if it is equipped with a k-bilinear operation { , } : A ⊗ A → A such that the following conditions are satisfied: 1. A is a Lie algebra with the bracket { , }; 2. the Leibniz rule are satisfied, i.e. for any a, b, c ∈ A, we have {ab, c} = a {b, c} + {a, c}b. If these conditions are satisfied, the operation { , } is called Poisson bracket, and ξa = {a, ·} is called Hamiltonian derivation. Definition 1.1.2. Let A and B be a Poisson algebra over k. An algebra homomorphism f : A → B is called Poisson homomorphism if f ({a, b}) = {f(a), f(b)} . Poisson algebras form a category, with morphism being Poisson homomorphism. Definition 1.1.3. A smooth manifold M is called a smooth Poisson manifold if the algebra A = C ∞ (M) of smooth complex value function on M is equipped with a structure of Poisson algebra over C Definition 1.1.4. An affine algebraic k-variety M is called an affine algebraic Poisson k-variety if the algebra A = k[M] of regular function on M is equipped with a structure of Poisson algebra over k
1. Poisson algebraic Groups 3 Definition 1.1.5. 1. Let M and N be smooth Poisson manifolds. A smooth map f : M → N is called a Poisson map if the induced map f ∗ : C ∞ (N) → C ∞ (M) is a Poisson homomorphism. 2. Let M and N be algebraic Poisson k-variety. An algebraic map f : M → N is called a Poisson map if the induced map f ∗ : k[N] → k[M] is a Poisson homomorphism. It is clear that smooth, and algebraic, Poisson manifolds form a category, with morphisms being Poisson map. Definition 1.1.6. Suppose that M is a smooth Poisson manifold, let A = C ∞ (M). 1. Given φ ∈ A, the vector field ξφ associated to the Hamiltonian derivation {φ, } of A is called Hamiltonian vector field. 2. A submanifold (not necessarily closed) N ⊂ M is called Poisson submanifold if the vector ψφ(n) is tangent to N for any n ∈ N and φ ∈ A. Let M a smooth Poisson manifold, then there is an equivalent definition of Poisson manifold in term of bivector fields. Recall that an n-vector field is a section of the bundle n TM where TM is the tangent vector bundle of M. In particular, we call 2-vector fields bivector fields. Recall also the definition of the Schouten bracket of n-vector fields which generalize the usual Lie bracket on vector fields. The Schouten bracket of an m-vector field with an n vector field is an (m + n − 1)-vector field which is locally defined by [u1 ∧ . . . ∧ um, v1 ∧ . . . ∧ vm] = (−1) i+j [ui, vj] ∧ u1 ∧ . . . ∧ ûi ∧ um ∧ v1 ∧ . . . ∧ ˆvj ∧ . . . ∧ vn i,j where u1, . . .,um, v1, . . .,vn ∈ TmM, m ∈ M, and [, ] denote the Lie bracket of vector fields. Denote by T ∗ M the cotangent bundle to M. Given a bivector field π on M, we define a bilinear operation { , } on C ∞ (M) by {φ, ψ} = 〈π, dφ ∧ dψ〉 (1.1) where 〈, 〉 is the natural paring between the sections of the bundle T ∗ M ∧ T ∗ M and TM ∧ TM. Proposition 1.1.7. The bracket 1.1 defines a Poisson manifold structure on M if and only if [π, π] = 0
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: Part I USEFUL ALGEBRAIC AND GEOMETR
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 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
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5. Quantum universal enveloping alg
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BIBLIOGRAPHY [Abe80] Eiichi Abe. Ho
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Bibliography 75 [DCP97] C. De Conci
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Bibliography 77 [LS91a] S. Z. Leven
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A Algebra with trace ........... 19
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