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

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2. Algebras with trace 20 and the power sums function ψk = xk i . It is easy to prove that, for every k ≤ n, is a polynomial pk(y1, . . .,yk) with rational coefficient, independent of n and such that: σk = pk(ψ1, . . .,ψk). We then set σk(a) = pk(tr(a), . . .,tr(a k )). Next we can formally define for every element a ∈ A and for every integer d a d th -characteristic polynomial: χd,a[t] = d (−1) i σi(a)t d−i i=0 Definition 2.1.2. 1. We say that an algebra A with trace satisfies the d th -formal Cayley-Hamilton theorem if χd,a[a] = 0 for all a ∈ A. 2. We say that A has degree d if it satisfies the d th -formal Cayley-Hamilton theorem and tr(1) = d. Note that algebra with trace of degree d form a category with morphisms being algebra morphisms compatible with trace, which will be denoted by Cd. 2.2 Representations We are interested in a representation theory of algebra with trace of degree d. Let give an example Example 2.2.1. Let A be a commutative algebra, then Md(A) with the ordinary trace is an algebra with trace of degree d. Definition 2.2.2. A n dimensional representation of an algebra with trace R with value in a commutative algebra A is an homomorphism ρ : R → Mn(A) compatible with the trace map. If A = k we think of this representation as a geometric point. Remark 2.2.3. We have necessarily n = d, since d = ρ(tr(1)) = tr(ρ(1)) = tr(I) = n, where I is the identity matrix of Mn(A). Before stating the main theorem of this section, we will give some examples of algebra with trace. In order to simplify the treatment and stick to a geometric language we assume, from now until the end of the chapter, that k is algebraically closed and of characteristic 0.
2. Algebras with trace 21 Example 2.2.4. Consider A to be an order in a finite dimensional central simple algebra D. This means that the center of A is a domain, A is torsion free over Z and, we have D = A⊗Z Q(Z), where Q(Z) is the field of fractions of Z. In other words, A embeds naturally in D which is its ring of fractions. If Q(Z) is the algebraic closure of Q(Z), we have that A ⊗Z Q(Z) is the full ring Md(Q(Z). Hence we have on D, and on A, the usual reduced trace map tr : D → Q(Z). It is well known that tr(A) = Z, if A is a finite Z module, Z is integrally closed and the characteristic is 0. So under this hypotheses A is an algebra of degree d. For more details cf. [Pro73] or [MR87]. Example 2.2.5. The second example is given by Azumaya algebras (cf [Art69]). Recall that: Definition 2.2.6. An algebra R over a commutative ring A is called an Azumaya algebra of degree d over A, if there exists a faithfully flat extension B of A such that R ⊗A B is isomorphic to the algebra Md(B). In this case it’s easy to show that the ordinary trace maps R into A. Let R be a finitely generated algebra, we want to describe it’s d dimensional representation. Theorem 2.2.7. Assumes that R ∈ Cd is a finitely generated algebra. Set T = tr(R). (i) T is a finitely generated algebra, and R is s finite module over T. In particular T is the coordinate ring of an affine algebraic variety XT = Maxspec(T). (ii) The points of XT parameterize equivalence classes of d-dimensional, trace compatible, and semisimple representations of R (iii) Set Spec(R) equivalences classes of irreducible representations of R. The canonical map Spec(R) χ → XT, induced by the central characters, is surjective and each fiber consists of all those irreducible representations of R which are irreducible components of the corresponding semisimple representation. In particular each irreducible representation of R has dimension at most d. (iv) The set ΩR = {a ∈ Spec(T),such that the corresponding semisimple representation is irreducible} is a Zariski open set. This is exactly the part of Spec(T) over which R is an Azumaya algebra of degree d. Proof. See [DCP93] theorem 4.5, page 48.
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: 2. ALGEBRAS WITH TRACE In this chap
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
Page 59 and 60: 5. Quantum universal enveloping alg
Page 77 and 78:
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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