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

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.