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

2. ALGEBRAS WITH TRACE
In this chapter we will require some slight knowledge of the theory of algebras
with trace that will be useful, in the next chapters, for the study of quantum
groups and quasi polynomial algebras. More details and a more general
approach in order to study these algebras can be found in [Pro87], [Pro73],
[Pro74] or [Pro79].
2.1 Definition and properties
Let A be an associative algebra with an unit element 1 over a field k of
characteristic 0 and let us denote the algebraic closure of k by k.
Definition 2.1.1. A trace map in an algebra A is a linear map
tr : A → A
satisfying the following axioms: for all pairs of element a, b ∈ A
1. tr(ab) = tr(ba)
2. tr(a)b = btr(a)
3. tr(tr(a)b) = tr(a)tr(b)
An algebra with a trace map is called algebra with trace
Note. The value of the trace is a subalgebra of the center of A (by condition
2).
An ideal I of A algebra with trace is an ordinary ideal closed under trace,
so that A/I inherits a trace.
We are interested in a particular family of algebra with trace as in [Pro87].
Once we have a trace map we want to define for all a ∈ A the element σk(a)
"the symmetric function over the eigenvalue of a", by declaring that tr(ak )
should be the sum of the kth power of the eigenvalues of a. To do this recall
that in the ring Q[x1, . . .,xn] it defines the elementary symmetric function
by the identity
n
(t − xi) = (−1) i σit d−i
i=0