trace forms on algebras
The trace form of is defined as :
The trace form is a symmetric bilinear form.
Given and then . So . So we have
Furthermore, is general property of traces, thus
So the trace form is a symmetric bilinear form. ∎
The symmetric property can be interpreted as a weak form of commutativity of the product: commute within their trace from. A more essential property arises for certain algebras and can be interpreted as “the product is associative within the trace” and written as
We shall call such an algebra weakly associative though the term is not standard.
This property is clear for all associative algebras as:
Given a weakly associative algebra, then the radical of the trace form is an ideal of the algebra.
From this result many authors define an algebra to be semi-simple if its trace form is non-degenerate. In this way, , the radical of , is semi-simple. [Some variations on this definition are often required over small fields/characteristics, especially when characteristic is 2.]
More can be said when ideals are considered.
Given a weakly associative algebra , then if is an ideal of then so is .
Given , then for all and , then as is an ideal and so as . This makes so is a right ideal. Likewise so and thus is an ideal of . ∎
To proceed one factors out the radical so that is semisimple. Then given an ideal of , if then as the trace form is a non-degenerate bilinear from, , and so by iterating we produce a decomposition of into minimal ideals:
Hence we arrive at the alternative definition of a semisimple algebra: that the algebra be a direct product of simple algebras. To obtain the property it is sufficient to assume has not ideal such that . This is the content of the proof in
Alternatively any bilinear form with (1) can be used. However, the trace form is always definable and the desired properties are easily translated into implications about the multiplication of the algebra.
Jacobson, Nathan Lie Algebras, Interscience Publishers, New York, 1962.
- 2 Koecher, Max, The Minnesota notes on Jordan algebras and their applications. Edited and annotated by Aloys Krieg and Sebastian Walcher. [B] Lecture Notes in Mathematics 1710. Berlin: Springer. (1999).
|Title||trace forms on algebras|
|Date of creation||2013-03-22 16:28:01|
|Last modified on||2013-03-22 16:28:01|
|Last modified by||Algeboy (12884)|