trace forms on algebras

Given an finite dimensional algebraPlanetmathPlanetmath A over a field k we define the left(right) regular representationPlanetmathPlanetmath of A as the map L:AEndkA given by Lab:=ab (Rab:=ba).

Example 1.

In a Lie algebraMathworldPlanetmath the left representation is called the adjointPlanetmathPlanetmathPlanetmath representation and denoted adx and defined (adx)(y)=[x,y]. Because [x,y]=-[y,x] in characteristicPlanetmathPlanetmath not 2, there is generally no distinction of left/right adjoint representations.

The trace form of A is defined as ,:T×Tk:

Proposition 2.

The trace form is a symmetric bilinear formMathworldPlanetmath.


Given a,b,xA and lk then La+lbx=(a+lb)x=ax+lbx=Lax+lLbx. So La+lb=La+lLb. So we have


Furthermore, tr(fg)=tr(gf) is general property of traces, thus


So the trace form is a symmetric bilinear form. ∎

The symmetricPlanetmathPlanetmath property can be interpreted as a weak form of commutativity of the product: a,bA 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

ab,c=a,bc. (1)

We shall call such an algebra weakly associative though the term is not standard.

This property is clear for all associative algebras as:


When we use a Lie algebra, the trace form is commonly called the Killing formMathworldPlanetmathPlanetmath which has property (1). A result of Koecher shows that Jordan algebrasMathworldPlanetmath also have this property.

Proposition 3.

Given a weakly associative algebra, then the radicalPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of the trace form is an ideal of the algebra.


We know the radical of form R is a subspacePlanetmathPlanetmath so we must simply show that R is an ideal. Given xR and yA then for all zA, xy,z=x,yz=0. Thus xyR. Likewise yzR so R is a two-sided idealMathworldPlanetmath of A. ∎

From this result many authors define an algebra to be semi-simplePlanetmathPlanetmath if its trace form is non-degenerate. In this way, A/R, R the radical of A, 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.

Proposition 4.

Given a weakly associative algebra A, then if I is an ideal of A then so is I.


Given aI, then for all bA and cI, then bcI as I is an ideal and so ab,c=a,bc=0 as aI. This makes abI so I is a right idealMathworldPlanetmath. Likewise c,ba=cb,a=0 so baI and thus I is an ideal of A. ∎

To proceed one factors out the radical so that A is semisimplePlanetmathPlanetmath. Then given an ideal I of A, if II=0 then as the trace form is a non-degenerate bilinearPlanetmathPlanetmath from, A=II, and so by iterating we produce a decomposition of A into minimal ideals:


Hence we arrive at the alternative definition of a semisimple algebra: that the algebra be a direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmath of simple algebras. To obtain the property II=0 it is sufficient to assume A has not ideal I such that I2=0. This is the content of the proof in

Theorem 5.

[1, Thm III.3] Let A be a finite-dimensional weakly associative (trace) semisimple algebra over a field k in which no ideal I0 of A has I2=0, then A is a direct product of minimal ideals, that is, of simple algebras.

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.


  • 1 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
Canonical name TraceFormsOnAlgebras
Date of creation 2013-03-22 16:28:01
Last modified on 2013-03-22 16:28:01
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 4
Author Algeboy (12884)
Entry type Topic
Classification msc 17A01
Defines regular representation
Defines trace form