Trace form

Let $(𝐴, \cdot)$ be an algebra over $𝕂$ and $𝜌 : 𝐴 \to 𝑉$ be a finite-dimensional representation of $𝜌$ . The trace form is the bilinear form on $𝐴$ defined by¹ #m/def/ralg

⟨ 𝑥, 𝑦 ⟩ 𝜌 = t r 𝜌 (𝑥, 𝑦) = t r (𝜌 (𝑥) 𝜌 (𝑦))

for $𝑥, 𝑦 \in 𝐴$ .

#state/tidy | #lang/en | #SemBr

Note the product of the algebra might not be represented as composition in the representation, e.g. for a Lie algebra representation. ↩

Trace form

Footnotes