[[Algebra theory MOC]]
# Trace form

Let $(A, \cdot)$ be an [[K-algebra|algebra]] over $\mathbb{K}$ and $\rho : A \to V$ be a finite-dimensional [[representation]] of $\rho$.
The **trace form** is the [[bilinear form]] on $A$ defined by[^not] #m/def/falg 
$$
\begin{align*}
\langle x,y \rangle _{\rho} = \Tr_{\rho}(x, y) = \Tr(\rho(x)\rho(y))
\end{align*}
$$
for $x,y \in A$.

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

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