I am new to the Frobenius Algebra course. One of my textbook exercises ask to prove:
A matrix ring over $\mathbb{C}$ is also a Frobenius algebra over $\mathbb{R}$
My attempt:
I know that I can give a Frobenius form (i.e, nullspace of $M_{n\times n}(\mathbb{C})$ contains no nontrivial left ideal) $ \epsilon_{1}: M_{n\times n}(\mathbb{C}) \to \mathbb{C}$ by trace map, but I can't see why it is a Frobenius using definition that nullspace of $M_{n\times n}(\mathbb{C})$ contains no nontrivial left ideal.
Further, I can give a Frobenius form, $\epsilon_{2}:\mathbb{C} \to \mathbb{R}$ by sending $a+ib$ to $a$. this is trivially a Frobenius form as field $\mathbb{C}$ have no nontrivial ideals.
Actual Question:
Now in order to show that there is a Frobenius form $M_{n\times n}(\mathbb{C}) \to \mathbb{R}$, I want to show $\epsilon_{1} \circ \epsilon_{2}$ is a required Frobenius form.
Any solution/reference will be a great help!!
Further could you please refer to some elementary book on Frobenius algebra other than a book by "Joachim Kock" (which is not elementary for me, it skips a lot of things)
Firstly let's show that $M_{n\times n}(\mathbb{C})$ is a Frobenius algebra over $\mathbb{C}$.
Any non-trivial left ideal will contain a non-zero matrix $b\in M_{n\times n}(\mathbb{C})$. We need to find a matrix $a\in M_{n\times n}(\mathbb{C})$ such that ${\rm tr}(ab)\neq 0$, thus proving that the left ideal does not lie in the nullspace of tr.
Multiplication by a matrix on the left corresponds to row operations. As $b\neq 0$ we know some entry $b_{ij}\neq 0$. So pick $a$ so that left multiplication by $a$ takes the $i^{\rm th}$ row to the $j^{\rm th}$ row, and makes every other row $0$. Explicitly $a_{ji}=1$ and every other entry of $a$ is zero.
Then ${\rm tr}(ab)=b_{ij}\neq 0$ as required.
Now consider the form coming from the map $x\mapsto {\rm Re(tr(}x))$. We repeat the above argument, now taking $a_{ji}=\overline{b_{ij}}$, and all other entries of $a$ equal to $0$. Then ${\rm tr}(ab)=|b_{ij}|^2\neq 0$, as required.