# Explicit isomorphism between two Jordan algebras

I have seen written that the space of $$m\times m$$-complex matrices $$M_m(\mathbb C)$$ endowed with the usual Jordan product is isomorphic, as a Jordan algebra to the complexification of the space $$Herm(m,\mathbb C)$$ of hermitian $$m\times m$$ matrices.
Is it possible (easy) to write down an explicit isomorphism between the two?

For example, for the $$2\times 2$$ matrices, the elements of $$Herm(2,\mathbb C)\otimes_{\mathbb R}\mathbb C$$ are of the form $$\left(\begin{smallmatrix}\alpha &\lambda + I\mu\\\lambda-I\mu &\beta\end{smallmatrix}\right)$$ with $$I=i\otimes_{\mathbb R}1$$ and $$\alpha,\beta,\lambda,\mu\in \mathbb C$$ and the elements of $$M_2(\mathbb C)$$ are of the form $$\left(\begin{smallmatrix}\alpha &c\\d &b\end{smallmatrix}\right)$$ with $$a,b,c,d\in \mathbb C$$. The question is: is there a nice expression of $$a,b,c,d$$ in terms of $$\alpha,\beta,\lambda,\mu$$?

• Try the canonical map induced by inclusion. – YCor Mar 1 at 6:15
• @Ycor: Thank you for your answer but I cannot see it: (complexified) hermitian matrices are of the form (for example $2\times 2$ $\left(\begin{smallmatrix}\alpha &a+Ib \\a-Ib &\beta\end{smallmatrix}\right)$ where $I$ stands for $i\otimes_{\mathbb R} \mathbb C$ and $\alpha,\beta,a,b\in \mathbb C$ but the $2\times 2$-matrices are just of the form $\left(\begin{smallmatrix}m &n \\o &p\end{smallmatrix}\right)$. What I would like to see is an expression of $m,n,o,p$ in terms of $\alpha,\beta, a,b$. – pi_1 Mar 1 at 12:28
• @dodd: They have the same dimension over $\mathbb C$ ($Herm(m,\mathbb C)$ is complexified) $dim_{\mathbb C}(M_m(\mathbb C))=m^2$ and for $dim(Herm(m,\mathbb C)\otimes_{\mathbb R}\mathbb C$, there the $m$ complex numbers on the diagonal and the $\frac{m(m-1)}{2}$ pair of complex numbers outside the daigonal (see for example p.158 of Analysis on symmetric cones by Faraut and Koranyi) – pi_1 Mar 1 at 12:33
• View an $n$-dimensional algebra as an $n^3$-tuple of numbers, namely coefficients of the law in a basis [and isomorphism means equivalence up to a suitable rule basis change]. The space $M_m(C)$ has a $C$-basis consisting of Hermitian matrices, in which the structure constants are real (and even integral). This basis is also a basis of the real algebra of Hermitian matrices, and the coefficients are just the same. This is all what you need. – YCor Mar 1 at 13:12
• In concrete terms, take $c = \lambda + i \mu$ and $d = \lambda- i \mu$. – Will Sawin Mar 3 at 21:53