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The exceptional complex simple Lie algebra $F_4$ has an irreducible 26-dimensional representation $V$ with Dynkin label [0,0,0,1] in the usual ordering of the simple roots one can find, say, in Humphreys's book on Lie algebras and representation theory. In fact, $F_4$ can be defined as the Lie subalgebra of $\mathfrak{sl}(V)$ which preserves a symmetric inner product and a certain cubic form on $V$.

Now there are three different real forms of $F_4$ and my question is about what happens to $V$ when restricting to these real forms. The three real forms are the compact real form, the split real form and a third form. They can be distinguished by the 'index' of the Killing form; i.e., if the Killing form $\kappa(X,Y) = \operatorname{Tr} \operatorname{ad}_X \operatorname{ad}_Y$ has signature $(p,q)$, its index is $p-q$. I am most familiar with the compact real form, for which the Killing form is negative-definite, whence of index $-52$. The split real form has index $4$ and the third real form has index $-20$, and are denoted $F_4^4$ and $F_4^{-20}$, respectively.

I would like to know the following (pointers to the literature would also be greatly appreciated):

Questions

What is the type of $V$ under the different real forms? I know that for the compact real form it is real, but I would like to know also for $F_4^{-20}$ and $F_4^4$.

 

And if the type is real (as I suspect is the case), what is the signature of the invariant inner product on the underlying real representation $V_{\mathbb{R}}$?

Thanks in advance!

Edit

Based on Jim's answer below, the representations are of real type in all cases. From Bruce's answer, it would seem that for the split case $F_4^4$ the signature is (14,12).

In a rather convoluted calculation, I seem to find that for $F_4^{-20}$ the signature is (16,10), but I would like confirmation since I have seen at least one claim in the Physics literature (see last equation in §4) that it is (25,1).

The exceptional complex simple Lie algebra $F_4$ has an irreducible 26-dimensional representation $V$ with Dynkin label [0,0,0,1] in the usual ordering of the simple roots one can find, say, in Humphreys's book on Lie algebras and representation theory. In fact, $F_4$ can be defined as the Lie subalgebra of $\mathfrak{sl}(V)$ which preserves a symmetric inner product and a certain cubic form on $V$.

Now there are three different real forms of $F_4$ and my question is about what happens to $V$ when restricting to these real forms. The three real forms are the compact real form, the split real form and a third form. They can be distinguished by the 'index' of the Killing form; i.e., if the Killing form $\kappa(X,Y) = \operatorname{Tr} \operatorname{ad}_X \operatorname{ad}_Y$ has signature $(p,q)$, its index is $p-q$. I am most familiar with the compact real form, for which the Killing form is negative-definite, whence of index $-52$. The split real form has index $4$ and the third real form has index $-20$, and are denoted $F_4^4$ and $F_4^{-20}$, respectively.

I would like to know the following (pointers to the literature would also be greatly appreciated):

Questions

What is the type of $V$ under the different real forms? I know that for the compact real form it is real, but I would like to know also for $F_4^{-20}$ and $F_4^4$.

And if the type is real (as I suspect is the case), what is the signature of the invariant inner product on the underlying real representation $V_{\mathbb{R}}$?

Thanks in advance!

Edit

Based on Jim's answer below, the representations are of real type in all cases. From Bruce's answer, it would seem that for the split case $F_4^4$ the signature is (14,12).

In a rather convoluted calculation, I seem to find that for $F_4^{-20}$ the signature is (16,10), but I would like confirmation since I have seen at least one claim in the Physics literature (see last equation in §4) that it is (25,1).

The exceptional complex simple Lie algebra $F_4$ has an irreducible 26-dimensional representation $V$ with Dynkin label [0,0,0,1] in the usual ordering of the simple roots one can find, say, in Humphreys's book on Lie algebras and representation theory. In fact, $F_4$ can be defined as the Lie subalgebra of $\mathfrak{sl}(V)$ which preserves a symmetric inner product and a certain cubic form on $V$.

Now there are three different real forms of $F_4$ and my question is about what happens to $V$ when restricting to these real forms. The three real forms are the compact real form, the split real form and a third form. They can be distinguished by the 'index' of the Killing form; i.e., if the Killing form $\kappa(X,Y) = \operatorname{Tr} \operatorname{ad}_X \operatorname{ad}_Y$ has signature $(p,q)$, its index is $p-q$. I am most familiar with the compact real form, for which the Killing form is negative-definite, whence of index $-52$. The split real form has index $4$ and the third real form has index $-20$, and are denoted $F_4^4$ and $F_4^{-20}$, respectively.

I would like to know the following (pointers to the literature would also be greatly appreciated):

Questions

What is the type of $V$ under the different real forms? I know that for the compact real form it is real, but I would like to know also for $F_4^{-20}$ and $F_4^4$.

And if the type is real (as I suspect is the case), what is the signature of the invariant inner product on the underlying real representation $V_{\mathbb{R}}$?

Thanks in advance!

The exceptional complex simple Lie algebra $F_4$ has an irreducible 26-dimensional representation $V$ with Dynkin label [0,0,0,1] in the usual ordering of the simple roots one can find, say, in Humphreys's book on Lie algebras and representation theory. In fact, $F_4$ can be defined as the Lie subalgebra of $\mathfrak{sl}(V)$ which preserves a symmetric inner product and a certain cubic form on $V$.

Now there are three different real forms of $F_4$ and my question is about what happens to $V$ when restricting to these real forms. The three real forms are the compact real form, the split real form and a third form. They can be distinguished by the 'index' of the Killing form; i.e., if the Killing form $\kappa(X,Y) = \operatorname{Tr} \operatorname{ad}_X \operatorname{ad}_Y$ has signature $(p,q)$, its index is $p-q$. I am most familiar with the compact real form, for which the Killing form is negative-definite, whence of index $-52$. The split real form has index $4$ and the third real form has index $-20$, and are denoted $F_4^4$ and $F_4^{-20}$, respectively.

I would like to know the following (pointers to the literature would also be greatly appreciated):

Questions

What is the type of $V$ under the different real forms? I know that for the compact real form it is real, but I would like to know also for $F_4^{-20}$ and $F_4^4$.

And if the type is real (as I suspect is the case), what is the signature of the invariant inner product on the underlying real representation $V_{\mathbb{R}}$?

Thanks in advance!

Edit

Based on Jim's answer below, the representations are of real type in all cases. From Bruce's answer, it would seem that for the split case $F_4^4$ the signature is (14,12).

In a rather convoluted calculation, I seem to find that for $F_4^{-20}$ the signature is (16,10), but I would like confirmation since I have seen at least one claim in the Physics literature (see last equation in §4) that it is (25,1).

The exceptional complex simple Lie algebra $F_4$ has an irreducible 26-dimensional representation $V$ with Dynkin label [0,0,0,1] in the usual ordering of the simple roots one can find, say, in Humphreys's book on Lie algebras and representation theory. In fact, $F_4$ can be defined as the Lie subalgebra of $\mathfrak{sl}(V)$ which preserves a symmetric inner product and a certain cubic form on $V$.

Now there are three different real forms of $F_4$ and my question is about what happens to $V$ when restricting to these real forms. The three real forms are the compact real form, the split real form and a third form. They can be distinguished by the 'index' of the Killing form; i.e., if the Killing form $\kappa(X,Y) = \mathrm{Tr} \mathrm{ad}_X \mathrm{ad}_Y$ has signature $(p,q)$, its index is $p-q$. I am most familiar with the compact real form, for which the Killing form is negative-definite, whence of index $-52$. The split real form has index $4$ and the third real form has index $-20$, and are denoted $F_4^4$ and $F_4^{-20}$, respectively.

I would like to know the following (pointers to the literature would also be greatly appreciated):

Questions

What is the type of $V$ under the different real forms? I know that for the compact real form it is real, but I would like to know also for $F_4^{-20}$ and $F_4^4$.

And if the type is real (as I suspect is the case), what is the signature of the invariant inner product on the underlying real representation $V_{\mathbb{R}}$?

Thanks in advance!

The exceptional complex simple Lie algebra $F_4$ has an irreducible 26-dimensional representation $V$ with Dynkin label [0,0,0,1] in the usual ordering of the simple roots one can find, say, in Humphreys's book on Lie algebras and representation theory. In fact, $F_4$ can be defined as the Lie subalgebra of $\mathfrak{sl}(V)$ which preserves a symmetric inner product and a certain cubic form on $V$.

Now there are three different real forms of $F_4$ and my question is about what happens to $V$ when restricting to these real forms. The three real forms are the compact real form, the split real form and a third form. They can be distinguished by the 'index' of the Killing form; i.e., if the Killing form $\kappa(X,Y) = \operatorname{Tr} \operatorname{ad}_X \operatorname{ad}_Y$ has signature $(p,q)$, its index is $p-q$. I am most familiar with the compact real form, for which the Killing form is negative-definite, whence of index $-52$. The split real form has index $4$ and the third real form has index $-20$, and are denoted $F_4^4$ and $F_4^{-20}$, respectively.

I would like to know the following (pointers to the literature would also be greatly appreciated):

Questions

What is the type of $V$ under the different real forms? I know that for the compact real form it is real, but I would like to know also for $F_4^{-20}$ and $F_4^4$.

And if the type is real (as I suspect is the case), what is the signature of the invariant inner product on the underlying real representation $V_{\mathbb{R}}$?

Thanks in advance!