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The two compact real form Lie algebras $\mathfrak{so}(16)$ and $\mathfrak{su}(11)$ have the same dimension (120).

They are certainly not isomorphic, but does there exist some kind of algebraic procedure (applying to the generators of these Lie algebras) to go from one to the other?

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    well, I guess the mapping $\phi(f_i)=g_i$ where $\{f_i \}$ is the basis for $so(16)$ and $\{g_i \}$ is the basis for $su(11)$ gives a natural mapping. But, this is just linear algebra. I'm guessing you want something more interesting.2012-10-02
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    The mapping must transform the commutation relations of one Lie algebra to the commutation relations of the other Lie algebra, so it is not trivial at all.2012-10-02
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    but isn't that would be an isomorphism?2012-10-02
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    Well, I use the term "algebraic procedure" which is very vague and in fact this "procedure" could be very complex. So maybe there is a duality, if you want, between the two Lie algebras, but certainly not isomorphism.2012-10-02
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    For instance, from the real form of su(2) Lie algebra, if you multiply 2 generators by the unit complex i, you get an other Lie algebra which is not isomorphic to su(2).2012-10-02
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    This question really lacks a precise definition of *algebraic procedure*.2012-10-02
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    I agree, but this is precisely to keep the door open to a wide range of maps. But the answer to my question is maybe simply no...2012-10-02

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I do know of one strange thing that is kind of like this!

There are two real Clifford algebras corresponding to two different signature choices (+--- and -+++). Both of these signature conventions can be used to do things with Minkowski spacetime.

They are the same dimension, and have complementary signatures... but the two Clifford algebras are nonisomorphic!

However Pertti Lounesto shows in Clifford Algebras and Spinors that there is a "tilting map" that connects the two algebras. I thought I remember scanning it in detail once, but now I can't seem to access that page on googlebooks.

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    Yes, you are right, the clifford algebras $Cl(p,q)$ et $Cl(q,p)$, where $p$ et $q$ are the dimension of space and time, are different, but the even part of these algebras (even product of gamma matrices), which is the spinor representation algebra, is the same. But my question is different.2012-10-03