Let the linear conformal transformations (homotetics and orthogonal transformation) act on the real plane (x,y) by the real matrix
a b
-b a
If such a matrix acts on a pair of planes, we will talk about the pair conformal mapping. In the case where there is both a pair conformal map and a conformal map in each plane, we obtain an algebra isomorphic to the algebra of quaternions from these maps.
Similarly, if such a matrix acts on a pair of pairs of planes, then we will talk about a two-pair conformal map. Then the algebra of octonions is interpreted by us as the algebra of simultaneously two-pair, pair and simply conformal maps.
It is now clear that the Lie algebras of the octonion algebra g_2 is simply a 14-dimensional algebra of pair rotations of an 8-dimensional Euclidean space.
It would be possible to write out here and generators of this algebra, but we will not clutter the screen. However, if you wish, I will show the generators of another exceptional algebra e_8 that perform rotation in a 16-dimensional space with Euclidean and neutral metric.
The generators of e_8 Lie algebra:
28 type generators
(1_{i j } - 1_{j i}) (x_1,...,x_16)[\partial x_1,...,\partial x_16]
where i,j = 1,...,8
28 type generators
(1_{i+8 j+8} - 1_{j+8 i+8})(x_1,...,x_16)[\partial x_1,...,\partial x_16]
where i,j = 1,...,8
64 type generators
(1_{i j+8} - 1_{j+8 i})(x_1,...,x_16)[\partial x_1,...,\partial x_16]
where i,j = 1,...,8
64 type generators
(1_{i j+8} + 1_{j+8 i})(x_1,...,x_16)[\partial x_1,...,\partial x_16]
where i,j = 1,...,8
64 type generators
(1_{i i} - 1_{j+8 j+8})(x_1,...,x_16)[\partial x_1,...,\partial x_16]
where i,j = 1,...,8
Total: 28 + 28 + 64 + 64 +64 = 248
where 28 + 28 + 64 = 120 and 64 + 64 = 128
The generators of g_2 Lie algebras:
(1_{i j} - 1_{j i} + 1_{i+4 j+4} - 1_{j+4 i+4}) (x_1,...,x_8)[\partial x_1,...,\partial x_8]
where i,j =1,...,8 and i+4 = (i+4) mod 8
Let me explain. This means that the generators of the algebra, we use a vector field lying on a torus and is closed in Villarceau circle. In this case, the torus itself lies in an 8-dimensional Euclidean space.
As for the generators of e_8, they all tangent to the torus S7 x S7 (which is the intersection of the sphere of 16-dimensional Euclidean space and the hypersphere of space with a neutral metric) and rotate in 16-dimensiodal Euclidean space.
The question is that these algebras are interesting to physicists, and if I am not mistaken, there is an alternative not only in mathematics but also in elementary particle physics
I just noticed now.
Are your 64 type generators commutative ?
(1_{i i} - 1_{j+8 j+8})(x_1,...,x_16)[\partial x_1,...,\partial x_16]
If so they're not e8.
Igor,
Don't mind, your idea was interesting, but if it was right E8 atlas project would have been simply melted down. (Corresponding my one is p.20 in my "An E8 Primer". I hope it will be sufficiently simple for your purpose of physics.)
Next time, do think TWICE and TWICE and one more TWICE !
Shuichiro
@Shuichiro, And yet, where does it follow that in this basis the structural constants of all possible pairs of these 64 generators should not be zero?
Just think TWICE !!
There is no E64 ( you know Cartan subalgebras ).
Shuichiro Teramachi , Thanks. Understood this so, that in e_8 there is no E64.
Can't it be that it's there, but we can't see it? For example, you see the subalgebra g_2 in the algebra given by my generators?
Here is another option. Since 142+52 = 248, one would expect that (g_2 x g_2) + f_4 = e_8, and try to verify this on the algebra presented here.
P.S. True, I have not thought about it twice.
Since the Lie algebra g_2 was described earlier and g_2 x g_2 is generated by pairs of generators of the upper (x_1,...,x_8) and lower (x_9,...,x_16) half-spaces, then it remains to determine Lie algebra f_4.
The 52 generators of Lie algebra f_4:
36 generators
1_{i j} - 1_{j i}
where i,j = 1,...,9
8 generators
1_{i 9} + 1_{9 i}
where i = 1,...,8
and 8 generators
1_{i i} - 1_{9 9}
where i = 1,...,8
And now we have another representation of exceptional algebra e_8 = (g_2 x g_2) + f_4 . In my opinion it is no different from what we had before.
@Shuichiro Teramachi, such a coincidence can not be just an accident.
e8 is a simple Lie algebra, while (g_2 x g_2) + f_4 is not.
Did you really think twice ?
@Shuichiro, You think very straightforwardly. In this case, the plus (+) does not mean the direct sum, but is used to indicate the need to add 52 algebra generators f_4 to construct the algebra e_8.
Sorry, then 8 generators 1_{i i} - 1_{9 9} where i = 1,...,8
are already commutative, and g_2 x g_2 has 4 more commutative one.
It's hopeless, there is no E12.
Of course, I know bad (you can tell do not know) the structure of the algebra e_8, but I don't understand what's wrong diagonal generators. Let's take a closer look at them. I understand it as follows. The generator
1_{12} - 1_{21}
generates an algebra of complex numbers and a Lie algebra tangent to the Euclidean circle
x^2 + y^2 = const
The generator
1_{12} + 1_{21}
generates an algebra of double numbers and a Lie algebra tangent to a pseudo-Euclidean circle
x^2 - y^2 = const
In turn, the generator
1_{11} - 1_{22}
generates an algebra of dual numbers and a Lie algebra tangent to a pseudo-Euclidean circle in an isotropic basis
xy = const
In the multidimensional case, it is easy to notice that the commutator of two arbitrary diagonal generators is always zero. What's wrong with these generators?
> What's wrong with these generators?
The most wrong is your ignorance of Cartan subalgebras
(a basic of basics of Lie algebras).
And I guess you have serious misunderstanding in your g_2 x g_2.
Are you confusing the following cases ?
Case1 x : tensor product , Case2 x : direct product.
Case1
dim = 248 is OK, but you lose Lie brackets between g_2 x g_2 and f4,
then your fake e8 becomes not simple Lie algebra and has 12 commutative generators, so it's not e8.
Case2
Certainly it has at most 8 commutative generators, but in this case
dim = (14+14) + 52 < 248, it's hopeless.
Further, their is no F8 nor G4, hence all of your constructions faded away like castle of sand.
As far as I understand in simple algebra there should not be a commutative subalgebra. Thank you for the lesson, I will think how to correct the error.
>
Case1
dim = 248 is OK, but you lose Lie brackets between g_2 x g_2 and f4
You are right, the key here -- f_4 should be expanded and rid of the Cartan subalgebra.
That's ok.
Before that, I recommend you to read some basic text book of Lie algebras thoroughly, though I don't know which one is suitable for you.
You seem to have me completely confused. On the one hand, you argue that there should not be Cartan subalgebras in a simple algebra, and on the other hand, simple algebras are just classified (as I understand from public sources) by their Cartan subalgebras.
I'm not saying such that.
What you say is simply that you are not understanding even the definition of e8.
Found an elementary introduction
[MK]
https://docs.wixstatic.com/ugd/ad9bab_687a860fe5c347dd95902adc5b974361.pdf
written about Cartan subalgebras of a1=su2 and a2=su3, see Ch7 (§7.3).
After that you can read Ch8 & Ch20 of a famous book.
[G]
http://mural.uv.es/rusanra/Lie%20Algebras%20in%20Particle%20Physics%202ª%20ed%20-%20From%20Isospin%20to%20Unified%20Theories%20(Georgi,%201999).pdf
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