What Are Tri-Linear Invariants in M-Theory and 3-Algebras?

[tom.stoer]

After some hints regarding M-theory I tried to understand 3-algebras.

I asked this question here some months ago and I asked some other experts (J. Baez was one of them) regarding an E(n) invariant scalar product (n=6, 7, 8); unfortunately nobody had a convincng idea.

In Lubosz's blog I found a hint that in M-theory everything is (magically) related to the number 3; he mentions 3-algebras and exceptional symmetries because of their tri-linear invariants.

Does anybody know what these invariants are and how they can be interpreted?

 

[AndyW]

Hello,

I am glad to see that you are interested in M-theory and 3-algebras. As you mentioned, these are important concepts in understanding the fundamental structure of our universe. Let me try to provide some clarification regarding your question about the E(n) invariant scalar product and the tri-linear invariants of 3-algebras.

In M-theory, the number 3 is indeed a special number as it represents the number of spatial dimensions in our universe. This is why 3-algebras play a crucial role in this theory. A 3-algebra is a mathematical structure that generalizes the concept of a vector space to three dimensions. It has three basis elements, which can be interpreted as the three spatial dimensions.

Now, the tri-linear invariants that you mentioned are related to the exceptional symmetries in M-theory, also known as E(n) symmetries. These symmetries arise from the existence of 3-algebras and are characterized by their tri-linear invariants. These invariants are special mathematical quantities that remain unchanged under certain transformations of the 3-algebra. They can be interpreted as fundamental building blocks of the universe, similar to how elementary particles are the building blocks of matter.

The E(n) invariant scalar product that you mentioned is a special type of inner product that is invariant under E(n) transformations. This means that the value of the scalar product remains the same even if the 3-algebra is transformed using an E(n) symmetry. This scalar product is important in M-theory as it allows us to define a notion of distance and angles in the three-dimensional space described by the 3-algebra.

I hope this helps to answer your question. It is worth noting that M-theory is a highly complex and still developing theory, so there is still much to be discovered and understood about 3-algebras and their role in this theory. But they are definitely a fascinating area of study and I encourage you to continue exploring and learning about them. If you have any further questions, please do not hesitate to ask.