Dimension of an algebra/vector space

Does the dimension of an algebra/vector space have any connection to Euclidian spacial dimensions, for all algebras/vector spaces?

I know some algebras/vector spaces can be represented in Euclidian space.

Thank you

Solution 1:

For all real vector spaces, the dimension is equal to the Euclidean dimension. For all complex vector spaces, the dimension is equal to half the Euclidean dimension.

For algebras with commutative multiplication, we often look at the Krull dimension. In the case of finitely-generated algebras, algebraic geometry gives us a tight relationship between the Krull dimension and the Euclidean dimension of a corresponding affine space (for example, $\mathbb{R}[X,Y]/(XY-1)$ corresponds to the hyperbola $XY=1$).

As for (real) Lie algebras, they can be identified with the tangent space of a Lie group. The dimension equals both the Euclidean dimension of this Lie group, and the Euclidean dimension of the tangent space. Again, when working over $\mathbb{C}$, the Euclidean dimension doubles.

I'm not sure what it means to say that $E_8$ is 248-dimensional, other than as a vector space. We could say that the corresponding Lie group has dimension $248$, but this is mostly just a restatement. Any consistent notion of dimension should give us the same answer here.