groups products vs vector space products

Question

I started from wandering if the cross product (a product between two vectors that gives a vector) can be abstracted like dot product (a product between two vectors that gives a scalar) is abstracted in abstract inner spaces in linear algebra.

In abstract algebra (Gallian) a vector space V over a field F is defined as an Abelian group, so it seems that it has to inherit group internal and external products. Groups internal and external products are operations between groups and subgroups, not between elements of them, but in abstract algebra even componentwise operations of addition and multiplication are defined for group and ring elements.

So: - in which relation are groups internal and external products with vector space products? - in which relation are group/ring componentwise operations with dot and inner products in vector spaces?

Answer 1

Regarding the first part about the cross product - the cross product is the Lie bracket in the Lie algebra $\mathfrak{so}(3)$ of the Lie group $SO(3)$. The "group product" of a Lie group $G$ can be tranferred to the "vector space product" of its Lie algebra $\mathfrak{g}$, the Lie bracket. Conversely, the map $\exp: \mathfrak{g}\rightarrow G$ goes back to the Lie group from the Lie algebra. This may be helpful for the phrase "group products vs vector space products" in the title.

Answer 2

So, in general I'm pretty sure the answer is no. First, I just want to make sure you understand that while for most of group theory we write groups as if the binary operation was multiplication, the group operation on a vector space $V$ is the addition operation.

Second, regarding internal and external products of groups, your first recollection is right. Those are operations on the subgroups. In the case of vector spaces, these are just internal and external sums of spaces. So if $U,W \subset V$ are subspaces of $V$, then the internal direct sum is a new subspace $U + W = \{ u+ w | u \in U, w \in W\}$. If $W,V$ are vector spaces, not necessarily one a subset of the other, the external sum is formally simply:

$$V \oplus W = \{(v,w) : v \in V, w \in W\}$$

With component wise operations. These are in fact exactly the constructions you encounter in group theory. However, they do not give us any way to construct a product of two elements. Sometimes we have an additional structure of multiplication that plays nice, in which case we have what's called an algebra.