Subalgebras and direct sums

  • A
  • Thread starter mnb96
  • Start date
  • Tags
    Sums
In summary, the Lie algebra ##\mathfrak{sl}(2)## with Lie multiplication is a counterexample to the statement that the product of two subspaces is a subalgebra of the product.
  • #1
715
5
Hi,

consider a (finite dimensional) vector space ##V=U\oplus W##, where the subspaces ##U## and ##V## are not necessarily orthogonal, equipped with a bilinear product ##*:V\times V \rightarrow V##.

The subspace ##U## is closed under multiplication ##*##, thus ##U## is a subalgebra of ##V##.

Does this imply that also ##W## is a subalgebra of ##V##?

(Note, I can already prove the special case that if U and W are orthogonal, then both U and W are indeed subalgebras).
 
Physics news on Phys.org
  • #2
mnb96 said:
Hi,

consider a (finite dimensional) vector space ##V=U\oplus W##, where the subspaces ##U## and ##V## are not necessarily orthogonal, equipped with a bilinear product ##*:V\times V \rightarrow V##.

The subspace ##U## is closed under multiplication ##*##, thus ##U## is a subalgebra of ##V##.

Does this imply that also ##W## is a subalgebra of ##V##?

(Note, I can already prove that if U and W are orthogonal, then both U and W are indeed subalgebras, but I am interested in the general case).
Just as a side note: orthogonal doesn't make sense, as long as you don't specify the quadratic form and the field. Vector spaces in general don't automatically allow inner products.

The answer to your question is no. Example:
##h=\begin{bmatrix}1&0\\0&-1\end{bmatrix}\; , \;x=\begin{bmatrix}0&1\\0&0\end{bmatrix}\; , \;y=\begin{bmatrix}0&0\\1&0\end{bmatrix}## with ##V=\operatorname{span}_\mathbb{F}\{\,h,x,y\,\}\; , \;U=\mathbb{F}\cdot h\; , \; W=\operatorname{span}_\mathbb{F}\{\,x,y\,\}##. With the multiplication ##v*w= v\cdot w - w \cdot v## we have ##h*h=0\in U## and ##x*y=h \notin W##.

Edit: Typo corrected. ##h_{21}=0## not ##1##.
 
Last edited:
  • #3
Hi fresh_42,

you gave a very interesting counterexample of my statement that is actually too inspiring to close the discussion here :)

In fact, let's define the "product of two subspaces" as ##UV=\left \{uv\;|\; u\in U, \, v\in V \right \}##, and notice that in your construction ##H^2=0##. In other words, ##H## (as a set) acted as a nilpotent w.r.t. the product.

I am wondering if it is possible to find a similar counterexample, in which ##H^2=H##, i.e. the closure of ##H## w.r.t. the product is ##H## itself.
 
  • #4
mnb96 said:
Hi fresh_42,

you gave a very interesting counterexample of my statement that is actually too inspiring to close the discussion here :)
It is the Lie algebra ##\mathfrak{sl}(2)## with ##.*. =[.,.]## as Lie multiplication.
In fact, let's define the "product of two subspaces" as ##UV=\left \{uv\;|\; u\in U, \, v\in V \right \}##, and notice that in your construction ##H^2=0##. In other words, ##H## (as a set) acted as a nilpotent w.r.t. the product.
What does "as a set" mean? ##H=h\cdot \mathbb{F}\,##? That's the heritage of the Lie algebra structure, where ##[X,X]=X*X=0## holds for any element.
I am wondering if it is possible to find a similar counterexample, in which ##H^2=H##, i.e. the closure of ##H## w.r.t. the product is ##H## itself.
We don't have any restrictions for the multiplication. So we can simply define a multiplication by ##A^2=A## and leave all other as they are: ##H*X=2X, H*Y=-2Y,X*Y=H##. I don't see any obvious reasons, why this shouldn't work. However, to find a realization by matrices or similar could take a moment, at least if we don't want to use the tensor algebra and its universal property. I would look among genetic algebras for an example.
 

1. What is a subalgebra?

A subalgebra is a subset of a larger algebra that forms a smaller algebra itself. It contains all the operations and elements of the larger algebra, but with a restricted domain.

2. How is a subalgebra different from a subgroup?

A subgroup is a subset of a group that is closed under the group operation. In contrast, a subalgebra is a subset of an algebra that is closed under the algebra's operations.

3. What is a direct sum?

A direct sum is a mathematical operation that combines two algebraic structures, such as groups or vector spaces, into a single larger structure. It is denoted by a plus sign with a circle around it (⊕).

4. How are direct sums related to subalgebras?

A subalgebra of a direct sum is a subset of the direct sum that is itself a direct sum of substructures. In other words, the subalgebra is a smaller version of the direct sum that retains the same structure and operations.

5. Can a subalgebra be a direct sum?

Yes, a subalgebra can be a direct sum if it satisfies the necessary conditions. For example, if a subalgebra A is a subset of a direct sum B, and A and B share the same operations and elements, then A can be considered a direct sum of smaller substructures.

Similar threads

  • Linear and Abstract Algebra
Replies
7
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
934
  • Linear and Abstract Algebra
Replies
19
Views
4K
  • Linear and Abstract Algebra
Replies
7
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
917
  • Linear and Abstract Algebra
Replies
2
Views
797
  • Linear and Abstract Algebra
Replies
3
Views
871
  • Linear and Abstract Algebra
Replies
3
Views
2K
  • Linear and Abstract Algebra
Replies
32
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
1K
Back
Top