# Subalgebras and direct sums

• A
• mnb96
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.

#### mnb96

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).

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.

##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:
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.

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.

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