Example Of An Algebraic $F$-group $G$ With Non-centralizing Maximal Torus.

Introduction

In the realm of algebraic groups, a maximal torus is a subtorus that is maximal with respect to inclusion. A centralizing maximal torus is one where the centralizer of the torus is the torus itself. However, there are cases where the centralizer of a maximal torus is not the torus itself, leading to a non-centralizing maximal torus. In this article, we will explore an example of an algebraic $F$-group $G$ with a non-centralizing maximal torus $T$, where $\text{Cent}_{G}(T) \neq T$ holds.

Background

To understand the concept of a non-centralizing maximal torus, we need to delve into the basics of algebraic groups. An algebraic group is a group that is also an algebraic variety, with the group operation and inversion being morphisms of varieties. A reductive group is a connected algebraic group with a maximal torus whose centralizer is the group itself. However, not all algebraic groups are reductive, and in this case, we are interested in a non-reductive group.

Non-Reductive Groups

A non-reductive group is an algebraic group that is not reductive. This means that the group has a maximal torus whose centralizer is not the group itself. In other words, there exists a maximal torus $T$ such that $\text{Cent}_{G}(T) \neq G$. This is the key property that we need to construct an example of an algebraic $F$-group $G$ with a non-centralizing maximal torus.

Example

Let $G$ be the group of upper triangular matrices with entries in a field $F$. Specifically, let $G = \begin{pmatrix} F & F & F \\ 0 & F & F \\ 0 & 0 & F \end{pmatrix}$. This group is not reductive because it has a maximal torus whose centralizer is not the group itself.

To see this, let $T$ be the subgroup of diagonal matrices in $G$. Then $T$ is a maximal torus in $G$. However, the centralizer of $T$ in $G$ is not $G$ itself. In fact, the centralizer of $T$ is the subgroup of upper triangular matrices with zeros on the diagonal. This is a proper subgroup of $G$, and therefore, $\text{Cent}_{G}(T) \neq G$.

Properties of the Example

The example we constructed has several interesting properties. First, it shows that there exists an algebraic $F$-group $G$ with a non-centralizing maximal torus $T$. This is a non-trivial result, as it requires the group to be non-reductive.

Second, the example shows that the centralizer of a maximal torus can be a proper subgroup of the group itself. This is a key property that distinguishes non-reductive groups from reductive groups.

Conclusion

In this article, we constructed an example of an algebraic $F$-group $G$ with a non-centralizing maximal torus $T$. The group $G$ is a non-reductive group, and the maximal torus $T$ is a subgroup of diagonal matrices. The centralizer of $T$ in $G$ is a proper subgroup of $G$, and therefore, $\text{Cent}_{G}(T) \neq G$.

This example has several interesting properties, including the fact that it shows the existence of an algebraic $F$-group $G$ with a non-centralizing maximal torus $T$. It also shows that the centralizer of a maximal torus can be a proper subgroup of the group itself.

Future Directions

There are several directions that this research can take. One possible direction is to study the properties of non-reductive groups in more detail. This could include studying the structure of the centralizer of a maximal torus, as well as the properties of the group itself.

Another possible direction is to study the relationship between non-reductive groups and reductive groups. This could include studying the properties of the group that distinguishes it from being reductive, as well as the properties of the maximal torus that is centralizing.

References

• [1] Borel, A. (1968). Linear algebraic groups. Benjamin.
• [2] Humphreys, J. E. (1995). Linear algebraic groups. Springer-Verlag.
• [3] Springer, T. A. (1977). Linear algebraic groups. Birkhäuser.

Introduction

In our previous article, we explored an example of an algebraic $F$-group $G$ with a non-centralizing maximal torus $T$. In this article, we will answer some of the most frequently asked questions about this topic.

Q: What is a non-centralizing maximal torus?

A: A non-centralizing maximal torus is a maximal torus $T$ in an algebraic group $G$ such that the centralizer of $T$ in $G$ is not the group itself. In other words, there exists an element $g \in G$ such that $g \notin \text{Cent}_{G}(T)$.

Q: What is the difference between a centralizing and non-centralizing maximal torus?

A: A centralizing maximal torus is a maximal torus $T$ in an algebraic group $G$ such that the centralizer of $T$ in $G$ is the group itself. In other words, for any element $g \in G$, we have $g \in \text{Cent}_{G}(T)$. A non-centralizing maximal torus, on the other hand, is a maximal torus $T$ in an algebraic group $G$ such that the centralizer of $T$ in $G$ is not the group itself.

Q: What is an example of an algebraic $F$-group $G$ with a non-centralizing maximal torus $T$?

A: One example of an algebraic $F$-group $G$ with a non-centralizing maximal torus $T$ is the group of upper triangular matrices with entries in a field $F$. Specifically, let $G = \begin{pmatrix} F & F & F \\ 0 & F & F \\ 0 & 0 & F \end{pmatrix}$. This group is not reductive because it has a maximal torus whose centralizer is not the group itself.

Q: What are the properties of the centralizer of a maximal torus?

A: The centralizer of a maximal torus $T$ in an algebraic group $G$ is a subgroup of $G$ that contains $T$. In fact, the centralizer of $T$ is the largest subgroup of $G$ that contains $T$ and is stable under the action of $T$.

Q: What is the relationship between non-reductive groups and reductive groups?

A: Non-reductive groups are algebraic groups that are not reductive. A reductive group is a connected algebraic group with a maximal torus whose centralizer is the group itself. In other words, a reductive group is a group that has a maximal torus whose centralizer is the group itself.

Q: What are some of the key properties of non-reductive groups?

A: Some of the key properties of non-reductive groups include:

• They have a maximal torus whose centralizer is not the group itself.
• They have a subgroup that is stable under the action of the maximal torus.
• They have a subgroup that is not stable under the action of the maximal torus.

Q: What are some of the key properties of reductive groups?

A: Some of the key properties of reductive groups include:

• They have a maximal torus whose centralizer is the group itself.
• They have a subgroup that is stable under the action of the maximal torus.
• They have a subgroup that is not stable under the action of the maximal torus.

Conclusion

In this article, we answered some of the most frequently asked questions about algebraic $F$-groups with non-centralizing maximal torus. We hope that this article has provided a clear and concise overview of this topic and has helped to clarify some of the key concepts.

References

• [1] Borel, A. (1968). Linear algebraic groups. Benjamin.
• [2] Humphreys, J. E. (1995). Linear algebraic groups. Springer-Verlag.
• [3] Springer, T. A. (1977). Linear algebraic groups. Birkhäuser.

Note: The references provided are a selection of the most relevant and influential works in the field of algebraic groups. They provide a comprehensive introduction to the subject and are a good starting point for further reading.