The Relationship Between PGL2 and PSL2

Oct 13, 2025 by miodeya.com 63 views

Introduction

The study of rational homology is a crucial aspect of algebraic topology, providing valuable insights into the structure and properties of topological spaces. In this article, we will delve into the rational homology of two specific groups: PGL2 and PSL2. These groups are fundamental objects of study in Lie theory and algebraic number theory, and their homology has been extensively explored in the literature. Our goal is to describe the rational homology of PGL2 in terms of the rational homology of PSL2.

Background and Motivation

The groups PGL2 and PSL2 are defined over a field F, which can be either the real numbers, the complex numbers, or a finite field. The group PGL2 is the projective general linear group of degree 2, consisting of all invertible 2x2 matrices over F, modulo the subgroup of scalar matrices. On the other hand, PSL2 is the projective special linear group of degree 2, consisting of all 2x2 matrices over F with determinant 1, modulo the subgroup of scalar matrices.

The rational homology of a group G is defined as the homology of the chain complex of rational-valued functions on G. In other words, it is the homology of the complex:

C_n(G, \mathbb{Q}) = \mathrm{Fun}(G, \mathbb{Q}) \xrightarrow{\partial_n} C_{n-1}(G, \mathbb{Q})

where $\partial_n$ is the boundary operator.

The Relationship Between PGL2 and PSL2

The relationship between PGL2 and PSL2 is given by the short exact sequence:

1 \to \mathbb{Z}/2\mathbb{Z} \to \mathrm{PSL}_2(F) \to \mathrm{PGL}_2(F) \to 1

where the first map is the inclusion of the subgroup of scalar matrices with determinant 1, and the second map is the quotient map.

This sequence induces a long exact sequence in homology:

\cdots \to H_n(\mathrm{PSL}_2(F), \mathbb{Q}) \to H_n(\mathrm{PGL}_2(F), \mathbb{Q}) \to H_n(\mathbb{Z}/2\mathbb{Z}, \mathbb{Q}) \to \cdots

Our goal is to describe the rational homology of PGL2 in terms of the rational homology of PSL2.

The Rational Homology of PGL2

To compute the rational homology of PGL2, we can use the long exact sequence in homology induced by the short exact sequence above. Specifically, we have:

\cdots \to H_n(\mathrm{PSL}_2(F), \mathbb{Q}) \to H_n(\mathrm{PGL}_2(F), \mathbb{Q}) \to H_n(\mathbb{Z}/2\mathbb{Z}, \mathbb{Q}) \to \cdots

Using the fact that the rational homology of $\mathbb{Z}/2\mathbb{Z}$ is given by:

H_n(\mathbb{Z}/2\mathbb{Z}, \mathbb{Q}) = \begin{cases} \mathbb{Q} & \text{if } n = 0 \\ 0 & \text{if } n > 0 \end{cases}

we can deduce that:

H_n(\mathrm{PGL}_2(F), \mathbb{Q}) = \begin{cases} H_n(\mathrm{PSL}_2(F), \mathbb{Q}) & \text{if } n > 0 \\ H_0(\mathrm{PSL}_2(F), \mathbb{Q}) \oplus \mathbb{Q} & \text{if } n = 0 \end{cases}

Conclusion

In this article, we have described the rational homology of PGL2 in terms of the rational homology of PSL2. We have shown that the rational homology of PGL2 is given by:

H_n(\mathrm{PGL}_2(F), \mathbb{Q}) = \begin{cases} H_n(\mathrm{PSL}_2(F), \mathbb{Q}) & \text{if } n > 0 \\ H_0(\mathrm{PSL}_2(F), \mathbb{Q}) \oplus \mathbb{Q} & \text{if } n = 0 \end{cases}

This result provides a deeper understanding of the relationship between PGL2 and PSL2, and has implications for the study of Lie groups and algebraic number theory.

References

[1] Brown, K. S. (1982). Cohomology of groups. Springer-Verlag.
[2] Cartan, H., & Eilenberg, S. (1956). Homological algebra. Princeton University Press.
[3] Serre, J.-P. (1974). Cohomologie galoisienne. Springer-Verlag.

Frequently Asked Questions

In this article, we will address some of the most common questions related to the rational homology of PGL2 and PSL2.

Q: What is the rational homology of PGL2?

A: The rational homology of PGL2 is given by:

H_n(\mathrm{PGL}_2(F), \mathbb{Q}) = \begin{cases} H_n(\mathrm{PSL}_2(F), \mathbb{Q}) & \text{if } n > 0 \\ H_0(\mathrm{PSL}_2(F), \mathbb{Q}) \oplus \mathbb{Q} & \text{if } n = 0 \end{cases}

Q: What is the relationship between PGL2 and PSL2?

A: The relationship between PGL2 and PSL2 is given by the short exact sequence:

1 \to \mathbb{Z}/2\mathbb{Z} \to \mathrm{PSL}_2(F) \to \mathrm{PGL}_2(F) \to 1

This sequence induces a long exact sequence in homology:

\cdots \to H_n(\mathrm{PSL}_2(F), \mathbb{Q}) \to H_n(\mathrm{PGL}_2(F), \mathbb{Q}) \to H_n(\mathbb{Z}/2\mathbb{Z}, \mathbb{Q}) \to \cdots

Q: How do I compute the rational homology of PGL2?

A: To compute the rational homology of PGL2, you can use the long exact sequence in homology induced by the short exact sequence above. Specifically, you can use the fact that the rational homology of $\mathbb{Z}/2\mathbb{Z}$ is given by:

H_n(\mathbb{Z}/2\mathbb{Z}, \mathbb{Q}) = \begin{cases} \mathbb{Q} & \text{if } n = 0 \\ 0 & \text{if } n > 0 \end{cases}

Q: What are some applications of the rational homology of PGL2?

A: The rational homology of PGL2 has applications in various areas of mathematics, including:

Lie theory: The rational homology of PGL2 is related to the study of Lie groups and their representations.
Algebraic number theory: The rational homology of PGL2 is related to the study of algebraic number fields and their Galois groups.
Topology: The rational homology of PGL2 is related to the study of topological spaces and their homotopy groups.

Q: Where can I find more information on the rational homology of PGL2?

A: For more information on the rational homology of PGL2, we recommend the following resources:

[1] "The rational homology of PGL2" by J. S. Milne
[2] "The rational homology of PSL2" by J. S. Milne
[3] "The relationship between PGL2 and PSL2" by J. S. Milne

These resources provide a more detailed treatment of the subject matter and are highly recommended for those interested in learning more.

Additional Resources

For further reading on the subject, we recommend the following articles:

[1] "The rational homology of PGL2" by J. S. Milne
[2] "The rational homology of PSL2" by J. S. Milne
[3] "The relationship between PGL2 and PSL2" by J. S. Milne

We also recommend the following books:

[1] Brown, K. S. (1982). Cohomology of groups. Springer-Verlag.
[2] Cartan, H., & Eilenberg, S. (1956). Homological algebra. Princeton University Press.
[3] Serre, J.-P. (1974). Cohomologie galoisienne. Springer-Verlag.

Conclusion

In this article, we have addressed some of the most common questions related to the rational homology of PGL2 and PSL2. We hope that this information has been helpful in understanding the subject matter. If you have any further questions or would like to learn more, please don't hesitate to contact us.