The Commuting Variety of the Projective Linear Lie Algebra $\mathfrak{pgl}_n$ in Positive Characteristic

Core Concepts

In positive characteristic p, the commuting variety of the projective linear Lie algebra $\mathfrak{pgl}_n$ has two irreducible components when p divides n, with dimensions determined by n and the highest power of p dividing n.

Abstract

• Bibliographic Information: Roman, V. (2024). The commuting variety of $\mathfrak{pgl}_n$. arXiv preprint arXiv:2402.11106v3.

• Research Objective: This paper investigates the structure of the commuting variety for the Lie algebra $\mathfrak{pgl}_n$ over an algebraically closed field of characteristic p > 0, specifically when p divides n. The goal is to determine the number of irreducible components of this variety and their dimensions.

• Methodology: The author employs techniques from algebraic geometry and representation theory. A key approach is to relate the problem to representations of the first Weyl algebra. By analyzing the irreducible representations of this algebra, the author gains insights into the structure of the commuting variety.

• Key Findings:

  • When the characteristic p divides n, the commuting variety of $\mathfrak{pgl}_n$ has precisely two irreducible components.
  • The dimensions of these components are n² + r - 1 and n² + n - 2, where r is the highest power of p that divides n.
  • The variety of pairs of matrices in GLn(k) with commutator equal to a scalar multiple of the identity is irreducible and its dimension is determined.

• Main Conclusions: This paper provides a complete characterization of the commuting variety for $\mathfrak{pgl}_n$ in positive characteristic when p divides n. The presence of two irreducible components contrasts with the characteristic zero case, where the commuting variety is irreducible.

• Significance: This research enhances the understanding of commuting varieties, which are important objects in Lie theory and representation theory. The results contribute to the broader study of algebraic structures in positive characteristic, where behavior can differ significantly from characteristic zero.

• Limitations and Future Research: The paper focuses specifically on the case of $\mathfrak{pgl}_n$. Exploring similar questions for other Lie algebras in positive characteristic could be a fruitful direction for future research. Additionally, investigating the geometric properties of the irreducible components of the commuting variety in more detail would be of interest.

Stats

If n = pr, then the commuting variety of $\mathfrak{pgl}_n$ has two irreducible components, of dimensions n² + r −1 and n² + n −2.
The variety {(x, y) ∈ GLn(k) × GLn(k) | [x, y] = ζI} is irreducible of dimension n² + n/d, where ζ is a root of unity of order d with d dividing n.

Quotes

"Commuting varieties have been studied ever since the 1950s, the first result being attributed to Motzkin and Taussky [MT55], who showed that in the case of the Lie algebra gln, the commuting variety is irreducible of dimension n2+n (over a field of any characteristic)."
"In the 1970s, Richardson [Ric79] generalized the problem and proved that in characteristic zero, for a reductive Lie algebra g, the commuting variety is irreducible and moreover, its dimension is given by the formula dim(g) + rank(g)."

Key Insights Distilled From

The commuting variety of $\mathfrak{pgl}_n$

by Vlad Roman at arxiv.org 11-19-2024

https://arxiv.org/pdf/2402.11106.pdf

Deeper Inquiries

How do the methods used in this paper generalize to studying commuting varieties of other Lie algebras in positive characteristic?

While the paper focuses on the commuting variety of $\mathfrak{pgl}_n$ in positive characteristic, some of its methods and ideas can be extended to study commuting varieties of other Lie algebras. Here's a breakdown:
Generalizations:

Jordan Decomposition and Regular Elements: The use of Jordan decomposition, particularly focusing on regular nilpotent elements, is a key aspect of the paper's approach. This technique can be applied to other classical Lie algebras, as their structure and representation theory often rely on similar concepts. Identifying and understanding the regular elements in these algebras would be crucial for extending the results.
Weyl Algebra Representations: The paper cleverly connects the problem to the representation theory of the first Weyl algebra ($A_1$). This connection arises from the commutation relation defining $A_1$, which mirrors the defining relation of the commuting variety. For other Lie algebras, exploring analogous algebraic structures with similar defining relations to their respective commuting varieties could provide valuable insights. However, the representation theory of such algebras might be more intricate than that of $A_1$.
Dominant Morphisms and Dimension Arguments: The paper utilizes dominant morphisms and fiber dimension arguments to determine the irreducibility and dimension of the commuting variety. These are standard tools in algebraic geometry and can be applied to study other varieties, including commuting varieties of different Lie algebras. However, constructing explicit dominant maps and understanding their fibers might be more challenging depending on the complexity of the Lie algebra.
Challenges for Other Lie Algebras:

More Complex Structure: Lie algebras other than $\mathfrak{pgl}_n$ often have more intricate structures, including different root systems, Dynkin diagrams, and representation theories. This complexity can make it significantly harder to generalize the specific arguments used for $\mathfrak{pgl}_n$.
Distinguished Nilpotent Orbits: The classification of nilpotent orbits, particularly the distinguished ones, plays a crucial role in understanding commuting varieties. In general, the classification of nilpotent orbits for exceptional Lie algebras is more involved than for classical types, potentially leading to a more intricate analysis of the commuting variety.

Could there be a deeper connection between the representation theory of the first Weyl algebra and the geometry of the commuting variety that explains the appearance of two components?

Yes, there's a strong possibility of a deeper connection. Here's why and how it might manifest:

Commutation Relation as a Bridge: The defining relation of the first Weyl algebra, [xy - yx = 1,] is fundamentally a statement about commutation. This directly mirrors the defining property of elements in the commuting variety, where the commutator vanishes. This shared algebraic structure suggests a deeper link.
Representation Decomposition and Components: The paper shows that irreducible representations of $A_1$ are directly related to the structure of the commuting variety of $\mathfrak{pgl}_n$. The appearance of two components in the commuting variety might be a geometric reflection of different "types" of representations or a decomposition of the representation space of $A_1$.
Geometric Invariants from Representations: Representation theory provides powerful tools for studying algebraic structures. It's conceivable that invariants or properties of $A_1$ representations, such as their dimensions, characters, or centralizer subgroups, could translate into geometric invariants of the commuting variety, potentially explaining the emergence of two components.
Further Investigation:

Analyze $A_1$ Representations: A detailed study of the representation theory of $A_1$, particularly in positive characteristic, could reveal specific properties or decompositions that correspond to the two components of the commuting variety.
Geometric Interpretation of Representations: Explore how representations of $A_1$ can be interpreted geometrically, perhaps as spaces with specific transformations or actions. This geometric viewpoint might provide a more intuitive understanding of the connection to the commuting variety.

What insights from the study of commuting varieties in Lie theory can be applied to understanding other algebraic structures or problems in different areas of mathematics?

The study of commuting varieties, while deeply rooted in Lie theory, offers insights and techniques applicable to a broader range of mathematical areas:
1. Non-Commutative Algebra:

Quantization and Deformation: Commuting varieties can be viewed as "classical" objects. Their study might provide insights into the process of quantization, where you deform a commutative algebra into a non-commutative one. Understanding how the geometry of the commuting variety changes under such deformations could be valuable.
Representation Varieties: The techniques used to study commuting varieties, such as analyzing orbits and invariants, can be applied to other representation varieties, which parametrize representations of algebras or groups.
2. Algebraic Geometry:

Nilpotent Orbits and Resolutions: The study of nilpotent orbits, crucial for understanding commuting varieties, has connections to singularity theory and resolutions of singularities in algebraic geometry. Techniques from commuting variety analysis might offer new approaches to these problems.
Moduli Spaces: Commuting varieties can be seen as specific examples of moduli spaces, which parametrize geometric objects. The methods used to study their geometry and topology could have implications for understanding moduli spaces in other contexts.
3. Integrable Systems:

Lax Pairs and Hamiltonian Systems: Commuting varieties naturally arise in the study of integrable systems. The existence of Lax pairs, which involve commuting matrices, is a key characteristic of many integrable systems. Insights from commuting varieties could lead to new methods for constructing or analyzing such systems.
4. Quantum Algebra:

Quantum Groups and Deformations: The study of commuting varieties might have analogues in the context of quantum groups, which are deformations of classical Lie algebras. Understanding how the geometry of commuting varieties changes under quantum deformation could be an interesting avenue of research.