[Paper Review] Defining relations associated with the principal sl(2)-subalgebras of simple Lie algebras

Pavel Grozman, Dimitry Leites|arXiv (Cornell University)|Oct 3, 2005

Algebraic structures and combinatorial models4 references4 citations

TL;DR

This paper presents a novel, minimal presentation of finite-dimensional simple Lie algebras using Jacobson's generators—two elements derived from the principal $σ\mathfrak{sl}(2)$-subalgebra. By exploiting the decomposition of the algebra into irreducible $σ\mathfrak{sl}(2)$-modules, the authors derive a compact set of defining relations (only 9 for $σ\mathfrak{sl}(\lambda)$ series), significantly simpler than the standard Serre relations. The method enables cleaner presentations for integrable systems and $q$-quantization of infinite-dimensional matrix algebras.

ABSTRACT

The notion of defining relations is well-defined for any nilpotent Lie algebra. Therefore a conventional way to present a simple Lie algebra G is by splitting it into the direct sum of a commutative Cartan subalgebra and two maximal nilpotent subalgebras (positive and negative) and together the generators of both these nilpotent subalgebras together generate G. Though there are many relations between these generators, they are neat (Serre relations). It is possible to determine the relations for generators of different type, e.g, with the principal embeddings of sl(2) into G one can associate only TWO elements that generate G. We explicitly describe the corresponding presentations of simple Lie algebras, for all finite dimensional and certain infinite dimensional ones; namely, for the Lie algebra "of matrices of a complex size" realized as a subalgebra of the Lie algebra of differential operators in 1 indeterminate. The relations obtained are rather simple. Our results might be of interest in applications to integrable systems (like vector-valued Liouville (or Leznov-Saveliev, or 2-dimensional Toda) equations and KdV-type equations). They also indicate how to q-quantize the Lie algebra of matrices of complex size.

Motivation & Objective

• To provide a clean, minimal presentation of finite-dimensional simple Lie algebras using generators derived from the principal $σ\mathfrak{sl}(2)$-subalgebra.
• To identify a canonical pair (and a third generator for simplification) that generate any finite-dimensional simple Lie algebra over $ ℂ$.
• To derive defining relations between these generators that are significantly simpler than the standard Serre relations.
• To extend the framework to certain infinite-dimensional Lie algebras, such as the Lie algebra of matrices of complex size realized as differential operators.
• To facilitate applications in integrable systems and $q$-quantization by providing a computationally tractable and structurally transparent presentation.

Proposed method

• Identify the principal $σ\mathfrak{sl}(2)$-embedding of a simple Lie algebra ${\mathfrak{g}}$, which decomposes ${\mathfrak{g}}$ into irreducible $σ\mathfrak{sl}(2)$-modules with highest weights $2, k_1, k_2, \dots$, as listed in Table 1.1.
• Select Jacobson's generators: $x = \nabla_+ \in L^2$ (the $σ\mathfrak{sl}(2)$-generator), $z = l_{-r}$ a lowest weight vector from the highest weight module $L^r$ (with $r = k_1$ for ${\mathfrak{g}} \neq \u03c3\mathfrak{o}(2n)$, or $r = 2n-2$ for ${\mathfrak{g}} = \u03c3\mathfrak{o}(2n)$), and $u = \nabla_-$ for simplifying relations.
• Use a computer algebra system (Mathematica-based package by Grozman) to compute the defining relations between these generators, ensuring they are homogeneous in degree and minimal in number.
• Introduce a third generator $u$ to simplify the form of the relations, even though it increases their count slightly, while avoiding symmetric generators like $l_r$ which complicate the relations.
• Derive the defining relations by analyzing the action of $X^\pm$ and $H$ on the weight vectors of the irreducible $σ\mathfrak{sl}(2)$-modules, using the standard $\u03c3\mathfrak{sl}(2)$-basis: $X^-, H, X^+$.
• Extend the presentation to infinite-dimensional Lie algebras, such as the Lie algebra of matrices of complex size, realized as a subalgebra of differential operators in one indeterminate.

Experimental results

Research questions

• RQ1 Can a simple finite-dimensional Lie algebra be generated by only two elements derived from its principal $σ\mathfrak{sl}(2)$-subalgebra, and if so, what are the defining relations between them?
• RQ2 Are the defining relations between these generators simpler than the standard Serre relations, especially in terms of number and form?
• RQ3 Can the same generator and relation framework be extended to infinite-dimensional Lie algebras, such as the Lie algebra of matrices of complex size?
• RQ4 Why is the inclusion of a third generator $u = \nabla_-$ beneficial for simplifying the defining relations, despite increasing their count?
• RQ5 How does this presentation facilitate applications in integrable systems and $q$-quantization of infinite-dimensional Lie algebras?

Key findings

• The authors explicitly construct a presentation of any finite-dimensional simple Lie algebra using only two generators—$x$ and $z$—derived from the principal $σ\mathfrak{sl}(2)$-subalgebra, with a third generator $u$ introduced for simplification.
• For the $σ\mathfrak{sl}(\lambda)$ series, only 9 defining relations are required between Jacobson's generators, a dramatic simplification compared to the standard Serre relations.
• The defining relations are homogeneous and significantly more compact than those from the Chevalley presentation, especially for non-exceptional Lie algebras.
• The method successfully extends to the infinite-dimensional Lie algebra of matrices of complex size, realized as a subalgebra of differential operators in one indeterminate.
• The computer-assisted derivation confirms that alternative generator choices lead to more complex relations, validating the choice of Jacobson's generators as canonical.
• The results provide a foundation for $q$-quantization of the Lie algebra of matrices of complex size and offer a cleaner framework for studying integrable systems based on principal $σ\mathfrak{sl}(2)$-subalgebras.