[Paper Review] On the stochastic Lie algebra

Manuel Guerra, Andrey Sarychev|arXiv (Cornell University)|May 18, 2018

Advanced Topics in Algebra6 references3 citations

TL;DR

This paper provides a rigorous analysis of the stochastic Lie algebra 𝔰(n,ℝ), establishing its Levi decomposition and proving that it is isomorphic to 𝔰𝔩(n−1,ℝ) via an orthonormal basis. The authors explicitly compute the Killing form using this basis, confirm the semisimplicity of the Levi factor, and demonstrate that 𝔰(n,ℝ) is generated by just two generic matrices, resolving a gap in prior work and clarifying the algebraic structure of the stochastic group 𝒮(n,ℝ).

ABSTRACT

We study the structure of the Lie algebra $\mathfrak{s}(n,\mathbb R)$ corresponding to the so-called stochastic Lie group $\mathcal{S} (n,\mathbb R)$. We obtain the Levi decomposition of the Lie algebra, classify Levi factor and classify the representation of the factor in $\mathbb{R}^n$. We discuss isomorphism of $\mathcal{S}(n,\mathbb R)$ with the group of invertible affine maps ${\it Aff}(n-1,\mathbb R)$. We prove that $\mathfrak s(n, \mathbb R)$ is generated by two generic elements.

Motivation & Objective

• To rigorously establish the Levi decomposition of the stochastic Lie algebra 𝔰(n,ℝ), correcting a logical gap in prior work on the semisimplicity of the Levi factor and maximality of the radical.
• To classify the Levi factor of 𝔰(n,ℝ) and show it is isomorphic to 𝔰𝔩(n−1,ℝ) via explicit computation of the Killing form and Dynkin diagram.
• To prove that the stochastic Lie algebra 𝔰(n,ℝ) is generated by two generic matrices, resolving a minimal generation problem.
• To clarify the isomorphism between the stochastic group 𝒮(n,ℝ) and the group of invertible affine maps Aff(n−1,ℝ), and to analyze the corresponding Lie algebra structure.
• To construct an orthonormal basis for 𝔰(n,ℝ) that facilitates explicit computation of Lie algebraic invariants such as the Killing form and enables structural classification.

Proposed method

• Construct an orthonormal basis for 𝔰(n,ℝ) using vectors in the hyperplane Πₙ = {x ∈ ℝⁿ : ∑xᵢ = 0} and the normalized all-ones vector v₀ = (1,…,1)/√n.
• Define basis matrices: Z = (1/√(n−1))(Iₙ − v₀⊗v₀), Rᵢ = v₀⊗vᵢ for i=1,…,n−1, Aᵢⱼ = vᵢ⊗vⱼ for i≠j, and H ∈ ℋ with ∑γₗ = 0.
• Use the orthonormal basis to compute the Killing form explicitly, then apply Cartan’s criterion to prove semisimplicity of the Levi subalgebra 𝔩.
• Show that the Levi subalgebra 𝔩 is isomorphic to 𝔰𝔩(n−1,ℝ) by deriving its Dynkin diagram and confirming the root system matches Aₙ₋₂.
• Construct two generic matrices X and Y in 𝔰(n,ℝ) such that iterated adjoint actions adᵏXY generate all basis elements of the algebra.
• Use a vector γ ∈ ℝⁿ⁻¹ satisfying specific non-degeneracy conditions (γᵢ ≠ 0, γᵢ ≠ γⱼ, γᵢ−γⱼ ≠ γₖ−γₗ) to ensure the matrix (γᵢ−γⱼ)ᵏ has full rank, enabling generation of the entire algebra.

Experimental results

Research questions

• RQ1 What is the correct Levi decomposition of the stochastic Lie algebra 𝔰(n,ℝ), and how can the semisimplicity of the Levi factor be rigorously established?
• RQ2 Is the Levi factor of 𝔰(n,ℝ) isomorphic to 𝔰𝔩(n−1,ℝ), and what is the explicit structure of this isomorphism?
• RQ3 Can the stochastic Lie algebra 𝔰(n,ℝ) be generated by only two elements, and if so, what conditions ensure this minimal generation?
• RQ4 How does the structure of the stochastic group 𝒮(n,ℝ) relate to the group of invertible affine transformations Aff(n−1,ℝ), and what is the corresponding Lie algebra isomorphism?
• RQ5 What is the role of the orthonormal basis in enabling explicit computation of the Killing form and classification of the algebraic structure?

Key findings

• The stochastic Lie algebra 𝔰(n,ℝ) admits a Levi decomposition 𝔰(n,ℝ) = 𝔩 ⊕ 𝔯, where 𝔯 is the radical generated by matrices Ĥᵢ = Eᵢ(n) − Eₙ(n) and Ẑ = Iₙ − (1/n)Jₙ.
• The Levi subalgebra 𝔩 is isomorphic to 𝔰𝔩(n−1,ℝ), as confirmed by the Dynkin diagram Aₙ₋₂ and explicit computation of the Killing form.
• The Killing form on 𝔰(n,ℝ) is computed explicitly using the orthonormal basis, and Cartan’s criterion confirms the semisimplicity of 𝔩.
• The stochastic group 𝒮(n,ℝ) is isomorphic to the group of invertible affine maps Aff(n−1,ℝ), with the corresponding Lie algebra isomorphism realized via conjugation by a matrix M.
• The Lie algebra 𝔰(n,ℝ) is generated by two generic matrices X and Y, as shown by the fact that iterated adjoint actions adᵏXY span the entire algebra when γ satisfies the non-degeneracy conditions.
• The minimal number of generators for 𝔰(n,ℝ) is two, and this is proven by showing that the matrix of differences (γᵢ−γⱼ)ᵏ has full rank m = (n−1)(n−2), ensuring linear independence of the generated basis elements.