[Paper Review] A universal formula for the volume of compact Lie groups

R. L. Mkrtchyan, А. П. Веселов|arXiv (Cornell University)|Apr 10, 2013

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper presents a universal formula for the volume of compact simple Lie groups using Vogel's universal parameters α, β, γ, expressing the volume as (2√2π)^dim G × e^−Φ(α,β,γ), where Φ is an integral involving hyperbolic functions of the parameters. The key result is that on the SUₙ line (α = −2, β = 2, γ = z), this formula analytically continues the Barnes G-function, reconciling the volume formula with classical special functions.

ABSTRACT

We provide a closed formula for the volume of a simple compact Lie group in terms of the universal Vogel parameters. For the unitary groups SU_n this reduces to the integral representation of the classical Barnes G-function.

Motivation & Objective

• To derive a universal closed-form expression for the volume of any compact, simply-connected simple Lie group.
• To express the volume in terms of Vogel’s universal parameters α, β, γ, enabling a unified treatment across all simple Lie groups.
• To establish a connection between the volume function and the Barnes G-function through analytic continuation.
• To demonstrate that the proposed integral formula for Φ(α,β,γ) reduces to the Barnes G-function in the case of unitary groups SUₙ.

Proposed method

• Derive the volume using the Cartan-Killing metric and the root system structure of the Lie algebra.
• Express the volume as a product over positive roots involving sine and inner products with the Weyl vector.
• Apply Malmsten’s integral representation of the logarithm of the gamma function to transform the product into an integral over x.
• Use a key identity from previous work to express the sum over roots as a combination of hyperbolic functions and a rational term involving α, β, γ.
• Define Φ(α,β,γ) as the integral of a function F(x;α,β,γ) over (0,∞), which captures the group volume via e^−Φ.
• Verify the consistency of the formula with Macdonald’s volume formula and show agreement with the Barnes G-function for SUₙ via analytic continuation.

Experimental results

Research questions

• RQ1 Can a single universal formula express the volume of all compact simple Lie groups using Vogel’s parameters?
• RQ2 How does the volume formula relate to the Barnes G-function in the case of unitary groups SUₙ?
• RQ3 Does the integral representation of Φ(α,β,γ) provide a consistent analytic continuation of the volume function beyond integer ranks?
• RQ4 What is the role of the universal parameters α, β, γ in encoding the geometric and algebraic structure of the Lie group?

Key findings

• The volume of a compact Lie group G is given by Vol(G) = (2√2π)^dim G × e^−Φ(α,β,γ), where Φ is an integral over x involving hyperbolic ratios of the Vogel parameters.
• For SUₙ, the function Φ(−2,2,z) matches the logarithm of the Barnes G-function: Φ(−2,2,z) = ln G(z+1) − ½z² ln z + ½(z² − z) ln(2π).
• The formula provides a consistent analytic continuation of the product 1!2!⋯(n−1)! to complex z, agreeing with Barnes’ definition.
• The integral Φ(α,β,γ) converges for all complex α,β,γ outside the region where Re(α/t), Re(β/t), Re(γ/t) ≥ 0, which excludes the physical Lie algebra parameters.
• The derivation establishes a deep link between Lie group geometry and multiple Barnes gamma functions, suggesting broader applications in mathematical physics.