[Paper Review] Evaluation of composition function along with invariant vector fields on Lie group using structural constants of corresponding Lie algebra

А. А. Магазев, V. V. Mikheyev+1 more|arXiv (Cornell University)|Dec 2, 2013

Numerical methods for differential equations17 references5 citations

TL;DR

This paper proposes a method to construct the composition function, left- and right-invariant vector fields, and differential 1-forms on a Lie group using the structure constants of its Lie algebra. By employing second canonical coordinates, the approach reduces the problem to matrix inversion and matrix exponentiation, enabling the composition function to be expressed in quadratures and the transition between canonical coordinate systems to be computed via quadrature.

ABSTRACT

Methods of construction of the composition function, left- and right-invariant vector fields and differential 1-forms of a Lie group from the structure constants of the associated Lie algebra are proposed. It is shown that in the second canonical coordinates these problems are reduced to the matrix inversions and matrix exponentiations, and the composition function can be represented in quadratures. Moreover, it is proven that the transition function from the first canonical coordinates to the second canonical coordinates can be found by quadratures.

Motivation & Objective

• To develop systematic methods for constructing the composition function, left- and right-invariant vector fields, and differential 1-forms on a Lie group.
• To leverage the structure constants of the associated Lie algebra as the fundamental input for these constructions.
• To simplify the computation of these geometric objects by transforming them into matrix operations in second canonical coordinates.
• To establish that the transition between first and second canonical coordinates can be computed via quadrature.
• To provide a constructive framework for Lie group analysis based on algebraic data (structure constants) rather than coordinate charts.

Proposed method

• Utilize the structure constants of the Lie algebra to define the Lie bracket relations of left-invariant vector fields.
• Introduce second canonical coordinates to linearize the group multiplication problem and reduce it to matrix operations.
• Apply matrix inversion to compute the inverse elements in the second canonical coordinate system.
• Use matrix exponentiation to reconstruct the composition function from the structure constants.
• Express the composition function in quadrature form, enabling analytical integration of group operations.
• Derive the transition function from first to second canonical coordinates through integration (quadrature), based on the structure constants.

Experimental results

Research questions

• RQ1 How can the composition function of a Lie group be systematically derived from the structure constants of its Lie algebra?
• RQ2 What role do second canonical coordinates play in simplifying the computation of invariant vector fields and differential forms?
• RQ3 Can the transition between first and second canonical coordinate systems be expressed in closed form using integration?
• RQ4 To what extent can matrix operations (inversion and exponentiation) replace coordinate-based constructions in Lie group theory?
• RQ5 What is the analytical tractability of invariant geometric objects when derived solely from structure constants?

Key findings

• The composition function of a Lie group can be represented in quadratures using the structure constants and matrix exponentiation in second canonical coordinates.
• Left- and right-invariant vector fields and differential 1-forms can be constructed directly from the structure constants via matrix operations.
• The transition function from first to second canonical coordinates is computable by quadrature, establishing a closed-form relationship between the two coordinate systems.
• The use of second canonical coordinates transforms the nonlinear group multiplication problem into a sequence of linear algebra operations, significantly simplifying computation.
• Matrix inversion and matrix exponentiation serve as the core computational engines for reconstructing the group structure from algebraic data.
• The entire framework is consistent and constructive, enabling analytical treatment of geometric objects on Lie groups from purely algebraic inputs.