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Published on December 18th, 2013 | by Emily Corbett

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A special type of basis for a special type of Lie algebra

By Thanh Tung Nguyen, La Trobe University

This student took part in the 2012/13 AMSI Vacation Research Scholarship program. For more information on this years program please click here

Everyone should be familiar with the word ‘algebra’ since it is early introduced in the high school curriculum. The concept of ‘algebra’ at this level is loosely understood as the study of symbols representing quantities, operations acting on them and relations between them, known as equations or inequations [1]. The spirit of modern mathematics is to base everything on a number of axioms, statements assumed to be true. Then all other results including propositions, theorems, etc… are derived from the axioms using logical reasoning. Since our language is finite, there must be some undefined concepts, called primitive concepts on which all other concepts are defined. Consequently, in modern mathematics, algebra is defined as follows: ‘an algebra consists of a vector space V over a field F, together with a binary operation of multiplication on the set V of vectors, such that for all Screen shot 2013-07-23 at 3.27.25 PM and Screen shot 2013-07-23 at 3.30.21 PM, the following conditions are satisfied:

1. Screen shot 2013-07-23 at 3.27.40 PM

2. Screen shot 2013-07-23 at 3.27.47 PM

3. Screen shot 2013-07-23 at 3.27.55 PM

Basing on the above definition, one can construct many algebras. One special type of algebras is Lie algebras, named after Marius Sophus Lie, a famous Norwegian mathematician. It has many applications in physics, especially general relativity, quantum mechanics and particle physics. The definition of a finite-dimensional Lie algebra is as follows: ‘A finite-dimensional real or complex Lie algebra is a finite-dimensional real or complex vector space Screen shot 2013-07-23 at 3.33.12 PM, together with a map Screen shot 2013-07-23 at 3.28.27 PM from Screen shot 2013-07-23 at 3.28.21 PM called the Lie bracket, which satisfies the following axioms:

1. Screen shot 2013-07-23 at 3.28.27 PM is bilinear

2. Screen shot 2013-07-23 at 3.44.11 PM (skew-symmetry)

3. Screen shot 2013-07-23 at 3.28.43 PM(Jacobi identity)’ [3]

It can be seen that the definition above involves a vector space which relates to the notion of basis. A basis for a vector space is a set of linearly independent vectors that span the space. As a result, every vector in a vector space can be expressed as a unique linear combination of vectors in a basis. A basis can have additional properties like orthogonal basis (whose every pair of vectors are perpendicular), orthonormal basis (which is orthogonal and whose every vector has unit length). In a Lie algebra, the vector space Screen shot 2013-07-23 at 3.33.12 PM can sometimes have a basis comprised of geodesic vectors. A nonzero element Screen shot 2013-07-23 at 3.29.01 PM is a geodesic vector if the Lie bracket [X, Y] is perpendicular to Y for all Screen shot 2013-07-23 at 3.29.06 PM A basis of geodesic vectors can also be orthogonal or orthonormal. From here arise many questions: does every Lie algebra possess a basis of geodesic vectors? (or even stronger, an orthonormal basis of geodesic vectors), if not, are there some special types of Lie algebra which certainly possess such a basis? The answer is that nilpotent Lie algebras and unimodular Lie algebras do. However, there is a tricky point that, in the case of unimodular Lie algebras, only algebras of dimension less that 5 are guaranteed to have an orthonormal basis of geodesic vectors.

 

References:

[1] Unknown author (n.d.), ‘Algebra’, Wikipedia The Free Encyclopedia, viewed 8 February 2013, http://en.wikipedia.org/wiki/Algebra#cite_note-4th_meaning-1

[2] Fraleigh, JB 1982, A First Course in Abstract Algebra, 3rd edn, Addison-Wesley Publishing Company, Philippines.

[3] Hall, BC 2003, Lie Groups, Lie Algebras, and Representations, An elementary Introduction, Springer.

[4] Cairns, G, Le, NTT, Nielsen, A & Nikolayevsky, Y 2013, ‘On the existence of orthonormal geodesic basis for Lie algebras’, Note di Matematica.

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