by Carl Pierer
It is often the case in mathematics that by noticing some kind of symmetry, a problem can be simplified substantially. This makes them very useful. But, much like in art, mathematical symmetries also have an air of beauty and harmony. Perhaps this is one of the reasons why their study is a particularly satisfying branch of mathematics. A symmetry in mathematics is considered as an invariance under certain transformations. Consider, for instance, a square. There are the four rotational symmetries and four symmetries of reflection. While the vertices get permuted under these transformations, the configuration of the triangle is invariant. What this means is that if vertex 1 is joined to the vertices 2 and 4, then no matter what transformation we apply, these vertices will be joined afterwards as well. A transformation that does not keep these joints fixed will not be a symmetry. All these transformations taken together form a mathematical group. These can be, for a lack of a better word, considered as discrete symmetries and belong to the theory of finite groups. A different approach is to consider symmetries of continuous transformations. These are somewhat harder to visualise. Consider a circle. A rotation of the circle by any degree, no matter how small, will preserve the configuration of the circle. This is an example of a continuous transformation. The groups of these kinds of transformation find a natural place in Lie theory.
The person responsible for launching the early investigation of groups of continuous symmetries was Sophus Lie (1842-1899). In the 1860ies, the study of finite groups became a solid part of mathematics; by this time, tools had been developed and mathematicians started using them for various problems. In 1870, Camille Jordan published his Traité des substitutions et des équations algébriques. This was the first detailed study and clarification of Galois Theory. Evariste Galois (1811-1832) had studied algebraic equations and solutions to them. By finding symmetries in the roots of a polynomial and associating a group, he launched a wholly new and fruitful field of mathematical research. His work, possibly due to the highly tumultuous circumstances of his life and his early death, remained in sketches and it was not until others cleared up the ideas that the full impact of the theory was appreciated. Jordan's Traité was an effort to showcase and elaborate on the earlier work on groups by, amongst others Galois and Cauchy. But it also included substantially new contributions, introducing the concept of solvable groups, composition series and proving part of what is today known as the Jordan-Hölder theorem. The work has been credited with being an inspiration for many mathematicians and bringing group theory into the focus of late 19th century and 20th century mathematics.
Jordan's contributions to group theory inspired many mathematicians, including Lie. Lie's research between 1869 and 1873 led the way to what would later become known as Lie theory. In 1869, Lie moved to Berlin, where there was a strong research focus on analysis, including such famous mathematicians as Kummer, Kronecker, and Weierstrass. In Berlin, he met the younger Felix Klein (1842-1899) and they soon formed a friendship. The two were brought together by a common interest in geometry and their dislike for the prevalence of analysis in Berlin at the time. They met on a daily basis from October 1869 – October 1872 and influenced each other substantially. Lie's first insight concerning the notion and relevance of groups of continuous symmetries was to have realised that the solution to certain types of differential equations depended entirely on their invariance under certain continuous transformations. He published a first work, which was edited by Klein, using this idea in 1869.
Their ideas developed somewhat differently on the course of the following years. Klein became interested in understanding different kinds of geometry through groups. He developed this idea, which was later to form the core of his Erlangen Programm (published in 1872). Lie, however, envisaged a Galois-style theory for differential equations. Eventually, this approach proved fruitless, as the group associated to a differential equation in this way is in general trivial. In spite of this, the project developed into a beautiful theory in its own right. In 1873, Lie had reached a major breakthrough. While he had realised early on that he was able to associate finite-dimensional transformation groups to certain differential equations, he lacked a way of determining them exactly. In that year, he discovered that "he could determine these transformation groups in one variable" via infinitesimal transformations, which are closed under a bilinear operator (the bracket) and which we now call Lie algebras. This means he was able to approximate the transformation groups by working with linear approximations.
This discovery is lauded as the mathematical origin of Lie theory. Lie would devote the rest of his life to a systematic development of this theory. In 1886, he was offered the chair of geometry in Göttingen. There, Friedrich Engel (1861-1941) would soon become his assistant. Together, they created "(…) a school dedicated to the further development, application and dissemination of his theory of transformation groups" (Hawkins, 2000). Their research flourished, and in the years 1888-1893 Lie and Engel published the Theorie der Transformationsgruppen. Their work soon attracted international attention. Amongst the most interested were mathematicians in Paris, partly due to Lie's connections and visits. Lie had visited Paris in 1887, by which time Poincaré and Picard were spearheading the mathematical community in Paris. His research was met with great sympathy. Poincaré and Picard send students to study with Lie in Leipzig. At this time of Franco-German tensions, such an exchange was rather unusual. It is surely a testament to the interest Lie's work aroused in Paris. Because of his bad health, Lie returned to Norway in 1898. He died the following year. But his school in Leipzig was thriving, and three mathematicians there (Friedrich Engel, Friedrich Schur, and Eduard Study) oriented their work to aspects of Lie's theory. But also elsewhere, Lie's theory attracted many mathematicians. Amongst others, Wilhelm Killing (1847-1923). Killing had studied in Berlin, where he had become acquainted with methods of linear algebra. Thanks to this Killing was able to contribute a new perspective to determining the classification of transformation groups. With this perspective, he was able to give a classification of simple Lie groups. This was a big first step into the direction of classifying more general Lie groups.
In Paris, Picard wrote to Lie: "Paris is becoming a center for groups; it is all fermenting in young minds, and one will have an excellent wine after the liquors have settled a bit.''[i] Élie Cartan (1869-1951) fulfilled Picard's prophesy. He carried Killing's initial research further and discovered many of the structures that are still being used. In 1913, he arrived at a complete classification of the complex finite-dimensional representations of semisimple Lie groups. A representation of a Lie group can be thought of as describing a way of a Lie group acting on a vector space. We say a representation is complex finite-dimensional if the vector space is complex and finite-dimensional. Cartan's classifying theorem, known as the Theorem of the Highest Weight, thus describes a way of understanding the ways in which a large class of Lie groups manifest themselves in their action on complex finite-dimensional vector spaces.
While the contributions of Lie, Killing, Cartan, and others provided a solid foundation for the study of Lie groups, the subject thrived in the 1920ies when Hermann Weyl, with an eye on developments in the theory of relativity and quantum theory, developed the subject further. Today, Lie groups pop up in many different aspects of mathematics. While a large interest comes from physics – representations of Lie groups are studied, for instance, in quantum theory – they are also of intrinsic interest in mathematics. Because of their rich structure, their study allows many different approaches: from geometry and algebra to analysis. This makes them into fascinating representatives of the beauty of symmetries.
Bourbaki, N. (1975). Lie Groups and Lie Algebras. Addison-Wesley.
Hawkins, T. (2000). Emergence of the Theory of Lie Groups (1869-1926). Springer.
[i] Quoted after (Hawkins, 2000) after a letter dated May 1893. Hawkins cites the French original as: "Violà [sic] Paris devenir un centre de groupes; tout cela fermente dans ces jeunes cerveaux, et on aura un excellent vin quand les liqueurs seront un peu reposées."