Lie Algebra Is Used to Help Solve Hilbert's Fifth Problem

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Lie Algebra Is Used to Help Solve Hilbert's Fifth Problem

Overview

Advances in the theory of Lie algebra (also called Lie groups), developed by Norwegian mathematician Sophus Lie (1842-1899), have long enriched mathematics, particularly in the area of group theory. In 1952 Lie algebra was used to help solve one of the most famous problems in mathematics: Hilbert's fifth problem, posed in the year 1900 by David Hilbert (1862-1943). In addition to its use in solving this problem, Lie algebra has been used to gain a better understanding of the properties of many-dimensional surfaces in general, helping to advance the mathematical discipline of topology as well.

Background

One of the keynote speakers at the 1900 International Congress of Mathematicians, David Hilbert spoke about the future of mathematics. In his address, the renowned mathematician posed 23 major unsolved problems in mathematics to the Congress, noting that their solutions would help to push mathematics forward. One of these problems, the fifth, was to challenge mathematicians for over half of the twentieth century.

Hilbert's fifth problem was phrased as such: "Can the assumption of differentiability for functions defining a continuous transformation group be avoided?" Another way to state this question is to ask if any locally Euclidean topological group can be given the structure of an analytic manifold to become a Lie group. These individual terms can be confusing and will be explained further before going on.

A topological group is any group of numbers that are points in topological space for which group operations are continuous. An example of topological groups is the set of real numbers where addition or subtraction are group products. Another is the set of rigid motions that a group of points can take in Euclidean space (that is, space that follows the geometry described by Euclid [c. 330-260 B.C.]) when one point remains fixed. In this case, if one visualizes a rotating block of wood as representing a group of points, these points would form a topological group because they stay together as the block rotates in space.

A manifold is another concept entirely. Put most simply, a manifold is a surface with a given number of dimensions. For example, the surface of a sphere is a manifold with two dimensions because there are only two directions an object on that surface can move. The fact that the surface encloses a three-dimensional structure is irrelevant in this case. An analytic manifold can be mathematically described. One interesting property of manifolds is that, in Euclidean space, small patches appear to be flat and can be treated as such. For example, even though Earth is a sphere, we treat a floor as flat because, for all practical purposes, it looks that way to us at our scale. This makes it possible to use "ordinary" mathematics to describe small sections of the manifold and any shapes or curves that might be drawn on them.

The final term to describe is Lie algebra or Lie group. In general, a group is a set (either finite or infinite) of items (called operands) that can be combined via some sort of mathematical operation to form defined products. A Lie group is a special type of group in which the underlying space is an analytic manifold and, on that manifold, group operations are analytic (that is, they are described by equations). At this point, too, it must be noted that a Lie group is not, in the strictest sense, a group at all. Rather, it is a series of operations that, together, comprise a system for finding solutions to problems. In other words, the more correct term for a Lie group is Lie algebra, and these terms are often used interchangeably.

This brings us back to Hilbert's fifth problem, phrased as "Can any locally Euclidean topological group can be given the structure of an analytic manifold to become a Lie group?" With the definitions mentioned above, we see that this is asking if there is a general way to take any group of numbers (which will define a surface or a shape in space) in Euclidean space, place them on a smooth surface, and make them into a Lie group, so that the surfaces and the underlying space are still analytic.

Finally, we need to introduce the concept of "compactness" for groups. A group is "compact" if all of the members are adjacent when plotted in space or on a surface. For example, a ball would be compact because all of the points that make up the ball lie together in space. On the other hand, the stars in the sky are not compact as we see them because they are sprinkled more or less randomly across the sky.

Hilbert's fifth problem was solved in part in 1929 by John von Neumann (1903-1957), when he demonstrated that functions could be integrated on general compact groups. However, this solution was only partial. Three years later, Alfred Haar added to von Neumann's proof, and in 1934 Soviet mathematician Lev Semyonovich Pontryagin proved Hilbert's conjecture for a different set of groups, the Abelian groups. Finally, this problem was completely solved by Gleason, Montgomery, and Zippin in 1952. They showed that any locally compact topological group is a limit of Lie groups, making this finding even more important.

Impact

In general, by posing his problems and daring the world's mathematicians to solve them, Hilbert contributed to many advances in mathematics in each of the areas upon which these problems touched. The field of mathematics is richer because of this, and those fields that use mathematics have more mathematical tools they can bring to bear on their problems. Thus, in a sense, all of mathematics, physics, computer science, and many other fields have benefited from the posing of and solutions to Hilbert's problems.

In this case, the solution to Hilbert's fifth problem has led to a greater understanding of the mathematical properties of surfaces. Such surfaces include the shape of machine parts, the shape of planets and black holes, or the shape of the universe in its entirety.

However, the impact of this line of inquiry is not limited to solving this specific problem. The application of Lie group theory to solving Hilbert's fifth problem is just one of the impacts of this field on the whole of mathematics. In particular, Lie examined "contact transformations," which have since become important. A contact transformation is a way of describing the mathematical properties of a very small part of a surface by determining the equation of a line or vector tangent to the surface (i.e. touching the surface at only a single point). This is analogous to drawing a line tangent to a curve to determine the slope of the curve at that point or, in three dimensions, constructing a plane that touches a sphere at only one location to better determine the properties of the sphere. In the case of Lie algebra, a surface of any number of dimensions (called an n-dimensional surface) can be described in terms of a field of such vectors, in the same way that a curve can be described in terms of all of the lines that are tangent to it. What Lie did was to generalize this concept and to provide a way to work with surfaces of any number of dimensions. This, in turn, has an application when mathematically describing multi-dimensional problems, such as the shape of the universe, super-string theory (which assumes ten dimensions in space), and some problems with a large number of variables.

P. ANDREW KARAM