Love and Math

by Frenkel, Edward

  Fuchs handed me an eighty-page research paper he and Feigin had written.

  “I first thought of giving you a textbook on Lie algebras,” he said. “But then I thought, why not just give you this paper?”

  I put the paper carefully in my backpack. It was still unpublished at the time, and, thanks to the tight controls that Soviet authorities (afraid that people would make copies of banned literature, like books of Solzhenitsyn or Doctor Zhivago) placed on photocopiers, there were only a handful of copies available in the entire world. Very few people had ever gotten to see this paper – Feigin later joked that I may have been the only one who had read it from beginning to end.

  It was written in English and was supposed to appear in a collection of papers published in the U.S. But the publisher badly mismanaged the book, and its publication was delayed for some fifteen years. By then, most of the results were reproduced elsewhere, so it wasn’t much read after it came out either. Nevertheless, the article became famous, and Feigin and Fuchs eventually got their due credit. Their paper has been widely cited in the literature (as a “Moscow Preprint”), and even a new term was coined, “Feigin–Fuchs representations,” to refer to the new representations of Lie algebras they studied in this paper.

  As I started reading the paper, my first question was: what are these objects that carry such a strange name, “Lie algebras”? The paper that Fuchs gave me assumed quite a bit of knowledge about topics I’d never studied, so I went to a bookstore and bought all the textbooks on Lie algebras I could find. Whatever I could not find, I borrowed from the library at Kerosinka. I was reading all these books in parallel with the Feigin–Fuchs article. This experience shaped my learning style. Since then, I’ve never been satisfied with one source; I try to find all available sources and devour them.

  To explain what Lie algebras are, I first need to tell you about “Lie groups.” Both are named after a Norwegian mathematician Sophus Lie (pronounced LEE) who invented them.

  Mathematical concepts populate the Kingdom of Mathematics, just like species of animals populating the Animal Kingdom: they are linked to each other, form families and subfamilies, and often two different concepts mate and produce an offspring.

  The concept of a group is a good example. Think of groups as analogues of birds, which form a class in the Animal Kingdom, or Animalia (called class Aves). That class is split into twenty-three orders; each order in turn splits into families, and each of those splits into genera. For example, the African fish eagle belongs to the order of Accipitriformes, the family of Accipitridae, and the genus of Haliaeetus (compared to these names, “Lie group” doesn’t sound so exotic!). Likewise, groups form a large class of mathematical concepts, and within this class there are different “orders,” “families,” and “genera.”

  For example, there is an order of finite groups that includes all groups with finitely many elements. The group of symmetries of a square table, which we discussed in Chapter 2, consists of four elements, so it is a finite group. Likewise, the Galois group of a number field obtained by adjoining the solutions of a polynomial equation to the rational numbers is a finite group (for example, in the case of a quadratic equation it has two elements). The class of finite groups is further subdivided into families, such as the family of Galois groups. Another family consists of the crystallographic groups, which are the groups of symmetries of various crystals.

  There is also another order, of infinite groups. For example, the group of integers is infinite, and so is the braid group Bn, which we discussed in Chapter 5, for each fixed n = 2, 3, 4,... (Bn consists of braids with n threads; there are infinitely many such braids). The group of rotations of a round table, which consists of all points on a circle, is also an infinite group.

  But there is an important difference between the group of integers and the circle group. The group of integers is discrete; that is to say, its elements do not combine into a continuous geometric shape in any natural sense. We can’t move continuously from one integer to the next; we jump from one to another. In contrast, we can change the angle of rotation continuously between 0 and 360 degrees. And together, these angles combine into a geometric shape: namely, the circle. Mathematicians call such shapes manifolds.

  The group of integers and the braid groups belong to the family of discrete infinite groups in the Kingdom of Mathematics. And the circle group belongs to another family, that of Lie groups. Put simply, a Lie group is a group whose elements are points of a manifold. So this concept is the offspring resulting from the marriage of two mathematical concepts: group and manifold.

  Here is the tree of group-related concepts that we will discuss in this chapter (some of these concepts have not yet been introduced, but will be later in the chapter).

  Many symmetries arising in nature are described by Lie groups, and that’s why they are so important to study. For example, the group SU(3) we talked about in Chapter 2, which is used to classify elementary particles, is a Lie group.

  Here is another example of a Lie group: the group of rotations of a sphere. A rotation of a round table is determined by its angle. But in the case of a sphere, there is more freedom: we have to specify the axis as well as the angle of rotation, as shown on the picture. The axis can be any line passing through the center of the sphere.

  The group of rotations of the sphere has a name in math: the special orthogonal group of the 3-dimensional space, or, as it is commonly abbreviated, SO(3). We can think of the symmetries of the sphere as transformations of the 3-dimensional space in which the sphere is embedded. These transformations are orthogonal, meaning that they preserve all distances.1 Incidentally, this gives us a 3-dimensional representation of the group SO(3), a concept we introduced in Chapter 2.

  Likewise, the group of rotations of the round table, which we have discussed above, is called SO(2); these rotations are special orthogonal transformations of the plane, which is 2-dimensional. Thus, we have a 2-dimensional representation of the group SO(2).

  The groups SO(2) and SO(3) are not only groups but also manifolds (that is, geometric shapes). The group SO(2) is the circle, which is a manifold. So SO(2) is a group and a manifold. That’s why we say that it is a Lie group. Likewise, elements of the group SO(3) are points of another manifold, but it is more tricky to visualize it. (Note that this manifold is not a sphere.) Recall that each rotation of the sphere is determined by the axis and the angle of rotation. Now observe that each point of the sphere gives rise to an axis of rotation: the line connecting this point and the center of the sphere. And the angle of rotation is the same as a point of a circle. So an element of the group SO(3) is determined by a point of the sphere (it defines the axis of rotation) together with a point of a circle (it defines the angle of rotation).

  Perhaps we should start with a simpler question: what is the dimension of SO(3)? To answer this question, we need to discuss the meaning of dimension more systematically. We have already mentioned in Chapter 2 that the world around us is three-dimensional. That is to say, in order to specify a position of a point in space, we need to specify three numbers, or coordinates, (x, y, z). A plane, on the other hand, is two-dimensional: a position on the plane is specified by two coordinates, (x, y). And a line is one-dimensional: there is only one coordinate.

  But what is the dimension of a circle? It is tempting to say that the circle is two-dimensional because we can draw the circle on the plane, which is two-dimensional. A circle is curved, and this creates the illusion of its dimension being two. But actually, its curvature has nothing to do with its dimension.

  Dimension is the number of independent coordinates within our geometric object, the circle, not within the landscape – in this case, the plane – into which the circle is embedded. In fact, the circle can also be embedded into a three-dimensional space (think of a hoop in rhythmic gymnastics), or a space of even larger dimension. The dimension of this ambient space is not relevant to the dimension of the circle itself. What matters is that a point on the circle i
s determined by one number – the angle. This is the sole coordinate on the circle. That’s why it is one-dimensional.

  To explain this slightly differently, let’s pick a point on the circle and ask ourselves: how many different paths within the circle are passing through our point? Clearly, there is only one such path, just like there is only one path through a point on the line. And there are only two directions in which we can move away from the point within the circle: back and forth along this path. Mathematicians say that there is one degree of freedom on the circle.

  Note that as we zoom in and look at a smaller and smaller neighborhood of a point of the circle, the curvature of the circle all but disappears. There is practically no difference between a small neighborhood of a point on the circle and a small neighborhood of the same point on the tangent line to the circle; this is the line that is the closest approximation to the circle near this point. This shows that the circle and the line have the same dimension.2

  As we zoom in on a point, the circle and the tangent line appear closer and closer to each other.

  Likewise, the sphere is embedded into a three-dimensional space, but its intrinsic dimension is two. Indeed, there are two independent coordinates on the sphere: latitude and longitude. We know them well because we use them to determine the position on the surface of the Earth, which is close to the shape of a sphere. The mesh on the sphere that we see on the above picture is made of the “parallels” and “meridians,” which correspond to fixed values of latitude and longitude. The fact that there are two coordinates on the sphere tells us that it is two-dimensional.

  What about the Lie group SO(3)? Every point of SO(3) is a rotation of the sphere, so we have three coordinates: the axis of rotation (which may be specified by a point at which the axis pierces the sphere) is described by two coordinates, and the angle of rotation gives rise to the third coordinate. Hence the dimension of the group SO(3) is equal to three.

  Thinking about a Lie group, or any manifold, of more than three dimensions can be very challenging. Our brain is wired in such a way that we can only imagine geometric shapes, or manifolds, in dimensions up to three. Even imagining the four-dimensional combination of space and time is a strenuous task: we just don’t perceive the time (which constitutes the fourth dimension) as an equivalent of a spatial dimension. What about higher dimensions? How can we analyze five- or six- or hundred-dimensional manifolds?

  Think about this in terms of the following analogy: works of art give us two-dimensional renderings of three-dimensional objects. Artists paint two-dimensional projections of those objects on the canvas and use the technique of perspective to create the illusion of depth, the third dimension, in their paintings. Likewise, we can imagine four-dimensional objects by analyzing their three-dimensional projections.

  Another, more efficient way to imagine a fourth dimension is to think of a four-dimensional object as a collection of its three-dimensional “slices.” This would be similar to slicing a loaf of bread, which is three-dimensional, into slices so thin that we could think of them as being two-dimensional.

  If the fourth dimension represents time, then this four-dimensional “slicing” is known as photography. Indeed, snapping a picture of a moving person gives us a three-dimensional slice of a four-dimensional object representing that person in the four-dimensional space-time (this slice is then projected onto a plane). Taking several pictures in succession, we obtain a collection of such slices. If we run these pictures quickly in front of our eyes, we can see that movement. This is of course the basic idea of cinema.

  We can also convey the impression of the person’s movement by juxtaposing the pictures. At the beginning of the twentieth century, artists got interested in this idea and used it as a way to include the fourth dimension into their paintings, to render them dynamic. A milestone in this direction was Marcel Duchamp’s 1912 painting Nude Descending a Staircase, No. 2.

  It is interesting to note that Einstein’s relativity theory, which demonstrated that space and time cannot be separated from each other, appeared around the same time. This brought the notion of the four-dimensional space-time continuum to the forefront of physics. In parallel, mathematicians such as Henri Poincaré were delving deeper into the mysteries of higher-dimensional geometry and transcending the Euclidean paradigm.

  Duchamp was fascinated with the idea of the fourth dimension as well as non-Euclidean geometry. Reading E.P. Jouffret’s book Elementary Treatise on Four-Dimensional Geometry and Introduction to the Geometry of n Dimensions, which in particular presented the groundbreaking ideas of Poincaré, Duchamp left the following note:3

  The shadow cast by a 4-dimensional figure on our space is a 3-dimensional shadow (see Jouffret – Geom. of 4-dim., page 186, last 3 lines).... by analogy with the method by which architects depict a plan of each story of a house, a 4-dimensional figure can be represented (in each one of its stories) by three-dimensional sections. These different stories will be bound to one another by the 4th dim.

  According to art historian Linda Dalrymple Henderson,4 “Duchamp found something deliciously subversive about the new geometries with their challenge to so many long-standing ‘truths.’ ” The interest of Duchamp and other artists of that era in the fourth dimension, she writes, was one of the elements that led to the birth of abstract art.

  Thus, mathematics informed art; it allowed artists to see hidden dimensions and inspired them to expose, in a tantalizing aesthetic form, some profound truths about our world. The works of modern art they created helped elevate our perception of reality, affecting our collective consciousness. This in turn influenced the next generations of mathematicians. Philosopher of science Gerald Holton put this eloquently:5

  Indeed, a culture is kept alive by the interaction of all its parts. Its progress is an alchemical process, in which all its varied ingredients can combine to form new jewels. On this point, I imagine that Poincaré and Duchamp are in agreement with me and with each other, both having by now undoubtedly met somewhere in that hyperspace which, in their different ways, they loved so well.

  Mathematics enables us to perceive geometry in all of its incarnations, shapes, and forms. It is a universal language that applies equally well in all dimensions, whether we can visualize the corresponding objects or not, and allows us to go far beyond our limited visual imagination. In fact, Charles Darwin wrote that mathematics endows us with “an extra sense.”6

  For example, though we cannot imagine a four-dimensional space, we can actualize it mathematically. We simply represent points of this space as quadruples of numbers (x, y, z, t), just like we represent points of the three-dimensional space by triples of numbers (x, y, z). In the same way, we can view points of an n-dimensional flat space, for any natural number n, as n-tuples of numbers (we can analyze these in the same way as the rows of a spreadsheet, as we discussed in Chapter 2).

  Perhaps I need to explain why I refer to these spaces as being flat. A line is clearly flat and so is a plane. But it’s not as obvious that we should think of the three-dimensional space as flat. (Note that I am not talking here about various curved manifolds embedded into the three-dimensional space, such as a sphere or a torus. I am talking about the three-dimensional space itself.) The reason is that it has no curvature. The precise mathematical definition of curvature is subtle (it was given by Bernhard Riemann, the creator of Riemann surfaces), and we won’t go into the details now as this is tangential to our immediate goals. A good way to think about the flatness of the three-dimensional space is to realize that it has three infinite coordinate axes that are perpendicular to each other, just as a plane has two perpendicular coordinate axes. Likewise, an n-dimensional space, with n perpendicular coordinate axes, has no curvature and hence is flat.

  Physicists have thought for centuries that we inhabit a flat three-dimensional space, but, as we discussed in the Preface, Einstein has shown in his general relativity theory that gravity causes space to curve (the curvature is small, so that we don’t notic
e it in our everyday life, but it is non-zero). Therefore our space is in fact an example of a curved three-dimensional manifold.

  This brings up the question of how a curved space could possibly exist by itself, without being embedded into a flat space of higher dimension, the way a sphere is embedded into a flat three-dimensional space. We are used to thinking that the space we live in is flat, and so in our everyday experience curved shapes seem to appear only within the confines of that flat space. But this is a misunderstanding, an artifact of our narrow perception of reality. And the irony is that the space we live in isn’t flat to begin with! Mathematics gives us a way out of this trap: as Riemann showed, curved spaces do exist intrinsically, as objects of their own making, without a flat space containing them. What we need to define such a space is a rule of measuring distances between any two points of this space (this rule must satisfy certain natural properties); this is what mathematicians call a metric. The mathematical concepts of metric and the curvature tensor, introduced by Riemann, are the cornerstones of Einstein’s general relativity theory.7

  Curved shapes, or manifolds, can have arbitrarily high dimensions. Recall that the circle is defined as the set of points on a plane equidistant from a given point (or, as my examiner at MGU insisted, the set of all such points!). Likewise, a sphere is the set of all points in the three-dimensional space equidistant from a given point. Now, define a higher-dimensional analogue of a sphere – some call it a hypersphere – as the set of points equidistant from a given point in the n-dimensional space. This condition gives us one constraint on the n coordinates. Therefore the dimension of the hypersphere inside the n-dimensional space is (n − 1). Further, we can study the Lie group of rotations of this hypersphere.8 It is denoted by SO(n).