Central extensions

1. Introduction

Affine Lie algebras are infinite dimensional generalizations of Lie algebras and are special cases of Kac-Moody algebras. They turn out to be extremely useful in describing the symmetry properties of conformal field theories–quantum field theories with conformal invariance. They can be defined in a couple of ways starting from a Lie algebra. One way is to consider the loop algebras associated with a Lie algebra and then consider central extensions of this loop algebra. The second and equivalent way is to define them using the Cartan matrix of a Lie algebra and extending it (by an additional row and column) and then generalize the Serre relations. Recall that a Lie algebra can be defined using its Cartan matrix and Serre relations. Extending Lie algebras in this way would lead to a modified Dynkin diagram where the diagram corresponding to the Lie algebra would have an added root. I plan to discuss both these approaches but in this post, I’ll discuss the central extension point of view as it leads naturally to geometric aspects of affine Lie algebras. This point of view is explained in the book by Pressley and Segal and most of what I have in this post is from this book. I also found notes by Ko Honda useful, which are based on the book by Kohno. Continue reading

Vector bundles and their classification

1. Introduction

One of the things I find most fascinating is the interplay between mathematics and physics. One direction of this interplay is well-known i.e., how mathematics forms the basis of all physical concepts. In fact, this is now so ingrained in our minds that we naturally expect any new physical concept to have an appropriate mathematical description. Wigner calls this the “unreasonable effectiveness of mathematics” in his famous article. What is surprising, perhaps, is the other direction i.e., how physical intuition leads to new developments in mathematics. This is especially true with quantum physics and geometry.

Quantum field theories such as Yang-Mills and Chern-Simons theories provide a deep understanding of certain aspects of topology and geometry. Starting around in the 80s, these developments have led to the creation of the field of quantum topology and quantum algebra. Some of these results, which took mathematicians by surprise, were obtained from deep physical intuition. These developments are beautifully explained in a 2010 article by Atiyah, Dijkgraaf and Hitchin. For instance, understanding the space of the so called self-dual solutions of Yang-Mills theories leads to powerful invariants of four manifolds. Similarly, studying the holonomy of Wilson loop operators and their vacuum expectation values gives a new description of the Jones polynomial.

A basic and extremely important structure in mathematical physics and in the results mentioned above is that of a vector bundle. Several physical theories such as Yang-Mills theory and Chern-Simons theory can be cast in terms of vector bundles. Moreover, natural bundles on manifolds such as the tangent and cotangent bundles are useful in descriptions of general relativity and the phase space of classical mechanical systems (the latter, in fact, leads to the very useful concept of geometric quantization, which produces a quantum mechanical Hilbert space from the classical mechanical phase space). The concept of holonomy in a vector bundle, which measures the deviation produced in a standard vector when it is parallel transported around a loop, turns out to be the underlying mathematical description of the Aharonov-Bohm effect and the quantum field theoretic formulation of the Jones polynomial. These are just some examples of the use of vector bundles.

In this post, I want to discuss some aspects of vector bundles that are useful in the above applications. Aside from the definition and examples, I will talk about how finite dimensional vector bundles can be classified using homotopy. Continue reading

The far right in the United States

I’ve never been this disappointed in an election outcome. Even when Bush was elected a second time, I don’t remember feeling this emotional. For a while preceding this election, I used to wonder how much sway the far right have in the US. A lot of countries in western Europe like France, Germany, UK, Austria, Switzerland all have far right parties, although they are mostly in the fringes. In the US, there is the Tea Party of course, but it operates within the Republican party. How much influence does it have? There was some reason prior to this election to think that its social conservative influence was waning.

The reason for optimism is that in 2008 and then again in 2012, when the Republicans lost the general election, they did a lot of soul searching. In fact, they hired a few consultants to look into their failure and provide recommendations. That resulted in a 100 page report (you can see it here), which the press has since called the autopsy report. The report was a result of analysis and data collected from thousands of interviews. The report makes several recommendations, the most important of which is that the RNC should do a lot more to embrace women and minorities, especially hispanic voters. The Republicans should make them feel more welcome and not appear to be a party that wants them to disappear. The autopsy report was taken very seriously by many in the party. Ted Cruz and Marco Rubio organized their campaign strategies around it. Paul Ryan and even Newt Gingrich praised it as a model for the party in the coming years. It seemed at the time that the Republican party is moving away from far right tendencies or even away from social conservatism (that always seemed like a bit of a stretch). Of course, this was just a strategic move and not a major idealogical shift for the party. But still, this is not the direction that a far right party or a party mostly influenced by a far right ideology would take.

Trump, who derided the report from the beginning (even before he entered the race), did everything that flies in the face of the report. I don’t have to recount all the ways in which he did this. This, of course, made many in his party very uncomfortable and it seemed for a while that the party was tearing itself apart. But when he started getting more and more support and eventually won the primary, it seemed clear that the far right movement is not that small. But how big is it? It wasn’t clear and back then, I don’t think anyone really knew. That was also the time when people talked of a Clinton landslide. If Clinton had won (landslide or not), the Republican party would have been madder than ever at these Trump nationalists and it seemed to me that Trump and his supporters would have ended up starting their own party. That made a lot of sense because how could the US not have a far right movement when it is only growing everywhere else. That would also weaken the republican party and they would have to do more soul searching to reinvent themselves. I thought and even hoped at the time that this is the direction things would move.

But this unexpected and unthinkable election outcome has shown how flawed that thinking is. It has become abundantly clear that the far right movement is much stronger than anyone imagined. In fact, the US has had a far right party all along—the Republican party and it is not in the fringes like in western Europe. The far right is now the mainstream in the United States and it has all three branches of the government at its mercy.

Quantum Schur-Weyl Duality

There are several generalizations of Schur-Weyl duality, which was discussed in the previous post. In this post, I am mainly interested in one of these generalizations, which is known as quantum Schur-Weyl duality. But before talking about it, I want to briefly talk about the other (non-quantum) generalizations. In the first generalization, we replace the diagonal action of the unitary group with the diagonal action of the orthogonal group, {O(n)}. Since the orthogonal group is a subgroup of the unitary group, one expects its full centralizer on {V^{\otimes n}} (denote it by {A}) to be larger than the symmetric group. It is easy to see that permuting tensor copies is still in {A}. So {A} contains the symmetric group. Now, consider the contraction map {T_{ij}}, where {i} and {j} are the indices of a pair of tensor copies. Let {B=\{v_1,\dots, v_d\}} be a basis for {V}. The action of this map on any basis vector is as follows (here {u_i\in B})

\displaystyle  T_{i,j}|u_1,\dots ,u_n\rangle = \langle u_i|u_j\rangle\sum_k|u_1,\dots, u_{i-1}, v_k, \dots , u_{j-1}, v_k, u_n\rangle\,. \ \ \ \ \ (1)

Since we are dealing with the orthogonal group (i.e., the fundamental representation is isomorphic to its conjugate), {\sum_k |v_k,v_k\rangle} is the trivial representation of the diagonal action. So, it is easy to see that the contraction map commutes with the diagonal action. It turns out that there are no more elements that generate this algebra. This algebra is called the Brauer algebra. It is a diagram algebra in the sense that its basis elements can be represented using diagrams. Another interesting generalization is the action of the unitary group, where it action on {V^{\otimes 2n}} as {U^{\otimes n}\otimes U^{\ast \otimes n}}. This is equivalent to conjugation by {U^{\otimes n}} using the Choi-Jamiolkowski isomorphism. It turns out that the centralizer algebra of this action of the unitary group is the so called walled Brauer algebra. This algebra is generated by the following elements: any permutation of the first {n} tensor copies, any permutation of the last {n} copies and contraction maps between pairs of tensor copies, where one of them is from the first half and the other from the second half. It is again easy to see that these elements (and the algebra generated by them) are in the centralizer. However, as always, proving that the centralizer is generated by these (and no more) elements is the hard part.

The generalization that I want to talk about next is the duality between the Hecke algebra (of type {A_{n-1}}) and the quantum group {U_q(\mathfrak{sl}_2)}. Before we talk about the duality itself, I want to talk about each of these algebras. Continue reading

Schur-Weyl Duality

In this post, I discuss Schur-Weyl duality, its proof and some of its applications to the field of quantum information. With all the background, this post ended up being longer than I anticipated. I might stick to shorter posts in the future. There is some material that may be new, such as the dual version for the Schur transform and the description of immanants. I have a feeling that this may be known to some experts and just not written up anywhere.

1. Statement of Schur-Weyl duality

Schur-Weyl duality (in one form) says that the diagonal action of the general linear group commutes with the action of the symmetric group on the tensor product of finite dimensional vector spaces. Consider the {n}-fold tensor product {V^{\otimes n}} of a finite dimensional vector space {V} of dimension {d}. We have the diagonal action of the general linear group {\text{GL}(V)}, which applies an invertible operator {A} on each tensor copy. For this action of the general linear group, we are interested in the centralizer algebra i.e., {\text{End}_{\text{GL}(V)}V^{\otimes n}}. It is easy to see that the set of operators that permute the tensor copies are in this algebra and the question is – are there more? Schur-Weyl duality is the statement that there are no more operators and so, this centralizer algebra is isomorphic to the symmetric group algebra. We will actually consider the action of the so-called universal enveloping algebra {U(\mathfrak{gl}(V))} of the Lie algebra {\mathfrak{gl}(V)} instead of the general linear group. Schur-Weyl duality can then be stated as

Theorem 1 The image of {\mathbb{C}[S_n]} and the image of {U(\mathfrak{gl}(V))} in {\text{End}_{\mathbb{C}}(V^{\otimes n})} are centralizers of each other.

Before we prove this theorem, let us recall some facts about universal enveloping algebras of Lie algebras. Continue reading