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, . Since the orthogonal group is a subgroup of the unitary group, one expects its full centralizer on (denote it by ) to be larger than the symmetric group. It is easy to see that permuting tensor copies is still in . So contains the symmetric group. Now, consider the contraction map , where and are the indices of a pair of tensor copies. Let be a basis for . The action of this map on any basis vector is as follows (here )
Since we are dealing with the orthogonal group (i.e., the fundamental representation is isomorphic to its conjugate), 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 as . This is equivalent to conjugation by 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 tensor copies, any permutation of the last 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 ) and the quantum group . Before we talk about the duality itself, I want to talk about each of these algebras.
1. Hecke algebra
A Hecke algebra is sometimes called a quantum generalization of the symmetric group although -deformation of , which is also used, is probably more appropriate. It is generated by elements (and their inverses) for , which satisfy the following relations.
which are the usual relations in the braid group and in addition, we have the relation
There are several types of Hecke algebras and the above one is usually called Iwahori-Hecke algebra. The irreducible representations of this algebra have been worked out by Wenzl. They bear a striking resemblance (perhaps not surprisingly) to the representation theory of . It turns out that the irreducible representations of can also be labeled by Young diagrams with boxes. When is not zero or a primitive root of unity, all the Young diagrams with boxes correspond to irreducible representations. For the case when is a root of unity, the irreducible representations belong to a restricted set. First, let us look at the case when is any complex number but not zero or a root of unity. In this case, it turns out that is isomorphic to the symmetric group algebra. The action of a generator inside an irreducible representation can be described as follows. Suppose is a standard Young tableau of shape and let be the tableau with and interchanged. Recall that a tableau can be identified with a path in the Bratelli diagram of the symmetric group. Let be the Manhattan distance between and in and let denote the quantum integer
Quantum numbers (or numbers) can be extended to operators, where is replaced by an operator . Using this notation, we have
if and are not in the same row or column. If they are in the same row or column, then one can put to be zero above. This action bears striking resemblance to the action of a transposition of the symmetric group inside an irreducible representation.
Now, moving onto the case when is a root of unity (say ), it turns out that the Hecke algebra is not semisimple. The irreducible representations of are still labeled by Young diagrams but with a restriction. One only considers the so-called Young diagrams, where . The Jones representations of the braid group are for . A Young diagram has at most parts and . Let the set of all box diagrams be . Now a standard tableau is one whose path in the Bratelli diagram only passes though other diagrams. It turns out that on the space of such paths, there is an irreducible representation of (denoted ). For a Young diagram, recall that the standard tableau of shape span the irreducible space of the symmetric group (or the Hecke algebra at non-zero or non roots of unity). For a root of unity, the irreducible space is a subspace of since we consider only standard tableau. In fact, it is an open problem (as far as I know) to obtain a closed form expression for this dimension (although recursive procedures to calculate it exist). The action of a generator in this irreducible space is given as follows ( and have the same meaning as before).
If and are in the same row, we get and if they are in the same column, . This description of the Hecke algebra has been worked on by Wenzl in this paper. Wocjan and Yard in this paper, use this to construct an efficient quantum algorithm to approximate the Jones and HOMFLYPT polynomials.
2. Hopf algebras and quantum groups
Now let us look at the algebras that make up the dual side. These are -deformations of as one might guess (since on the “primal” side, we have a deformation of the symmetric group). However, the way to -deform them is far from obvious. In order to describe this, let us look at Hopf algebras and quantum doubles of (compatible) Hopf algebras. It turns out that the quantum double construction gives us the appropriate -deformation of the universal enveloping algebra . Hopf algebras are the language in which one can describe the quantum double construction. This construction, proposed by Drinfeld, has the very important property of producing Hopf algebras that are braided. In other words, the doubled Hopf algebra comes equipped with so called universal matrices, which satisfy Yang-Baxter equations (more precisely their quantum versions). Kassel’s book has a nice description of Hopf algebras and quantum doubles and a lot of the discussion below is based on that.
A Hopf algebra is a vector space which has an (associative) algebra structure (i.e., with a multiplication map and a unit map defined in the usual way) and something called a (coassociative) coalgebra structure, which comes with comultiplication and a counit. Moreover, the algebra and coalgebra structures must be compatible in the sense that the coalgebra structure must be an algebra homomorphism (or vice versa). More precisely, a coalgebra is defined with respect to comultiplication and a counit . A comultiplication can be thought of as a description of how acts on the tensor product of two representations. For instance, for a finite group (which is also a Hopf algebra), , which corresponds to the diagonal action. For a general Hopf algebra, it can be a linear superposition of tensor product terms. The Sweedler notation is useful when dealing with the general case. In this notation, one denotes or sometimes written as . This stands for the sum . The point is that it gets very confusing to keep track of all the variables and indices if we introduce more than one for each . So in the Sweedler notation becomes
By coassociativity, this is equal to , which is
Since the above two equations are the same, we can drop the extra indices and write
In fact, sometimes the summation symbol itself is dropped. While there is no ambiguity in dropping the summation symbol because the indices are enough to remind us that there is a sum, it may be better to have the to keep things explicit.
The counit is a map from the coalgebra to the complex numbers i.e., . This can be thought of as being equivalent to the trivial representation. In other words, we have . The compatibility condition of the two structures implies that and i.e., they are algebra homomorphisms. We also need , where is the unit map of the algebra structure and is the identity of . In fact, so far we only have a bialgebra. In order make a Hopf algebra, one also needs an antipode with the property that . The antipode generalizes inverses in . In fact, in a group, is the usual inverse i.e., . A Hopf algebra is called commutative if the algebra structure is commutative i.e., and it is called cocommutative if the coalgebra structure is cocommutative i.e., , where is the operator that swaps the tensor copies.
Once we have a Hopf algebra, we can impose extra structure on it that makes it more interesting. One important class of Hopf algebras are called braided or quasitriangular Hopf algebras. These are interesting because they give rise to solutions of the Yang-Baxter equations and hence to representations of the braid group. A braided Hopf algebra has a so called matrix which is an invertible element of satisfying the following equations.
which says that is cocommutative up to conjugation by . In addition, we need
In the above, if , then and similarly for the others. It is easy to show that a braided Hopf algebra satisfies the quantum Yang-Baxter equations, which are given by
3. Quantum doubles
Now, I would like to describe a construction by Drinfeld that gives systematic solutions to Yang-Baxter equations. It also gave one of the first examples of Hopf algebras that are neither commutative nor co-commutative. The construction takes a Hopf algebra and its dual and combines them in a fashion that looks like a semidirect product on groups (actually it is closer to another operation on groups called bicrossed product). The result is a Hopf algebra that is braided and hence has matrices that give rise to braid group representations.
Suppose that is a Hopf algebra. Recall that and are the multiplication and unit maps of the algebra structure and and are the comultiplication and counit maps. Now, one can construct its dual and this turns out to be the Hopf algebra . Technically, the dual is not of , but of the so called opposite Hopf algebra . In other words, the multiplication in is , which takes to rather than . It’s not hard to see that for this dual, the algebra and coalgebra structures are exchanged – with a small change: the comultiplication of the coalgebra structure comes from the opposite algebra. On the tensor product of these two Hopf algebras denoted , one can obtain a Hopf algebra structure as follows. First, as a vector space, it is . It is easy to describe the coalgebra structure, so let us do that first. The comultiplication map is simply a tensor product of the individual comultiplication maps (and then a rearranging of the tensor copies). In other words,
where swaps the second and third tensor copies. The counit is also the tensor product of the individual counits i.e., , which is . One needs to work harder for the algebra structure. In order to define algebra structure, a construction known as bicrossed products of algebras is exploited. Bicrossed products can be defined on a pair of algebras that are compatible. This compatibility is the main reason we need the dual to be rather than . Another way to look at the product might be useful. We think of as consisting of creation and annihilation operators. The creation operators can be thought of as belonging to and the annihilation operators to . Then, defining the product means defining the normal order. So in order to define what we mean by , we have to define how to normal order the expression. Rewriting the expression in more shorthand notation as , we need to normal order it i.e., commute past . Using bicrossed products, this turns out to be
where and is the natural embedding of and into . Typically, in a normal ordering, one has annihilation operators to the right. This can also be done by modifying the above formulae appropriately.
where is some basis of and is the dual basis. Here even though we have considered the matrix in what can be thought of as the fundamental representation of the doubled Hopf algebra, one can extend this definition of the matrix to any pair of representations of the double. In other words, if and are two finite dimensional representation of , then on the tensor product of the two, the matrix will be
where the s are the identity on the respective spaces.
4. The quantum group and its representations
It turns out that one can construct -deformations of Lie algebras as quantum doubles of the Borel subalgebras. More precisely, let (resp. ) be the Borel subalgebra consisting of the Cartan subalgebra and the set of positive roots (resp. negative roots). One can show that the universal enveloping algebras and the (and their quantum versions) are Hopf algebras dual to each other. The quantum double of (the quantum versions of) these two is denoted . Let us now take a closer look at the structure of the quantum group .
is defined to be the following algebra generated by and .
A couple of remarks:
- While one cannot simply substitute in the above equations to obtain , it turns out that there is an equivalent formulation of the above algebra where one can substitute and obtain the usual universal enveloping algebra as .
- Sometimes is written as to obtain a definition that looks closer to the defining relations of .
The Hopf algebra structure on is given by
and the group like elements
The map is zero for and and for and . The antipode is defined as
The matrix can be obtained from the fact that it is a quantum double (i.e., using the Eq.~(14)) and turns out to be
where and . It is easy to see that the matrices generate the Hecke algebra. Identifying with , we can see that it satisfies the definition of the Hecke algebra (the Yang-Baxter equation turns into the braid group condition). Another consequence of the braiding is that the matrices and hence the Hecke algebra commutes with the action of the quantum group. This can be seen from the equation , which essentially expresses the fact that the action of the quantum group commutes with .
The irreducible representations of for a root of unity are different from the case when is a root of unity. First, let us look at the simpler case of when is not a root of unity. In this case, the irreducible representations can be labeled by the highest weight , which in turn depends on an integer as , where . Given an , one can construct an dimensional irreducible module with highest weight . If is the highest weight vector, then we have and . Defining , we obtain a basis of the finite dimensional module. The action of the generators on this space is as follows.
It turns out that this module is irreducible, any simple finite dimensional module is generated by a highest weight vector and any two finite dimensional modules with the same highest weight are isomorphic. In this sense, the structure of finite dimensional simple modules parallels that of Lie algebras.
The case when is a root of unity is more complicated since one can have nilpotent elements. First define to be the order of (smallest integer such that ). Then define if is odd and if it is even. The source of the complications comes from the fact that and, in fact, for any multiple of . In spite of this, it turns out that simple modules of dimension smaller than are the same as for the generic case and that there are no simple modules of dimension greater than . For dimension equal to , there turn out to be two kinds of simple modules. These are described in Kassel’s book.
5. Ribbon Hopf algebras
There is a special kind of structure one can impose on a Hopf algebra that makes it a ribbon Hopf algebra. The quantum groups discussed above happen to be ribbon Hopf algebras. Exploiting this structure makes it easy to calculate irreducible representations of centralizer algebras of quantum groups. This viewpoint is nicely developed in this paper by Leduc and Ram. A ribbon Hopf algebra is a braided Hopf algebra with a ribbon element i.e., an invertible element in the center of the algebra. Formally, a ribbon Hopf algebra is braided (i.e., has matrices) and has an element such that
In the above equations the element is
where and come from the matrix i.e.,
For a quantum group , where is a simple Lie algebra, the ribbon element is , where is the so called Weyl vector (half the sum of positive roots).
6. Quantum Schur-Weyl duality
It turns out that there is a Schur-Weyl duality between the quantum group and the Hecke algebra of type on the fold tensor product space , where is the dimensional representation of . However, when is a root of unity, the actions of and turns out to be non-semisimple even though they are full commutants of each other. But first, let us look at the more straightforward case (or generic case) when is not a root of unity. In general, this duality extends to the quantum group corresponding to a simple Lie algebra , where is the fundamental representation of . The full centralizer of the tensor product action of on copies of the fundamental representation of is either an Iwahori-Hecke algebra or the Birman-Wenzl-Murakami (BMW) algebra depending on whether is of type , , or . Type gives the Iwahori-Hecke algebra and other types give the BMW algebra.
6.1. Generic case
In this case, it is easy to show that any finite dimensional module is semisimple. It also turns out that is semisimple (although it follows from the facts that action is semisimple and that is the full centralizer of that action). This means that we can decompose the tensor product space into irreducible spaces corresponding to the two actions and these spaces can be labeled by Young diagrams just as in classical Schur-Weyl duality.
where and are irreducible spaces of and respectively. This duality can be extended to and . The proof of this in the generic case is not so hard. There are several ways to prove it. One uses semisimplicity of the two algebras. One can also use the fact that the Hecke algebra is isomorphic to the symmetric group algebra in this case, then use the fact that their dimensions are . We have seen that the Hecke algebra is generated by the universal matrices (composed by the tensor swap). This means that actions of the quantum group and the Hecke algebra are already in the centralizer of each other.
6.2. Roots of unity
Things are not so simple in this case. First, the algebras (Hecke and the quantum group) are not semisimple. Although they are still full commutants of each other, it is not very satisfying to have nonsemisimple algebras. Moreover, this is the case that seems interesting from the point of view of link invariants and topological quantum computing. This is because the link invariants at roots of unity are more powerful and include Jones and HOMFLYPT polynomials. From the point of view of topological quantum computing, the representations of the braid group that one obtains at the roots of unity are dense in the unitary group and hence are `universal’ in the sense that one can express any unitary as a sequence of braids to arbitrary precision. One can appeal to the Solovay-Kitaev theorem to show that the number of crossings is polynomial in the number of strands.
An interesting question is whether there are quotients of these algebras (say and ) that are full centralizers of each other and are semisimple. For instance, could be the following quotient of . Recall that we defined earlier. Let be the direct sum of , where runs over all diagrams. The action of restricted to this space is preserved because this space consists of simple modules. Because of non-semisimplicity, the action of on the complement of this space need not be preserved. In this paper, Goodman and Wenzl analyze the Littlewood-Richardson (LR) coefficients of this quotient and find that there is an interesting duality called level-rank duality (which one finds in the fusion rules of certain conformal field theories). These LR coefficients themselves occur in certain WZW models. In fact, the irreducible representations of can be used to express the Jones polynomial and the one variable HOMFLYPT polynomial of a the trace closure of any braid. Suppose is a braid on strands, then the Jones polynomial of the trace closure of the braid at a root of unity is given by
where is the sum of the exponents of the generators in the word for (recall that any element can be expressed as a word in and ). Here we denote by the element of the Hecke algebra corresponding to the braid (by a slight abuse of notation). The quantity has the following expression
where recall that is the representation of the Hecke algebra inside a Young diagram and Tr is the usual trace.
The Schur functions are given by the following formula.
where is the usual hook length of the box with coordinates . The above formula for the HOMFLYPT polynomial suggests that the Schur functions are proportional to the dimensions of irreducible representations of the commutant of the quotient . The following are some very interesting questions that I don’t know the answers to.
- What quotient of is the commutant of ? The dimensions of its irreducible representations should be proportional to . How does this quotient relate to the WZW models mentioned above?
- What is the space on which acts? Presumably, this space does not have a tensor product structure. Of course, one way to specify it is as the space of all paths on the Bratelli diagram, but it would be interesting to know how to project on to this space.
- How can we write in terms of generators and relations? Since it is a quotient of , there must extra relations that satisfy.