Abstract
Abstract. We discuss and compare the notions of braided and coboundary monoidal categories. Coboundary monoidal categories are analogues of braided monoidal categories in which the role of the braid group is replaced by the cactus group. We focus on the categories of representations of quantum groups and crystals and explain how while the former is a braided monoidal category, this structure does not pass to the crystal limit. However, the categories of representations of quantum groups of finite type also possess the structure of a coboundary category which does behave well in the crystal limit. We explain this construction and also a recent interpretation of the coboundary structure using quiver varieties. This geometric viewpoint allows one to show that the category of crystals is in fact a coboundary monoidal category for arbitrary symmetrizable Kac-Moody type.
Introduction
In this expository paper we discuss and contrast two types of categories -braided monoidal categories and coboundary monoidal categories -paying special attention to how the categories of representations of quantum groups and crystals fit into this framework. Monoidal categories are essentially categories with a tensor product, such as the categories of vector spaces, abelian groups, sets and topological spaces. Braided monoidal categories are wellstudied in the literature. They are monoidal categories with an action of the braid group on multiple tensor products. The example that interests us the most is the category of representations of a quantum group U q (g). Coboundary monoidal categories are perhaps less well known than their braided cousins. The concept is similar, the difference being that the role of the braid group is now played by the so-called cactus group. A key component in the definition of a coboundary monoidal category is the cactus commutor, which assumes the role of the braiding.
Introduction
The theory of crystals can be thought of as the q → ∞ (or q → 0) limit of the theory of quantum groups. In this limit, representations are replaced by combinatorial objects called crystal graphs. These are edge-colored directed graphs encoding important information about the representations from which they come. Developing concrete realizations of crystals is an active area of research and there exist many different models. It is interesting to ask if the structure of a braided monoidal category passes to the crystal limit. That is, does one have an induced structure of a braided monoidal category on the category of crystals. The answer is no. In fact, one can prove that it is impossible to give the category of crystals the structure of a braided monoidal category (see Proposition 5.6). However, the situation is more hopeful if one instead considers coboundary monoidal categories. For quantum groups of finite type, there is a way -a unitarization procedure introduced by Drinfel ′ d [5] -to use the braiding on the category of representations to define a cactus commutor on this category. This structure passes to the crystal limit and one can define a coboundary structure on the category of crystals in finite type (see [7, 10] ). Kamnitzer and Tingley [9] gave an alternative definition of the crystal commutor which makes sense for quantum groups of arbitrary symmetrizable Kac-Moody type. However, while this definition agrees with the previous one in finite type, it is not obvious that it satisfies the desired properties, giving the category of crystals the structure of a coboundary category, in other types.
Introduction
In [24] , the author gave a geometric realization of the cactus commutor using quiver varieties. In this setting, the commutor turns out to have a very simple interpretation -it corresponds to simply taking adjoints of quiver representations. Equipped with this geometric description, one is able to show that the crystal commutor satisfies the requisite properties and thus the category of crystals, in arbitrary symmetrizable Kac-Moody type, is a coboundary category.
Introduction
In the current paper, when discussing the topics of quantum groups and crystals, we will often restrict our attention to the quantum group U q (sl 2 ) and its crystals. This allows us to perform explicit computations illustrating the key concepts involved. The reader interested in the more general case can find the definitions in the references given throughout the paper.
Introduction
The organization of this paper is as follows. In Section 1, we introduce the braid and cactus groups that play an important role in the categories in which we are interested. In Sections 2 through 4, we define monoidal, braided monoidal, and coboundary monoidal categories. We review the theory of quantum groups and crystals in Section 5. In Section 6 we recall the various definitions of cactus commutors in the categories of representations of quantum groups and crystals. Finally, in Section 7, we give the geometric interpretation of the commutor.
Introduction
The author would like to thank J. Kamnitzer and P. Tingley for very useful discussions during the writing of this paper and for helpful comments on an earlier draft.
Introduction
We thus have a surjective group homomorphism B n ։ S n . The kernel of this map is called the pure braid group.
Introduction
The braid group has several geometric interpretations. The one from which its name is derived is the realization in terms of braids. An n-strand braid is an isotopy class of a union of n non-intersecting smooth curve segments (strands) in R 3 with end points {1, 2, . . . , n} × {0} × {0, 1}, such that the third coordinate is strictly increasing from 0 to 1 in each strand. The set of all braids with multiplication giving by placing one braid on top of another (and rescaling so that the third coordinate ranges from 0 to 1) is isomorphic to B n as defined algebraically above.
Introduction
The braid group is also isomorphic to the mapping class group of the n-punctured disk -the group of self-homeomorphisms of the punctured disk with n-punctures modulo the subgroup consisting of those homeomorphisms isotopic to the identity map. One can picture the isomorphism by thinking of each puncture being connected to the boundary of the disk by a string. Each homeomorphism of the n-punctured disk can then be seen to yield a braiding of these strings. The pure braid group corresponds to the classes of homeomorphisms that map each puncture to itself.
Introduction
A similar geometric realization of the braid group is as the fundamental group of the configuration space of n points in the unit disk D. A loop from one configuration to itself in this space defines an n-strand braid where each strand is the trajectory in D × [0, 1] traced out by one of the n points. If the points are labeled, then we require each point to end where it started and the corresponding fundamental group is isomorphic to the pure braid group.
Introduction
1.2. The cactus group. Fix a positive integer n. For 1 ≤ p < q ≤ n, let
Introduction
Since s i,i+1 = s i , these elements generate S n . If 1 ≤ p < q ≤ n and 1 ≤ k < l ≤ n, we say that p < q and k < l are disjoint if q < k or l < p. We say that p < q contains k < l if p ≤ k < l ≤ q.
Definition 1.2 (Cactus group).
For n a positive integer, the n-fruit cactus group J n is the group with generators s p,q for 1 ≤ p < q ≤ n and relations (1) s 2 p,q = 1, (2) s p,q s k,l = s k,l s p,q if p < q and k < l are disjoint, and (3) s p,q s k,l = s r,t s p,q if p < q contains k < l, where r =ŝ p,q (l) and t =ŝ p,q (k).
Definition 1.2 (Cactus group).
It is easily checked that the elementsŝ p,q of the symmetric group satisfy the relations defining the cactus group and thus the map s p,q →ŝ p,q extends to a surjective group homomorphism J n ։ S n . The kernel of this map is called the pure cactus group.
Definition 1.2 (Cactus group).
The cactus group also has a geometric interpretation. In particular, the kernel of the surjection J n ։ S n is isomorphic to the fundamental group of the Deligne-Mumford compactification M n+1 0 (R) of the moduli space of real genus zero curves with n + 1 marked points. The generator s p,q of the cactus group corresponds to a path in M n+1 0 (R) in which the marked points p, . . . , q balloon off into a new component, this components flips, and then the component collapses, the points returning in reversed order. Elements of M n+1 0 (R) look similar to cacti of the genus Opuntia (the marked points being flowers) which justifies the Figure 1) . Note the similarity with the last geometric realization of the braid group mentioned in Section 1.1. Both the pure braid group and the pure cactus group are fundamental groups of certain spaces with marked points. We refer the reader to [3, 4, 7] for further details on this aspect of the cactus group.
Definition 1.2 (Cactus group).
1.3. The relationship between the braid and cactus groups. The relationship between the braid group and the cactus group is not completely understood. While the symmetric group is a quotient of both, neither is a quotient of the other. However, there is a homomorphism from the cactus group into the pro-unipotent completion of the braid group (see the proof of Theorem 3.14 in [6] ). This map is closely related to the unitarization procedure of Drinfel ′ d to be discussed in Section 6.1. In the next few sections, we will define categories that are closely related to the braid and cactus groups. We will see that there are some connections between the two. In addition to the unitarization procedure, we will see that braidings satisfy the so-called cactus relation (see Proposition 4.3).
Monoidal categories
If the maps ϕ U are all isomorphisms, we call ϕ a natural isomorphism. We will sometimes refer to the ϕ U themselves as natural isomorphisms when the functors involved are clear. (1) a bifunctor ⊗ : C × C → C called the tensor product, (2) natural isomorphisms (the associator)
Monoidal categories
for all U, V, W ∈ Ob C satisfying the pentagon axiom: for all U, V, W, X ∈ Ob C, the diagram
Monoidal categories
commutes, and (3) a unit object 1 ∈ Ob C and natural isomorphisms
Monoidal categories
for every V ∈ Ob C, satisfying the triangle axiom: for all U, V ∈ Ob C, the diagram
Monoidal categories
A monoidal category is said to be strict if we can take α U,V,W , λ V and ρ V to be identity morphisms for all U, V, W ∈ Ob C. That is, a strict monoidal category is one in which
Monoidal categories
The MacLane Coherence Theorem [18, §VII.2] states that the pentagon and triangle axioms ensure that any for any two expressions obtained from V 1 ⊗ V 2 ⊗ · · · ⊗ V n by inserting 1's and parentheses, all isomorphisms of these two expressions consisting of compositions of α's, λ's and ρ's are equal. This condition is called the associativity axiom. In a monoidal category, we can use the natural isomorphisms to identify all expressions of the above type and so we often write multiple tensor products without brackets. In fact, every monoidal category is equivalent to a strict one [18, §XI.3].
Examples.
Most of the familiar tensor products yield monoidal categories. For instance, for a commutative ring R, the category of R-modules is a monoidal category. We have the usual tensor product A ⊗ R B of modules A and B. The unit object is R and we have the natural isomorphisms
Examples.
In particular, the categories of abelian groups (where R = Z) and vector spaces (where R is a field) are monoidal categories. In a similar fashion, the category of R-algebras is monoidal under the usual tensor product of algebras. For an arbitrary (not necessarily commutative) Therefore, if we write σ 1 for the map σ ⊗ id and σ 2 for the map id ⊗ σ, we have
Examples.
the Yang-Baxter relation for B 3 . It follows that in a braided monoidal category, the braid group B n acts on n-fold tensor products. That is, if we denote id 
Examples.
In any symmetric monoidal category, the symmetric group S n acts on n-fold tensor products in the same way that the braid group acts in a braided monoidal category. We note that there is conflicting terminology in the literature. For instance, some authors refer to monoidal categories as tensor categories while others (see, for instance, [2] ) refer to symmetric monoidal categories as tensor categories and braided monoidal categories as quasitensor categories.
Examples.
Many of the examples in Section 2.2 can in fact be given the structure of a symmetric monoidal category. In particular, the categories of R-modules and R-algebras over a commutative ring R, the category of sets, and the category of topological spaces are all symmetric monoidal categories. In all of these examples, the braiding is given by
Examples.
An example that shall be especially important to us is the category of representations of a quantum group. It can be given the structure of a braided monoidal category but is not a symmetric monoidal category.
Coboundary monoidal categories
4.1. Definitions. Coboundary monoidal categories are analogues of braided monoidal categories in which the role of the braid group is replaced by the cactus group. As we shall see, they are better suited to the theory of crystals than braided monoidal categories.
Definition 4.1 (Coboundary monoidal category). A coboundary monoidal category is a monoidal category C together with natural isomorphisms
and (2) the cactus relation: for all U, V, W ∈ Ob C, the diagram
Definition 4.1 (Coboundary monoidal category). A coboundary monoidal category is a monoidal category C together with natural isomorphisms
The collection of maps σ c U,V is called a cactus commutor.
Definition 4.1 (Coboundary monoidal category). A coboundary monoidal category is a monoidal category C together with natural isomorphisms
We will use the term commutor for a collection of natural isomorphisms σ U,V : U ⊗ V → V ⊗ U for all objects U, V ∈ Ob C (this is sometimes called a commutativity constraint), reserving the term cactus commutor for a commutor satisfying the conditions in the definition above.
Definition 4.1 (Coboundary monoidal category). A coboundary monoidal category is a monoidal category C together with natural isomorphisms
Suppose C is a coboundary category and U 1 , . . . , U n ∈ Ob C. For 1 ≤ p < q ≤ n, one defines natural isomorphisms 
Definition 4.1 (Coboundary monoidal category). A coboundary monoidal category is a monoidal category C together with natural isomorphisms
Therefore, in a coboundary monoidal category the cactus group J n acts on n-fold tensor products. This is analogous to the action of the braid group in a braided monoidal category. For this reason, the authors of [7] propose the name cactus category for a coboundary monoidal category. Proof. Suppose C is a braided monoidal category with braiding σ and U, V, W ∈ Ob C. By the axioms of a braided monoidal category, we have
Definition 4.1 (Coboundary monoidal category). A coboundary monoidal category is a monoidal category C together with natural isomorphisms
The first equality uses the definition of a braiding and the second is the Yang-Baxter equation.
Definition 4.1 (Coboundary monoidal category). A coboundary monoidal category is a monoidal category C together with natural isomorphisms
Note that Proposition 4.3 does not imply that every braided monoidal category is a coboundary monoidal category because we require cactus commutors to be involutions whereas braidings, in general, are not. In fact, any commutor that is both a braiding and a cactus commutor is in fact a symmetric commutor (that is, it endows the category in question with the structure of a symmetric monoidal category).
Examples.
The definition of coboundary monoidal categories was first given by Drinfel ′ d in [5] . The name was inspired by the fact that the representation categories of coboundary Hopf algebras are coboundary monoidal categories. Since the cactus group surjects onto the symmetric group, any symmetric monoidal category is a coboundary monoidal category. Our main example of coboundary categories which are not symmetric monoidal categories will be the categories of representations of quantum groups and crystals. Furthermore, we will see that the category of crystals cannot be given the structure of a braided monoidal category. Thus there exist examples of coboundary monoidal categories that are not braided.
Quantum groups and crystals
5.1. Quantum groups. Compact groups and semisimple Lie algebras are rigid objects in the sense that they cannot be deformed. However, if one considers the group algebra or universal enveloping algebra instead, a deformation is possible. Such a deformation can be carried out in the category of (noncommutative, noncocommutative) Hopf algebras. These deformations play an important role in the study of the quantum Yang-Baxter equation and the quantum inverse scattering method. Another benefit is that the structure of the deformations and their representations becomes more rigid and the concepts of canonical bases and crystals emerge.
Quantum groups and crystals
We introduce here the quantum group, or quantized enveloping algebra, defined by Drinfel ′ d and Jimbo. For further details we refer the reader to the many books on the subject (e.g. [2, 8, 17] ). Let g be a Kac-Moody algebra with symmetrizable generalized Cartan matrix A = (a ij ) i,j∈I and symmetrizing matrix D = diag(s i ∈ Z >0 | i ∈ I). Let P, P ∨ , and Q + be the weight lattice, coweight lattice and positive root lattice respectively. Let C q be the field C(q 1/2 ) where q is a formal variable. For n ∈ Z and any symbol x, we define
Quantum groups and crystals
The quantum group or quantized enveloping algebra U q (g) is the unital associative algebra over C q with generators e i , f i (i ∈ I) and q h (h ∈ P ∨ ) with defining relations
Quantum groups and crystals
As q → 1, the defining relations for U q (g) approach the usual relations for g in the following sense. Taking the "derivative" with respect to q in the second and third relations gives
Quantum groups and crystals
Furthermore, if we naively apply L'Hôpital's rule, we have
Quantum groups and crystals
and so the fourth defining relation of U q (g) becomes [e i , f j ] = δ ij h i in the q → 1 limitcalled the classical limit. Similarly, since we have
Quantum groups and crystals
the last two relations (called the quantum Serre relations) become the usual Serre relations in the classical limit. Thus, we can think of U q (g) as a deformation of g. For a more rigorous treatment of the classical limit, we refer the reader to [8, §3.4] . The algebra U q (g) has a Hopf algebra structure given by comultiplication
Quantum groups and crystals
and antipode
Quantum groups and crystals
There are other choices but we will use the above in what follows. The representations of U(g) can be q-deformed to representations of U q (g) in such a way that the dimensions of the weight spaces are invariant under the deformation (see [8, 15] ). The q-deformed notion of a weight space is as follows: for a U q (g)-module M and λ ∈ P , the λ-weight space of M is
Crystal bases.
In this section we introduce the theory of crystal bases, which can be thought of as the q → ∞ limit of the representation theory of quantum groups. In this limit, representations are replaced by combinatorial objects called crystal graphs. These objects, which are often much easier to compute with than the representations themselves, can be used to obtain such information as dimensions of weight spaces (characters) and the decomposition of tensor products into sums of irreducible representations. For details, we refer the reader to [8] . We note that in [8] , the limit q → 0 is used. This simply corresponds to a different choice of Hopf algebra structure on U q (g). We choose to consider q → ∞ to match the choices of [2] .
Crystal bases.
For n ∈ Z ≥0 and i ∈ I, define the divided powers
Crystal bases.
Let M be an integrable U q (g)-module and let M λ be the λ-weight space for λ ∈ P . For i ∈ I, any weight vector u ∈ M λ can be written uniquely in the form
Crystal bases.
Let A be the integral domain of all rational functions in C q that are regular at q = ∞. That is, A consists of all rational functions that can be written in the form g 1 (q
Crystal bases.
(one should think of these as rational functions whose limit exists as q → ∞). It was shown by Kashiwara [11] that all U q (g)-modules in the category O q int (integrable modules with weight space decompositions and weights lying in a union of sets of the form λ − Q + for λ ∈ P ) have unique crystal bases (up to isomorphism).
Crystal bases.
A crystal basis can be represented by a crystal graph. The crystal graph corresponding to a crystal basis (L, B) is an edge-colored (by I) directed graph with vertex set B and a i-colored directed edge from
Crystal bases.
Crystals can be defined in a more abstract setting where a crystal consists of such a graph along with maps wt : B → P and ϕ i , ε i : B → Z ≥0 satisfying certain axioms. In this paper, by the category of g-crystals for a symmetrizable Kac-Moody algebra g, we mean the category consisting of those crystal graphs B such that each connected component of B is isomorphic to some B λ , the crystal corresponding to the irreducible highest weight U q (g)-module of highest weight λ, where λ is a dominant integral weight. In this case
Crystal bases.
. For the rest of this paper, the word g-crystal means an object in this category.
Crystal bases.
Example 5.3 (Crystal bases of finite-dimensional representations of U q (sl 2 )). For n ∈ Z ≥0 , let V n be the irreducible U q (sl 2 )-module of highest weight n. It is a q-deformation of the corresponding sl 2 -module. Let v n be a highest weight vector of V n and define
Crystal bases.
Then {v n , v n−2 , . . . , v −n } is a basis of V n and v j has weight j. Let L = Span A {v n , v n−2 , . . . , v −n }, and
Crystal bases.
is a crystal basis of V n and the corresponding crystal graph is
Crystal bases.
(since there is only one simple root for sl 2 , we omit the edge-coloring).
Tensor products.
One of the nicest features of the theory of crystals is the existence of the tensor product rule which tells us how to form the crystal corresponding to the tensor product of two representations from the crystals corresponding to the two factors. 
where the action of the Kashiwara operatorsẽ i andf i are given bỹ
Here we write
where the action of the Kashiwara operatorsẽ i andf i are given bỹ
We write B 1 ⊗ B 2 for the crystal graph B 1 × B 2 of M 1 ⊗ M 2 with crystal operators defined by the formulas in Theorem 5.4. Note that even though we use a different coproduct than in [8] , the tensor product rule remains the same as seen in [2, Proposition 14.1.14].
5.4.
The braiding in the quantum group. The category of representations of U(g), the universal enveloping algebra of a symmetrizable Kac-Moody algebra g, is a symmetric monoidal category with braiding given by
5.4.
However, the analogous map is not a morphism in the category of representations of U q (g) and this category is not a symmetric monoidal category. However, it is a braided monoidal category with a braiding constructed as follows. The R-matrix is an invertible element in a certain completed tensor product U q (g) ⊗U q (g) (see [2, 10] ). It defines a map U ⊗V → U ⊗V for any representations U and V of U q (g). The map given by
5.4.
for representations U and V of U q (g) is a braiding.
5.4.
As an example, consider the representation V 1 ⊗ V 1 of U q (sl 2 ). In the basis Note that R| q=1 = id and so in the classical limit, the braiding becomes the map flip. In the basis S 1 , we have 
Now consider the basis
Thus S 2 is a basis of V 1 ⊗ V 1 compatible with the decomposition From this form, we easily see that flip •R is an isomorphism of U q (sl 2 )-modules
Now consider the basis
It acts as multiplication by q 1/2 on the summand V 2 and by −q −3/2 on the summand V 0 . Now, from Example 5.3 and the tensor product rule (Theorem 5.4), we see that the crystal basis of V 1 ⊗ V 1 is given by
Now consider the basis
From the matrix of flip •R in the basis S 1 given in (5.1), we see that it does not preserve the crystal lattice L since it involves positive powers of q. Furthermore, there is no C q -multiple of flip •R which preserves L and induces an isomorphism of L/q −1/2 L. To see this, note from (5.1) that in order for g(q) flip •R, with g(q) ∈ C q , to preserve the crystal lattice, we would need q 1/2 g(q) ∈ A and thus q −1/2 g(q) ∈ q −1 A ⊆ q −1/2 A. However, we would then have
Now consider the basis
and so g(q) flip •R would not induce an isomorphism of L/q −1/2 L. Therefore, we see that the braiding coming from the R-matrix does not pass to the q → ∞ limit. That is, it does not induce a braiding on the crystal B 1 ⊗ B 1 .
Now consider the basis
It turns out that the above phenomenon is unavoidable. That is, no braiding on the category of representations of a quantum group passes to the q → ∞ limit. In fact, we have the following even stronger results.
Now consider the basis
Lemma 5.5. The category of sl 2 -crystals cannot be given the structure of a braided monoidal category.
Now consider the basis
Proof. We prove the result by contradiction. Suppose the category of sl 2 -crystals is a braided monoidal category with braiding σ. Consider the crystal B 1 . It has crystal graph
Now consider the basis
The crystal graph of the tensor product B 1 ⊗ B 1 has two connected components:
Now consider the basis
Since σ B 1 ,B 1 is an crystal isomorphism, we see from the above that it must act as the identity. Therefore
Now consider the basis
Now, the graph of the crystal B 1 ⊗ B 2 has two connected components:
Now consider the basis
The graph of the crystal B 2 ⊗ B 1 also has two connected components:
Now consider the basis
Since σ B 1 ,B 2 is a crystal isomorphism, we must have
Now consider the basis
By the naturality of the braiding, the following diagram commutes:
Now consider the basis
We therefore have
Now consider the basis
Comparing to (5.2), we see that
Now consider the basis
contradicting the fact that σ is a braiding.
Now consider the basis
An alternative proof of Lemma 5.5 was given in [7] . We can use the fact that g-crystals, for g a symmetrizable Kac-Moody algebra, can be restricted to sl 2 -crystals to generalize this result.
Now consider the basis
Proposition 5.6. For any symmetrizable Kac-Moody algebra g, the category of g-crystals cannot be given the structure of a braided monoidal category.
Now consider the basis
Proof. We prove the result by contradiction. Suppose the category of g-crystals was a braided monoidal category with braiding σ for some symmetrizable Kac-Moody algebra g. Let α 1 and ω 1 be a simple root and fundamental weight (respectively) corresponding to some vertex in the Dynkin diagram of g. The restriction of a g-crystal to the color 1 yields an sl 2 -crystal. More precisely, one forgets the operatorsẽ i ,f i , ϕ i and ε i for i = 1 and projects the map wt to the one-dimensional sublattice Zω 1 ⊆ P . In general, even if the original g-crystal was connected (i.e. irreducible), the induced sl 2 -crystal will not be. However, any morphism of g-crystals induces a morphism of the restricted sl 2 -crystals.
Now consider the basis
Consider the g-crystal B kω 1 , k ≥ 1, corresponding to the irreducible U q (g)-module of highest weight kω 1 . If we restrict this to an sl 2 -crystal, the connected component of the sl 2 -crystal graph containing the highest weight element b kω 1 is isomorphic to the sl 2 -crystal B k . Now, since b ω 1 ⊗ b ω 1 is the unique element of B ω 1 ⊗ B ω 1 of weight 2ω 1 , we have
Now consider the basis
The connected sl 2 -subcrystal containing the element
Now consider the basis
(as in the proof of Lemma 5.5). Thus, since σ Bω 1 ⊗Bω 1 is a morphism of sl 2 -crystals, we must have
Now consider the basis
Now, the only element of B ω 1 ⊗ B ω 1 of weight 2ω 1 − α 1 not contained in the connected sl 2 -subcrystal mentioned above is b ω 1 ⊗f 1 b ω 1 . Therefore, we must also have
Now consider the basis
Thus,
Now consider the basis
and the connected sl 2 -subcrystal containing the element b ω 1 ⊗f 1 b 2ω 1 is
Now consider the basis
(as in the proof of Lemma 5.5). Similarly, we have the following sl 2 -subcrystals of B 2ω 1 ⊗B ω 1 :
Now consider the basis
Now, since b ω 1 ⊗ b 2ω 1 and b 2ω 1 ⊗ b ω 1 are the unique elements of B ω 1 ⊗ B 2ω 1 and B 2ω 1 ⊗ B ω 1 (respectively) of weight 3ω 1 , we must have
Now consider the basis
Also
Now consider the basis
Now, consider the inclusion of g-crystals j :
Now consider the basis
By the naturality of the braiding σ, the following diagram is commutative:
Now consider the basis
Comparing this to (5.4) we see that
Now consider the basis
This contradicts the fact that σ is a braiding.
Crystals and coboundary categories
In this section, we discuss how the categories of U q (g)-modules and g-crystals can be given the structure of a coboundary category. For the case of g-crystals, we mention several different constructions and note the relationship between them.
Crystals and coboundary categories
where R op = flip(R) (as an operator on M 1 ⊗M 2 , R op acts as flip •R•flip) and the square root is taken with respect to a certain filtration on the completed tensor product U q (g)⊗U q (g). He then showed that flip •R is a cactus commutor and so endows the category of U q (g)-modules with the structure of a coboundary category (see the comment after the proof of Proposition 3.3 in [5] ). That is, it satisfies the conditions of Definition 4.1.
Crystals and coboundary categories
Consider the representation V 1 ⊗ V 1 of U q (sl 2 ). In Section 5.4, we described the action of the R-matrix on this representation in two different bases S 1 and S 2 . In the basis S 1 , we have
Crystals and coboundary categories
One easily checks that in the basis S 2 ,
Crystals and coboundary categories
Therefore, we can take the (inverse of the) square root
Crystals and coboundary categories
In the basis S 1 , we then have
Crystals and coboundary categories
Therefore, in the basis S 1 ,
Crystals and coboundary categories
And in the basis S 2 , 
Crystals and coboundary categories
Thus, flip •R passes to the q → ∞ limit and, up to signs, induces an involution on the crystal B 1 ⊗ B 1 (see Section 6.4 for details). As noted in Section 5.4, the same is not true of flip •R.
Crystals and coboundary categories
The second important property of flip •R is that it is a cactus commutor. We see immediately from (6.2) that (flip •R) 2 = id. A straightforward (if somewhat lengthy) computation shows that flip •R also satisfies the cactus relation (see [5, §3] for the proof in a more general setting).
Crystals and coboundary categories
The unitarized R-matrix has shown up in several different places. In [1] it arose naturally in the development of the theory of braided symmetric and exterior algebras. The reason for this is that if one wants to have interesting symmetric or exterior algebras, one needs an operator with eigenvalues of positive or negative one. Notice from (6.2) that while flip •R does not have this property, the operator flip •R does. Essentially, in the basis S 2 , the matrix for flip •R is obtained from the matrix for flip •R by setting q = 1. Note that this does not imply that flip •R is an operator in the classical limit, merely that its matrix coefficients in a certain basis do not involve powers of q. The q → ∞ limit of the unitarized R-matrix also appeared independently in the study of cactus commutors for crystals. We discuss this in the next two subsections.
Crystals and coboundary categories
6.2. The crystal commutor using the Schützenberger involution. Let g be a simple complex Lie algebra and let I denote the set of vertices of the Dynkin graph of g. If w 0 is the long element in the Weyl group of g, let θ : I → I be the involution such that α θ(i) = −w 0 · α i . Define a crystal B λ with underlying set {b | b ∈ B λ } and
Crystals and coboundary categories
There is a crystal isomorphism B λ ∼ = B λ . We compose this isomorphism with the map of sets B λ → B λ given by b → b and denote the resulting map by ξ = ξ B λ : B λ → B λ . We call the map ξ the Schützenberger involution. When g = gl n , there is a realization of B λ using tableaux. In this realization, ξ is the usual Schützenberger involution on tableaux (see [14] ).
Crystals and coboundary categories
For an arbitrary g-crystal B,
Crystals and coboundary categories
For crystals A and B, define In other words, σ S endows the category of g-crystals with the structure of a coboundary category.
Crystals and coboundary categories
6.3. The crystal commutor using the Kashiwara involution. Let g be a symmetrizable Kac-Moody algebra and let B ∞ be the g-crystal corresponding to the lower half U − q (g) of the associated quantized universal enveloping algebra. Let * : U q (g) → U q (g) be the C q -linear anti-automorphism given by
Crystals and coboundary categories
The map * sends U − q (g) to U − q (g) and induces a map * : B ∞ → B ∞ (see [12, §8.3] ). We call the map * the Kashiwara involution.
Crystals and coboundary categories
Let B λ be the g-crystal corresponding to the irreducible highest weight U q (g)-module of highest weight λ and let b λ be its highest weight element. The maps ι λ+µ λ make the family of crystals B λ into a directed system and the crystal B ∞ can be viewed as the limit of this system. We haveẽ i -equivariant maps ι ∞ λ : B λ → B ∞ which we will simply denote by ι ∞ when it will cause no confusion. Define ε * :
Crystals and coboundary categories
where we put the usual order on P + , the positive weight lattice of g, given by λ ≥ µ if and only if λ − µ ∈ Q + . Recall that we also have the map ε :
Crystals and coboundary categories
Then by [12, Proposition 8.2] , the Kashiwara involution preserves weights and satisfies
Crystals and coboundary categories
Consider the crystal B λ ⊗ B µ . Since ϕ(b) = ε(b) + wt(b) for all b ∈ B λ , we have that ϕ(b λ ) = wt(b λ ) = λ. It follows from the tensor product rule for crystals that the highest weight elements of B λ ⊗ B µ are those elements of the form
Crystals and coboundary categories
as crystals, we can make the following definition. We call σ c the crystal commutor. Note that Theorem 6.3 only implies that it is a cactus commutor for g of finite type.
Crystals and coboundary categories
6.4. The relationship between the various commutors. We have described three ways of constructing commutors in the categories of U q (g)-modules or g-crystals. The three definitions are closely related. In [7] , Henriques and Kamnitzer defined a cactus commutor on the category of finite-dimensional U q (g)-modules when g is of finite type, using an analogue of the Schützenberger involution on U q (g). This definition of the commutor involves some choices of normalization. In [10] , Kamnitzer and Tingley showed that Drinfel ′ d's commutor coming from the unitarized R-matrix corresponds to the commutor coming from the Schützenberger involution up to normalization. From there it follows that Drinfel ′ d's commutor preserves crystal lattices and acts on crystal bases as the crystal commutor, up to signs. The precise statement is the following. We thus have essentially two definitions of the crystal commutor. The first, using the Schützenberger involution (and coinciding with Drinfel ′ d's commutor in the crystal limit) only applies to g of finite type but with this definition, it is apparent that the commutor satisfies the cactus relation. The second definition, using the Kashiwara involution, applies to g of arbitrary type but it is not easy to see that it satisfies the cactus relation. In the next section, we will explain how a geometric interpretation of this commutor using quiver varieties allows one to prove that this is indeed the case.
A geometric realization of the crystal commutor
In this section we describe a geometric realization of the crystal commutor defined in Section 6.3 in the language of quiver varieties. This realization yields new insight into the coboundary structure and equips us with new geometric tools. Using these tools, one is able to show that the category of g-crystals for an arbitrary symmetrizable Kac-Moody algebra g can be given the structure of a coboundary category. This extends the previously known result, which held for g of finite type.
A geometric realization of the crystal commutor
7.1. Quiver varieties. Lusztig [16] , Nakajima [20, 21, 22] and Malkin [19] have introduced varieties associated to quivers (directed graphs) built from the Dynkin graph of a Kac-Moody algebra g with symmetric Cartan matrix. These varieties yield geometric realizations of the quantum group U q (g), the representations of g, and tensor products of these representations in the homology (or category of perverse sheaves) of such varieties. In addition, Kashiwara and Saito [13, 23] , Nakajima [22] and Malkin [19] have used quiver varieties to give a geometric realization of the crystals of these objects. Namely, they defined geometric operators on the sets of irreducible components of quiver varieties, endowing these sets with the structure of crystals. In the current paper, we will focus on the sl 2 case of these varieties for simplicity. In this case, the quiver varieties are closely related to grassmannians and flag varieties.
A geometric realization of the crystal commutor
In this section, all vector spaces will be complex. Fix integers w ≥ 0 and n ≥ 1, and w = (w i ) n i=1 ∈ (Z ≥0 ) n such that n i=1 w i = w. Let W be a w-dimensional vector space and let 0 = W 0 ⊆ W 1 ⊆ · · · ⊆ W n = W, dim W i /W i−1 = w i for 1 ≤ i ≤ n, be an n-step partial flag in W . Define the tensor product quiver variety T(w) = {(U, t) | U ⊆ W, t ∈ End W, t(W i ) ⊆ W i−1 ∀ i, im t ⊆ U ⊆ ker t}.
A geometric realization of the crystal commutor
We use the notation T(w) since, up to isomorphism, this variety depends only on the dimensions of the subspaces W i , 0 ≤ i ≤ n. We have
A geometric realization of the crystal commutor
T(u, w), where T(u, w) = {(U, t) ∈ T(w) | dim U = u}.
A geometric realization of the crystal commutor
Let B(u, w) denote the set of irreducible components of T(u, w) and set B(w) = u B(u, w).
A geometric realization of the crystal commutor
Define wt : B(w) → P, wt(X) = w − 2u for X ∈ B(u, w), ε : T(w) → Z ≥0 , ε(U, t) = dim U/ im t, ϕ : T(w) → Z ≥0 , ϕ(U, t) = dim ker t/U = ε(U, t) + w − 2 dim U.
A geometric realization of the crystal commutor
For k ∈ Z ≥0 , let T(u, w) k = {(U, t) ∈ T(u, w) | ε(U, t) = k}, and for X ∈ B(u, w), define ε(X) = ε(U, t) and ϕ(X) = ε(U, t) for a generic point (U, t) of X. Let B(u, w) k = {X ∈ B(u, w) | ε(X) = k}, B(w) k = u B(u, w) k .
A geometric realization of the crystal commutor
The map T(u, w) k → T(u − k, w) 0 , (U, t) → (im t, t) is a Grassmann bundle and thus induces an isomorphism (7.1) B(u, w) k ∼ = B(u − k, w) 0 .
A geometric realization of the crystal commutor
We then define crystal operators on B(w) as follows. Suppose X ′ ∈ B(u−k, w) 0 corresponds to X ∈ B(u, w) k under the isomorphism (7.1). Definẽ and setf (X) = 0 for X ∈ B(u, w) k with k ≤ 2u − w. The mapsẽ k andf k defined above can be considered as the kth powers ofẽ andf respectively.
A geometric realization of the crystal commutor
One advantage of the geometric interpretation of the crystal commutor defined here is that it extends to any symmetrizable Kac-Moody algebra g. In particular, if g has symmetric Cartan matrix, then there exists a tensor product quiver variety whose irreducible components can be given the structure of a tensor product crystal. There then exists a map X → X † , which generalizes the map defined above. One can show that, in the case of two factors, this map corresponds to the crystal commutor. For three factors, the compositions σ

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