Quantum groups for restricted SOS models

We introduce the notion of restricted dynamical quantum groups through their category of representations, which are monoidal categories with a forgetful functor to the category of $\pi$-graded vector spaces for a groupoid $\pi$.


Introduction
The theory of quantum groups was designed in the 1980's as the algebraic object underlying the theory of exactly solvable models of statistical mechanics. Since then it has entered diverse fields of mathematics and mathematical physics and the world of exactly solvable models is entirely explained by quantum groups, in the guise of Yangians, quantum loop algebras, and elliptic quantum groups. Well, not entirely... One small village of indomitable models, called the RSOS models still holds out against the invaders. These Restricted Solid-On-Solid models, introduced in special cases by Baxter in his studies of the 8-vertex model and the hard hexagon model, and generalized by Andrews, Baxter and Forrester, are lattice models of twodimensional statistical mechanics for which the technology of exact solutions has provided some of the most spectacular results. They play a central role also in conformal field theory (CFT) as their critical behaviour is (or is conjectured to be) given by the universality classes of minimal unitary CFT models. While the unrestricted SOS models, whose local degrees of freedom take values in an infinite set, are by now well-described by the representation theory of dynamical elliptic quantum groups, the RSOS models, with finitely many allowed states at every lattice points, are much less understood.
We propose a theory of dynamical quantum groups with discrete dynamical parameter with the goal to establish the representation theory underlying RSOS models and their higher rank generalizations. We introduce a new approach to this problem, based on groupoid-graded vector spaces, which may be of independent interest and applicability in representation theory.
1.1. Quantum groups and solvable models of statistical mechanics. The notion of quantum group [11] emerged in the Leningrad school in the 1980s as the algebraic structure underlying exactly solvable models of statistical mechanics in 2 dimensions and integrable quantum field theory in 1+1 dimensions, see [17]. While "groups" may be a misnomer for these Hopf algebras, quantum groups share with groups the fact that they have an interesting representation theory for which tensor products of representations are defined. Excellent textbooks on quantum The authors are supported in part by the National Centre of Competence in Research SwissMAP-The Mathematics of Physics-of the Swiss National Science Foundation. They are also supported by the grants 196892 and 178794 of the Swiss National Science Foundation, respectively. groups are [7,30,36]. It soon appeared that quantum groups have a much wider scope of applications, ranging from low dimensional topology, conformal field theory, algebraic geometry, gauge theory, representation theory of affine Lie algebras etc.
Returning to the origin in statistical mechanics, the basic equation is the Yang-Baxter equation, which appeared in the 1960's in the work of C. N. Yang and R. Baxter on statistical mechanics and scattering theory in 1+1 dimensional quantum field theory [41]. In its basic form it is an equation for a meromorphic function z → R(z) ∈ End C (V ⊗ V ) of one complex variable (called the spectral parameter) with values in the linear endomorphisms of the tensor square of a finite dimensional complex vector space V . The Yang-Baxter equation is R(z − w) (12) R(z) (13) R(w) (23) = R(w) (23) R(z) (13) R(z − w) (12) in End(V ⊗ V ⊗ V ). Here the superscripts in the notation indicate the factors on which the endomorphisms act: for example R(w) (12) means R(w) ⊗ id. One also requires that R(z) is invertible for generic z. One of the simplest non-trivial solutions is Yang's R-matrix R(z) = Id+z −1 P VV where P V V is the flip v⊗w → w⊗v for any V .
As noted by Baxter, solutions of the Yang-Baxter equations give rise to families of commuting operators on n-fold tensor powers W = V ⊗n : fix complex numbers z 1 , . . . , z n and consider the operator valued function L(z) ∈ End(V ⊗ W ) given by the product R(z − z n ) (0n) · · · R(z − z 2 ) (02) R(z − z 1 ) (01) (we number the factors from 0 to n where 0 refers to the "auxiliary space" V in V ⊗ W ). Then the (row-to-row) transfer matrices defined by the partial traces T (z) = tr V L(z) are commuting endomorphisms of W = V ⊗n : T (z)T (w) = T (w)T (z) as the consequence of the Yang-Baxter equation. The relation to statistical mechanics is that the trace of T (z) m , say with all z i = 0, when written out using matrix multiplication, is a sum of products of matrix entries of R(z) over all the ways to assigning a basis vector to pairs of nearest neighbours of an n × m lattice with periodic boundary conditions. These assignments are configurations of local states of a system and the sum is the partition function of a statistical mechanics model.
The Bethe ansatz, invented by H. Bethe in 1931 [6] in the case of the Heisenberg spin chain, and further developed by E. Lieb, R. Baxter, C. N. and C. P. Yang and others in the 1960's, is a technique to find simultaneous eigenvectors and eigenvalues of the T (z). The Leningrad school, see [17] for a review of the results in the early phase, reformulated this technique under the names "Algebraic Bethe Ansatz" or "Quantum Inverse Scattering Method" in terms of representation theory of an algebra with quadratic relations (called RLL or RTT relations) whose coefficients are matrix entries of a solution of the Yang-Baxter equation. For example the Yangian Y (gl N ) corresponds to the Yang R-matrix with V = C N . It can be defined as the algebra with generators L ij;n , i, j = 1, . . . , N , n = 1, 2, . . . with relations It is a Hopf algebra deformation of the universal enveloping algebra of the current Lie algebra gl N [t] and has a universal R-matrix R in a completion of Y (gl N ) ⊗ Y (gl N ) relating the opposite coproduct ∆ ′ to the coproduct via ∆ ′ (x) = R∆(x)R −1 . Evaluating R in pairs V i ⊗ V j of finite dimensional representations of the Yangian yields solutions of the Yang-Baxter equation, in the generalized form: The spectral parameter may be viewed as a parameter of the representations V i , and in fact there is an issue of convergence of the action of R on finite dimensional representations, resulting in the fact that R Vi,Vj is a meromorphic function of the spectral parameters. To such a system of R-matrices we can associate corresponding transfer matrices T i = tr Vi R ViV3 , i = 1, 2, acting on V 3 and the Yang-Baxter equation with an invertible R V1V2 implies that T 1 T 2 = T 2 T 1 . Baxter's transfer matrices are the special case V 1,2 = V and V 3 a tensor product of vector representations.
This story extends to arbitrary semisimple (or reductive) Lie algebras g, and the Yangians Y (g) are the symmetry algebra of several integrable systems based on rational solutions of the Yang-Baxter equation such as the Heisenberg spin chain.
The theory admits a trigonometric version, leading to solutions of the Yang-Baxter equation with trigonometric coefficients. The corresponding quantum group is a Hopf algebra deformation of the loop Lie algebra g[t, t −1 ]. It is (a subquotient of) the Drinfeld-Jimbo quantum enveloping algebra U qĝ of the affine Kac-Moody Lie algebraĝ. The corresponding solvable models are the two-dimensional ice model (more generally, the six-vertex model) and the XXZ spin chain.
1.2. Elliptic quantum groups and dynamical Yang-Baxter equation. The next level, after the rational and trigonometric functions, are the elliptic functions, which are meromorphic functions which are periodic with respect to two independent periods. Several solvable models have an elliptic version and the trigonometric and rational versions are obtained as degenerate limits as the periods tend to infinity. The relation to quantum groups is more tricky in the elliptic case. On one hand there is a solution of the Yang-Baxter equation with elliptic coefficients due to Baxter, corresponding to the XYZ spin chain and the eight-vertex model, whose underlying algebraic structure is the Sklyanin algebra (which is not a Hopf algebra). On the other hand there are the SOS (solid-on-solid) also known as IRF (interaction-round-a-face) models. They are based on a variant of the Yang-Baxter equation, called the star-triangle relation. While the existence of the Baxter solution is special for g = sl N , we now know that SOS models exist for all semisimple Lie algebras. Also the Baxter solution can be related to a solution of the star-triangle relation by the so-called vertex-IRF transformation, also due to Baxter.
The elliptic quantum groups introduced in [20,21] provide a generalization of the theory of quantum groups that applies to elliptic SOS models. They are based on a modification of the Yang-Baxter equation, nowadays called the dynamical Yang-Baxter equation (1), which had previously been found by Gervais and Neveu in their study of the exchange relations of vertex operators in the Liouville conformal field theory [25]. The dynamical Yang-Baxter equation reappeared in various contexts since, e.g. [1, 8,9,14,15,27,38] A recent textbook on elliptic quantum groups is [33].
The unknown in the dynamical Yang-Baxter equation is a function R(z, a) ∈ End h (V ⊗ V ) of a second "dynamical" variable a ∈ h * with values in the dual vector space to an abelian Lie algebra h and V = ⊕ µ∈h * V µ is a finite dimensional semisimple h-module. The (quantum) dynamical Yang-Baxter equation is (23) (1) = R(w, a) (23) R(z, a + h (2) ) (13) R(z −w, a) (12) .
The "dynamical shift" notation is adopted here: R(w, a + h (3) ) (12) acts as R(w, a + µ 3 ) ⊗ Id on the product of weight subspaces The elliptic quantum group associated with a solution of the dynamical Yang-Baxter equation and its tensor category of representations can be again defined by quadratic relations similar to those of the Yangian but with dynamical shifts at the appropriate places, see [20,23,24]. The main new feature is that the representations are vector spaces over the field of meromorphic functions of the dynamical variables and the elements of the elliptic quantum group act as difference operators in these variables. The underlying generalization of the notion of Hopf algebra was formalized by Etingof and Varchenko [16] who called it h-Hopf algebroid.
The transfer matrix construction generalizes to the dynamical setting [22]: suppose that we have invertible operators R ViVj (z, λ) ∈ Hom(V i , V j ), i < j ∈ {1, 2, 3} obeying the dynamical Yang-Baxter equation (4) on V 1 ⊗ V 2 ⊗ V 3 , depending meromorphically on z ∈ C, λ ∈ h * . Then the transfer matrix is defined as an operator acting on meromorphic functions of λ with values in the zero-weight subspace of V 3 : Here the partial trace is over the weight-µ subspace of V i and the acts as a multiplication operator and (t µ f )(λ) = f (λ + µ).

1.3.
Restricted SOS models. The restricted solid-on-solid (RSOS) models introduced by Andrews, Baxter and Forrester [3], generalizing models previously considered by Baxter [4,5] in his study of the eight-vertex model and of the hard hexagon model, form a general class of models of statistical mechanics in two dimensions. A configuration of a solid-on-solid model on a subset M of a square lattice in the plane or 2-dimensional torus is described by assigning an integer l i (height) to each lattice site i ∈ M , with the restriction that |l i − l j | = 1 for neighbouring sites i, j. We can think of the graph of i → l i as a discrete random surface modeling the interface between two materials, whence the name. The probability of a configuration (l i ) is proportional to a product over the faces (unit squares with vertices in M ) of Boltzmann weights W (l i , l j , l k , l m ) depending on the heights on the corners i, j, k, l of the face. In the "solvable" SOS models the Boltzmann weights are part of a one-parameter family W (z; a, b, c, d) obeying the star-triangle relation g W (z − w; f, g, d, e) W (z; a, b, g, f ) W (w; b, c, d, g) = g W (w; a, g, e, f ) W (z; c, d, e, g) W (z − w; a, b, c, g) which is best understood graphically: For these families of models (depending essentially on an elliptic curve and a point of order r on it) they were able to compute several quantities in the thermodynamic limit M → Z 2 (under some physically motivated assumptions on the asymptotic behaviour), including the probability distribution of the height at the origin as a function of the boundary conditions in the ordered phase. One interesting mathematical outcome of this calculation is that it involves for r = 5 (Baxter's hard hexagon model) the celebrated Rogers-Ramanujan identity, which gets generalized to arbitrary r. From the point of view of statistical mechanics and conformal field theory, these models are interesting since their scaling limit at the critical point are conjectured [26] to be the unitary A-series of minimal models of Belavin-Polyakov-Zamolodchikov and Friedan-Qiu-Shenker. We will be mostly concerned with a generalization of the RSOS models in which the heights take values in the weight lattice of a simple Lie algebra, see [10,28,29]. The main difference is that in general the Boltzmann weights W (a, b, c, d) are no longer scalar-valued, but must be understood as linear operators.
The relation with the dynamical quantum groups comes from the simple observation that the star-triangle equation is essentially a rewriting of the dynamical Yang-Baxter equation. The row-to-row transfer matrix of the RSOS model is the transfer matrix (2) for suitable representations of the elliptic quantum group associated with gl 2 , acting on functions with support on a finite set.
This restriction of a difference operator such as (2) with meromorphic coefficients to a finite set is rather subtle as one needs to avoid the poles and check that the support condition is preserved. This was done in the case of the RSOS model in [19], where it was also shown that the gl 2 -elliptic weight functions of [18] obey "resonance conditions", guaranteeing that their restrictions to a suitable discrete or finite subsets of the values of the dynamical variable provide, via the Bethe ansatz, well-defined eigenvectors of the row-to-row transfer matrix of the RSOS model.

Categories of representations.
Instead of talking of quantum groups it is more convenient to talk about their tensor category of representations, and we take this approach in this paper. In the representation theory of elliptic quantum groups [13,23], the representation space of a representation is defined as a graded vector space over the field of meromorphic functions of the dynamical variables, where the grading is by weights of the underlying Lie algebra. The representation structure is defined by C-linear endomorphisms obeying quadratic relations and commutations relations with scalar multiplication by meromorphic functions.
For the application to RSOS models, where the dynamical variables take values in a discrete set, the approach with meromorphic functions is not suitable. In this paper we propose that the vector spaces underlying representations of quantum groups with discrete dynamical variables should be groupoid-graded vector spaces. More precisely we propose that representations of such quantum groups are monoidal categories equipped with a faithful monoidal functor to the category of π-graded vector spaces of finite type for a certain groupoid π (see Section 2 for the definitions). For applications to generalized RSOS models the groupoids are certain subgroupoids of the transformation groupoid for the translation action of the weight lattice of a semisimple Lie algebra. It turns out that in this approach the various shifts of dynamical variables appearing in the dynamical context appear naturally and one can immediately apply the standard technology of the Quantum Inverse Scattering Method (R-matrices, RLL relations, transfer matrices, Bethe ansatz). An instance of this is the fusion procedure, which consists in constructing solutions of the Yang-Baxter or star-triangle relations from known ones by taking subquotients of tensor products.
An interesting new feature in the groupoid-graded case is that the Grothendieck ring of the category of π-graded vector spaces is non-commutative in general, even in the case of action groupoids of abelian groups. Thus characters of representations of dynamical quantum groups live in a non-commutative ring. However if a collection of representations (V i ) admit R-matrices, which are isomorphisms V i ⊗V j ∼ = V j ⊗V i , then their characters generate commutative subring of the Grothendieck ring of π-graded vector spaces. In the case of transformation groupoids these rings are realized as rings of commuting difference operators.
1.5. Outline of the paper. We introduce the category Vect k (π) of π-graded vector spaces of finite type over a field k in Section 2. It is a variant of a special case of the category of π-graded modules considered in [35]. It is an abelian monoidal category with duality. We discuss the notion of character of a π-graded vector space taking values in the convolution ring of π. In Section 3 we adapt the machinery of Yang-Baxter equations and transfer matrices to case of π-graded vector spaces and explain the relation with star-triangle relation. We introduce the notion of partial traces in this context and prove that solutions of the Yang-Baxter equation give rise to commuting transfer matrices. In the case of transformation groupoids and their subgroupoids, we show that the Yang-Baxter equation can be written as a dynamical Yang-Baxter equation, and that transfer matrices produce commuting difference operators. In Section 4 we consider in more detail the example of the elliptic quantum group of type A n−1 , which admits a dynamical R-matrix with restricted dynamical variables and thus a monoidal category with a forgetful functor to π-graded vector spaces for a finite groupoid π. We compute a few characters, in particular the characters of (analogues of the) exterior powers of the vector representation, obtained by the fusion procedure. Finally in Section 5 we consider the case of dynamical R-matrices arising from quantum groups at root of unity, which may be viewed as a toy model for restricted models, with R-matrices that are independent of the spectral parameters. The construction uses a semisimple rigid braided category C q (g) of representations of quantum groups for each simple Lie algebra g and root of unity q. Technically it is a semisimple quotient of the category of tilting modules of the Lusztig quantum groups. It has finitely many isomorphism classes of simple objects. We construct a faithful monoidal functor from C q (g) to the categories of π-graded vector spaces of finite type for a suitable finite groupoid π. The braiding in C q (g) is then mapped to a system of dynamical R matrices with dynamical variable retricted to a finite set. This construction is a formalization of the "passage to the shadow world" of [31,39] and is a version with discrete dynamical variable of [12]. The characters of simple models define a representation of the Verlinde algebra by difference operators.

Grading by groupoids
2.1. Groupoids. A groupoid π on a set A is a small category with objects A whose morphisms, called arrows, are invertible. The set of morphisms from an object a to an object b is denoted by π(a, b). The composition of arrows γ ∈ π(a, b) and η ∈ π(b, c) is denoted by η •γ ∈ π(a, c) or by ηγ in case of typographical constraints. The inverse of γ ∈ π(a, b) is γ −1 ∈ π(b, a). We identify A with the subset of identity arrows and denote a groupoid by its set of arrows π when no confusion arises. The maps s, t : π → A sending γ ∈ π(a, b) to a and b, respectively, are called source and target map, respectively.
A subgroupoid of a groupoid π is a subset of (the set of arrows of) π that is closed under composition and inversion. It is a groupoid on the set of its identity arrows. The full subgroupoid of π on a subset B ⊂ A is the subgroupoid s −1 B ∩ t −1 B of arrows between objects of B.

2.2.
The convolution ring of a groupoid. To a groupoid π we associate the convolution ring of π, which is a unital associative ring Z(π) with an involutive anti-automorphism.
As an abelian group Z(π) consists of the maps n : π → Z such that for all a ∈ A, the set of arrows α ∈ s −1 (a) with n(α) = 0 is finite. The product is the convolution product The sum has finitely many non-zero terms because of the finiteness assumption. The unit is the characteristic function on identity arrows and the involutive antiautomorphism σ sends n to σ(n) : γ → n(γ −1 ). The assignment π → Z(π) is a contravariant functor from the category of groupoids to the category of unital involutive associative rings.
2.2.1. Remark. For any commutative ring R we have an R-algebra R(π) = R⊗ Z Z(π) obtained by extension of scalars. For transfer matrices we will need a more general construction where R is also π-graded, see 2.10 below.
2.3. Convolution rings of subgroupoids. It will be convenient to view the convolution ring of a subgroupoid π ′ ⊂ π as a subring of Z(π).
2.3.1. Lemma. The characteristic functions χ A ′ of subsets A ′ ⊂ A of the set of identity arrows are idempotents in Z(π).
Proof. By definition χ A ′ (γ) = 0 unless γ is an identity arrow a ∈ π(a, a) for a ∈ A ′ . In this case χ A ′ (a) = 1. Thus Lemma. Let π be a groupoid on A and π ′ the full subgroupoid on A ′ ⊂ A. Then the induced morphism Z(π) → Z(π ′ ) restricts to a unital ring isomorphism where χ A ′ is the unit element of the subring χ A ′ * Z(π) * χ A ′ . Moreover the left-hand side is the subring of functions vanishing on the complement of π ′ .
Proof. The map Z(π) → Z(π ′ ) is the restriction map r : n → n| π ′ . The extension by zero Z(π ′ ) → Z(π) is a right inverse. Its image consists of the functions vanishing outside π ′ . Thus r restricts to an isomorphism from the functions vanishing outside π ′ and Z(π). Now a function n ∈ Z(π) vanishing outside the full subgroupoid π ′ if and only it vanishes except on arrows between elements of A ′ . But this is equivalent

Action groupoids.
The main examples of groupoids for our purpose are action groupoids and their subgroupoids. Let G be a group with identity element e and A be a set with a right action A × G → A. The action groupoid A ⋊ G has objects A and an arrow a → a ′ for each g ∈ G such that a ′ = ag. Thus an arrow is described by a pair (a, g) ∈ A × G. The source and target are s(a, g) = a, t(a, g) = ag and the composition is The identity arrows are (a, e), a ∈ A and the inverse of (a, g) is (ag, g −1 ).
The convolution ring Z(A⋊G) contains the subring Z A of functions with support on the identity arrows as in the general case and a subring ZG isomorphic to the group ring of G via the injective ring homomorphism t : ZG → Z(π) sending g ∈ G to t g : (a, h) → δ g,h The right action of G defines a group homomorphism r : G → Aut(Z A ): for g ∈ G and f ∈ Z A , r g f (a) = f (ag).
2.4.1. Proposition. The convolution ring Z(A⋊G) is the crossed product Z A ⋊ r ZG of its subrings Z A and ZG. The involution acts trivially on Z A and as t g → t g −1 on ZG.
This means that Z(A ⋊ G) is isomorphic to the algebra generated by Z A and elements t g for g ∈ G with relations Explicitly, a function n ∈ Z(π) corresponds to the element g∈G n g t g where n g (a) = n(a, g).

2.5.
The category of π-graded vector spaces of finite type.
2.5.1. Definition. Let π be a groupoid with objects A. A π-graded vector space of finite type over a field k is a collection (V α ) α∈π of finite-dimensional vector spaces such that for each a ∈ A there are finitely many nonzero V α with source s(α) = a.
Note that we can equivalently say "target" instead of "source" in this definition, since arrows in a groupoid are invertible.
The π-graded vector spaces over k form an abelian category Vect k (π): the kvector space Hom(V, W ) of morphisms between objects V, W consists of families (f α ) α∈π of linear maps f α : V α → W α and the composition is defined componentwise.
2.6. Tensor product. The finite type condition allows us to define a monoidal structure (tensor product) on Vect k (π). The tensor product of objects is The direct sum is over all pairs of arrows whose composition is γ and has finitely many nonzero summands. Similarly the tensor product f ⊗ g of morphisms has components ⊕ β•α=γ f α ⊗ g β . For any three objects U, V, W of Vect k (π) and δ ∈ π, Therefore the associativity constraint in Vect k defines an associativity constraint in Vect k (π). The tensor unit in Vect k (π) is 1 = (1 γ ) γ∈π with 1 a = k for identity arrows a ∈ A and 1 γ = 0 for all other arrows. Then for every object V of Vect k (π), Recall that an object V of a monoidal category admits a left dual of an object V if there is an object V ∨ , called left dual of V , together with morphisms δ : are equal to the identity morphism. Similarly one has the notion of right dual object ∨ V with morphisms 1 → ∨ V ⊗ V , V ⊗ ∨ V → 1. Right and left duals of finite dimensional vector spaces coincide.
, and structure morphisms induced by those of the category of finite dimensional vector spaces is both left and right dual to V .
For example the morphism δ : Monoidal categories admitting left and right duals for all objects, which are then uniquely determined up to unique isomorphism, are called rigid. Left and right dualities are monoidal functors to the opposite categories with opposite tensor product. Rigid monoidal categories with coinciding left and right dual functors are called pivotal, see [40, Sections 1.6,1.7] for more details.

2.7.2.
Theorem. The k-additive category Vect k (π) with the tensor product ⊗, the tensor unit 1, associativity constraint α, left and right multiplication by the tensor unit λ, ρ, and duality ( ) ∨ is an abelian pivotal monoidal category. This is an immediate consequence of the fact that Vect k is a k-additive abelian monoidal category and Lemma 2.7.1.

2.7.3.
Remark. Contrary to the case of finite dimensional vector spaces, the monoidal category Vect k (π) is not symmetric or braided, so that V ⊗W is not isomorphic to W ⊗V in general. As we will see presently, the Grothendieck ring is not commutative in general.
2.7.4. Remark. The above construction works for any k-additive rigid monoidal category C over a commutative ring k instead of Vect k . The resulting category of π-graded objects of C of finite type is a k-additive monoidal category. For example, if we view a ring as a monoidal category with one object, the convolution ring Z(π) is the category of π-graded objects of finite type of Z. One can also replace π by a general small category, at the cost of giving up duality.
Since exact sequences of vector spaces split, the character map ch : V → ch V descends to a ring homomorphism from the Grothendieck ring K(Vect k (π)) to the convolution ring Z(π).
The inverse map sends n to the class of (k n(γ) ) γ∈π .
2.10. Convolution algebras with coefficients in π-graded algebras. In the setting of π-graded vector spaces the natural home for transfer matrices is convolution algebras with coefficients in π-graded algebras.
2.10.1. Definition. Let π be a groupoid. A π-graded algebra is an algebra object R in Vect k (π). This means that R = ⊕ γ∈π R γ is a π-graded vector space of finite type together with a product µ : R ⊗ R → R and a unit morphism u : 1 → R obeying the axioms of an associative algebra.
Thus the product xy = µ(x ⊗ y) of x ∈ R α , y ∈ R β is defined if β is composable with α, it belongs to R β•α ; the associativity (xy)z = x(yz) holds when defined; for each object a there is a unit object 1 a , the image by u of 1 ∈ 1 a = k, such that 1 b x = x = x1 a for x ∈ R γ of degree γ ∈ π(a, b).
2.10.3. Definition. Let R be a π-graded algebra. The convolution algebra Γ(π, R) with coefficients in R is the k-algebra of maps f : π → R such that f (γ) ∈ R γ for all γ ∈ π and with convolution product 2.10.4. Example. Let R = End 1 be the π-graded algebra of Example 2.10.2 for the tensor unit 1. Then R a = k1 a for a ∈ A and R α = 0 for non-identity arrows α and Γ(π, R) = k(π) = k ⊗ Z(π) is the extension of scalars of the convolution ring of π, see Remark 2.2.1.

2.10.5.
Lemma. Let π be a groupoid with object set A. The convolution algebra Γ(π, R) with coefficients in a π-graded algebra R is an associative unital k-algebra. The unit is the map a → 1 a for identity arrows a ∈ A and α → 0 for other arrows.
There is one such equation for all α, . . . , ζ so that γ • β • α = δ • ǫ • ζ and the sum is over arrows ρ, σ, τ for which all factors are defined, namely such that the diagrams are commutative in π.
It is convenient to have a graphical representation for these morphisms: We have also displayed in the corners the objects between which the arrows α, . . . , δ are defined. For example α is a morphism from a to b. In the literature one often considers the case where there is at most one arrow between any two objects, and it is then customary to label W by the four objects a, b, c, d instead of the morphisms.

RLL relations.
The machinery of the Quantum Inverse Scattering Method [17] can be applied: given a solutionŘ(z) ∈ End(V ⊗ V ), an L-operator on W ∈ Vect k is a meromorphic function z → L(z) ∈ Hom(V ⊗ W, W ⊗ V ) such that (12) , For exampleŘ(z) is an L-operator on V thanks to the Yang-Baxter equation.
Given a basis of V the RLL relations may be written as relations for the matrix entries L ij (z) ∈ End(W ). Thus L-operators may be understood as π-graded meromorphic representations of the quadratic algebra A R with generators L ij (z), and RLL relations. Here meromorphic refers to the required meromorphic dependence on z ∈ C. M (R, π). The L-operators form an abelian monoidal category M (R, π) of π-graded meromorphic representations of A R : an object (W, L W ) is a π-graded vector spaces W ∈ Vect k (π) endowed with an L-operators L W on W . A morphism from (W, L W ) to (Z, L Z ) is a morphism f : W → Z of π-graded vector spaces such that

The monoidal category
The tensor product (W ⊗ Z, L W ⊗Z ) is the tensor product in Vect k (π) endowed with the composition The fact that L V ⊗W is an L-operator is a straightforward consequence of the definitions. We have an action of C on the category M (R, π): for each u ∈ C let t u be the endofunctor sending an object (W, L W ) to (W, L W (· + u)) and a morphism f to f . Clearly t 0 is the identity endofunctor and t u t v = t u+v . Moreover t u is a monoidal functor: the obvious map 3.3.1. Example. Let V ∈ M (R, π) be the representation with L-operatorŘ. Then for each u ∈ C, the representation V (u) = t u V has L operator L V (u) (z) =Ř(z + u). This object of M (R, π) is called vector representation with evaluation point u.

3.3.2.
Example. Let 1 ∈ Vect k (π) be the tensor unit, see Section 2.6 and let L 1 (z) be the composition ρλ −1 : V ⊗ 1 → V → 1 ⊗ V of the structure isomorphisms. Then 1 with this L-operator is a representation, called the trivial representation. It is fixed by the action of t u .

3.3.3.
Example. The dual representation of a representation (W, L W ) admitting a dual is the representation (W ∨ , L W ∨ ) on the π-graded dual vector space W (see 2.7). Its L-operator L W ∨ (z) =L W (z) −1 is the inverse of the dual operatorL W (z) : with the structure maps defining the duality in the category of π-graded vector spaces, see 2.7. It exists wheneverL W (z) is invertible for generic z.

Proposition.
(i) IfŘ W,Z is an R-matrix for W, Z ∈ M (R, π) and u ∈ C, then the same isomorphismŘ W,Z of π-graded vector spaces is an R-matrix for t u V, t u W .
3.5. Partial traces and transfer matrices. The partial trace over V is the map defined as follows.
For f ∈ Hom(V ⊗ W, W ⊗ V ) and α ∈ π(a, b), γ ∈ π(a, a), let f (α, γ) be the component of f mapping for any basis e i of V α and dual basis e * i of the dual vector space (V α ) * . 3.5.1. Definition. The partial trace tr V f ∈ Γ(π, End W ) of f ∈ Hom Vect π (V ⊗ W, W ⊗ V ) over V is the section tr V f : α → ⊕ γ∈π(a,a) tr Vα f (α, γ) ∈ (End W ) α 3.5.2. Example. Let W = 1 be the tensor unit, with nonzero components W a = k, indexed by identity arrows a ∈ A. For α ∈ π(a, b), we have EndW α = Hom(W a , W b ) = k.
The convolution algebra Γ(π, End 1) is the extension of scalar of the convolution ring of π. The partial trace of the identity which is the (image in k of the) character ch V of V .

Lemma.
(i) If ϕ : V → V ′ is an isomorphism of π-graded vector spaces then vector spaces and let f be the composition Then Proof. Recall that a morphism ϕ is a collection of linear maps ϕ α : V α → V ′ α , so (i) is the standard property of the trace on each V α .
As for (ii) to compute tr V1⊗V2 we need to select for each γ ∈ π(a, b) and µ ∈ π(b, b) the component g(α) of g = f π(a, c) and β ∈ π(c, b), Thus the trace is non-trivial on the components of f mapping These components factor as The claim follows by taking tensor product of bases of V 1α of V 2β .
with invertibleŘ V1V2 . Then the transfer matrices Proof. We can write the Yang-Baxter equation as The claim then follows from Lemma 3.5.3 (ii).
3.5.5. Remark. For W = 1 the R-matrix R V,1 is the tautological map V ⊗1 → 1⊗V of Example 3.5.2. Thus the transfer matrix generalizes the notion of character.
3.6. Action groupoids and dynamical Yang-Baxter equation. If π = A ⋊ G is an action groupoid, R-matrices for π-graded vector spaces are expressed in terms of the graded components as dynamical R-matrices. Then the tensor product of π-graded modules is
Here we use the "dynamical" notation with the placeholder h (i) : If we compose on the left with the product p (23) p (13) p (12) = p (12) p (13) p (23) of flips p : v ⊗ w → w ⊗ v we get the YBE for R = p •Ř in the form:

3.7.
Transfer matrices in the case of action groupoids. In the case of action groupoid π = A ⋊ G we can identitify the convolution algebra as an algebra of difference operators (or discrete connections) acting on the sections of a sheaf over A. Let G(a) = {g ∈ G | ag = a} denote the stabilizer subgroup of an object a ∈ A and for W ∈ Vect k (π) let W G(a) = ⊕ g∈G(a) W (a,g) . Let Γ(A, W ) be the space of maps ψ : A → ∪ a∈A W G(a) such that for all a ∈ A, ψ(a) ∈ W G(a) . Then Γ(A, W ) is naturally a module over Γ(π, End W ) and the transfer matrices can be realized as linear operators on Γ(A, W ). Explicitly, we have End W (a,g) = ⊕ h∈G(a) Hom(W (ag,ghg −1 ) , W (a,h) ).

Elliptic quantum groups
As a class of examples of the above construction let us work out the case of the dynamical R-matrix defining the elliptic quantum group in the gl n -case, see [20].
Here k = C.

4.2.
Dynamical R-matrix. Fix two complex numbers τ, γ such that Im τ > 0 and γ ∈ Z + τ Z. Let be the odd Jacobi theta function and [z] = θ(γz, τ )/(γθ ′ (0, τ )) is normalized to have derivative 1 at z = 0. The function z → [z] of one complex variable is an odd entire function with first order zeros on the lattice Λ = Z 1 γ + Z τ γ . The defining representationV = C n of gl n has a weight decompositionV = ⊕ n i=1V ǫi whereV ǫi = Ce i is the span of the i-th standard basis vector.
Let E ij be the n × n matrix such that E ij e k = δ jk e i for all k ∈ {1, . . . , n}. The (unnormalized) elliptic dynamical R-matrix with spectral parameter z ∈ C iš It is a meromorphic function of z ∈ C, a ∈ C n and solves the dynamical Yang-Baxter equation in the additive form: (23) , and the inversion relationŘ The relation to the R matrix presented in [20] isŘ(z, a) = P R(z, λ) where λ = γa.
By restricting a to take values in an orbit O b we can construct R-matrices acting on groupoid-graded vector spaces. To do this we need to avoid the poles on the hyperplanes a i − a j ≡ 0 mod Λ, (i = j) of the R-matrix. We consider the case where r = 1/γ is a integer > n. Then for a ∈ P the poles are at a i − a j = mr, i = j ∈ {1, . . . , n}, m ∈ Z. Let ∆ ⊂ P be the union of these hyperplanes. The affine Weyl group W r is the group generated by orthogonal reflections at these hyperplane in the Euclidean space P ⊗ R = R n . It acts freely and transitively on the complement of ∆ ⊗ R and P r ++ is the set of weights in a connected component of the complement and is a fundamental domain for the action of W r on P ∆.
We distinguish two cases, named after the corresponding models of statistical mechanics.
If b does not lie in ∆ then R(z) is a well-defined endomorphism of V b ⊗ V b and obeys the Yang-Baxter equation.
(2) Generalized RSOS model. Let b = 0. The groupoid π is the full subgroupoid of O 0 ⋊ P on P r ++ : it consists of pairs (a, µ) ∈ O 0 ⋊ P such that both a and a + µ lie in P r ++ . The non-zero components are V RSOS (a,ǫi) =V ǫi = Ce i , a, a + ǫ i ∈ P r ++ .
To defineŘ(z) in the RSOS case we view V RSOS as a O 0 ⋊ P -graded subspace of V b=0 such that V RSOS γ = 0 for γ ∈ π, see Section 2.9.
Proof. WhileŘ(z) is not defined on all vectors in V ⊗ V , it is well-defined on V RSOS ⊗ V RSOS since by construction the denominators [a i − a j ] don't vanish for a ∈ P r ++ . Suppose (a + ǫ j , ǫ i ) and (a, ǫ j ) are composable arrows in the subgroupoid π, meaning that a, a + ǫ j and a + ǫ i + ǫ j belong to P r ++ . ThenŘ(z) maps V (a,ǫj ) ⊗ V (a+ǫj ,ǫi) to itself if i = j and to V (a,ǫj ) ⊗V (a+ǫj,ǫi) ⊕ V (a,ǫi) ⊗V (a+ǫi,ǫj ) if i = j. The first summand is indexed by a pair of arrows in the subgroupoids but the second is not if a + ǫ i ∈ P r ++ . ThusŘ(z) preserves V RSOS if and only if the component of the image ofŘ(z) in V (a,ǫi) ⊗ V (a+ǫi,ǫj ) for a + ǫ i ∈ P r ++ vanishes. So we need to check the vanishing of the component ofŘ(z, a) in the case where a + ǫ i ∈ P r ++ and a, a + ǫ j , a + ǫ i + ǫ j ∈ P r ++ . The condition for a being in P r ++ is a 1 > · · · > a n > a 1 − r. For a ∈ P r ++ , a + ǫ i violates an inequality if and only if i ≥ 2 and a i−1 = a i + 1 or i = 1 and a n = a 1 − r + 1. The condition that a + ǫ i + ǫ j ∈ P r ++ implies that j = i − 1 if i ≥ 2 and j = n if i = 1. In both cases a j = a i + 1 and thus (7)

Characters.
Here we consider the RSOS case and write V instead of V RSOS for the vector representation.
Out of the vector evaluation representation V (z) one can construct several new representations by the fusion or reproduction method [32,34], which admit pairwise R-matrices for generic values of the evaluation parameters. It follows that their characters form a unital commutative algebra of difference operators with integer coefficients. The unit element is the character of the trivial representation.
Here are some examples of character calculations. The groupoid π r (P ) of the restricted elliptic quantum group with weight lattice P is the full subgroupoid of O 0 ⋊ P on the set A = P r ++ . Its convolution ring is isomorphic to the subring 1 , . . . , t ±1 n ] is the ring of Laurent polynomials with generators t i = t ǫi , i = 1, . . . , n.
• The character of the trivial representation is the multiplication operator by the characteristic function of A = P r ++ : ch 1 = χ A • The character of the vector representation V (z).
The R-matrix (6) has a pole at z = 1 and is not invertible at z = −1. We set R reg (1, a) = res z=1 R(z, a). Then we have an exact sequence (see the Appendix) The analogue of the symmetric square of the vector representation, is S 2 V (z) = KerŘ reg (1) ∼ = CokerŘ(−1). Its character can be computed from the explicit basis of Lemma A.0.1 Similarly, the second exterior power 2 V (z) = CokerŘ reg (1) ∼ = KerŘ(−1) has character These are the famous fusion rules that first appeared in conformal field theory. The algebra with generators L p and relations above is called Verlinde algebra.

Exterior powers.
The kth exterior power k V (z) is defined as the quotient of V (z + k − 1) ⊗ · · · ⊗ V (z + 1) ⊗ V (z) by the sum of the images ofŘ reg (1) (j,j+1) for j = 1, . . . , k − 1. Its character is Here e k (t 1 , . . . , t n ) = 1≤i1<···<i k ≤n t i1 · · · t i k is the kth elementary symmetric polynomial. It follows from the existence of R-matrices for pairs of exterior powers that these characters commute.
The convolution ring Z[π r (P )] ∼ = χ A D P (O 0 )χ A acts naturally on functions on A = P r ++ . The characters can be simultaneously diagonalized.

4.4.1.
Theorem. Let q = e 2πi/r and pick any n-th root q 1/n of q. For each λ ∈ A = P r ++ let ψ λ be the function on A given by Proof. Let us first ignore the characteristic functions χ A and consider ψ λ as a function on the whole weight lattice. Then for any symmetric polynomial P (t) in t 1 , . . . , t n , P (t)ψ λ = P qλ 1 , . . . , qλ n ψ λ .

Quantum enveloping algebras at roots of unity
We are mainly concerned with dynamical R-matrices with non-trivial dependence of the spectral parameter, but it is instructive to consider the case of constant dynamical R-matrices arising from the representation theory of semisimple Lie algebras and their quantum versions. The main simplification in this case is that the category of finite dimensional modules is braided, namely for each pair of objects V, W there is an isomorphism τ V,W : V ⊗ W → W ⊗ V , obeying compatibility conditions with the structure of monoidal category, given by the evaluation of the universal R-matrix composed with the permutation of factors. This property fails for some pairs of objects in the case of quantum affine Lie algebras or Yangians, which is the case where the R-matrices have a non-trivial dependence on the spectral parameter.
We focus on the case of quantum groups at root of unity which is a toy model for restricted models. Strictly speaking the above has to be corrected in this case and one has to be more careful in the definition of the category of finite dimensional modules. We consider the semisimple quotient of the category of tilting modules [7,Sction 11.3], [2,37]. It is an abelian monoidal C-linear ribbon category C q (g) depending on a simple Lie algebra g and a primitive ℓ-th root of unity q. It has a finitely many equivalence classes of simple objects L λ labeled by dominant weights in a scaled Weyl alcove P ℓ + , and any object is isomorphic to a direct sum of simple modules. The alcove P ℓ + is a finite subset of the cone P + of dominant weight, bounded by a hyperplane as in 4.1. See [37] for a description in the most general case of Lie algebra and root of unity. Let P be the weight lattice and π be the full subgroupoid of P ⋊ P on P ℓ + . Its objects are dominant weights and there is exactly one arrow a → b for any two weights a, b ∈ P ℓ + . As usual we denote by (a, b − a) this arrow.
Proof. A morphism f : W → W ′ in U q g-mod induces a morphismf : ϕ → (f ⊗id)•ϕ fromŴ toŴ ′ and this assignment is compatible with compositions so that we get a well-defined functor. The trivial module k which is the tensor unit in U q g is mapped to the tensor unitk = 1 in Vect k (π), see 2.6. Moreover we have a natural transformationŴ ⊗Ẑ → W ⊗ Z whose restriction toŴ a,µ ⊗Ẑ a+µ,λ−µ is the composition Hom(L a , W ⊗ L a+µ ) ⊗ Hom(L a+µ , Z ⊗ L a+λ ) → Hom(L a , W ⊗ Z ⊗ L a+λ ), Since C q (g) is semisimple, by taking the direct sum over µ we get an isomorphism (Ŵ ⊗Ẑ) a,λ → ( W ⊗ Z) a,λ on each graded component. where n λ,µ (a) = N a+λ a,µ . The commutativity of the characters is the associativity of the Verlinde algebra.