Injective Comodules for 2 × 2 Quantum Matrices

William Chin and Leonid Krop

January 25, 1999

Abstract

We study the injective comodules for the coordinate bialgebra of quantum 2 × 2 matrices at a root of unity. The related GL_{2} and SL_{2} theories are also specified.

Introduction

Our main concern is a description of the injective indecomposable comodules for the bialgebra k_{ζ}[M(2)] = Mk and closely related Hopf algebras GLk and SLk where ζ ∈ k is a root of unity of odd order and k is a field of characteristic zero.

In the classical case, the modular representation theory of M(n) was developed by Green ([Gr 2]). Closely related to M(2), but somewhat more special case of the modular SL(2) was investigated by Winter ([Wi]). It seems that he was the first one to realize that certain indecomposable representations constructed by Jeyakumar ([Je]) and Humpreys ([Hu]) form the basic building blocks of the injective indecomposables for SL(2). Later, some structural information for injectives over higher Frobenius kernels using homological methods was given by [AJL], but a complete picture was not attained.

We follow the approach of Winter, and construct quantized Jeyakumar modules. Here, the absence of higher Frobenious kernels results in drastic simplification of the classical pattern, as the injectives turn out to be finite dimensional, and our results give a complete description of the three quantized coordinate algebras.

A major source of the paper is the work of Green ([Gr 1]) in which he developed block theory for comodules as well as an extension of modular set-up for algebraic groups. Green points out the role of injective comodules in the structure theory of coalgebras, which generalizes the role of projectives for finite dimensional algebras. This turns out to be an apt viewpoint for a study of comodules of quantum deformations of M(2) and its offsprings GL(2) and SL(2). The coalgebraic theory efficiently replaces some standard hyperalgebraic methods.

Our choice of Mk as our principal object of study is motivated by the convenience of having a large supply of injective finite - dimensional subcomodules in the form of left and right weight subspaces of Mk. On the other hand, being just a bialgebra, Mk has a somewhat more complex pattern of injectives than do the Hopf algebras GLk and SLk.

An outline of the contents of the paper is as follows. In the first two sections we fix some notation and describe the coordinate coalgebra of quantum matrices M, and the related coalgebras SL and GL. We then describe the theory of weight spaces and highest weight comodules, and the quantum symmetric algebra. In section 4, we show that simple comodules are self-dual. The quantized Jeyakumar modules are constructed in section 5. They are used in section 6 to determine the structure of the injective indecomposables for Mk. In section 7, we show how the theory restricts to GLk and SLk. Finally in section 8, we determine the Grothendieck ring for SLk, showing that G_{0}(SLk) is a polynomial ring generated by the class of the defining representation, and that K_{0}(SLk) is the ideal generated by the Steinberg comodule.

0 Notation and Conventions

ζ lth root of unity over Q, odd l.

k ground field of characteristic 0

q an indeterminate over k

M_{q} quantum coordinate bialgebra over k(q) associated to the monoid of 2 × 2 matrices (also known as k_{q}[M_{2}])

SL_{q} quantum coordinate Hopf algebra over k(q) associated to the special linear group SL(2)

U_{q} quantum Hopf algebra over k(q) associated to sl_{2}

Mk, SLk, Uk ”k - forms” of the above algebras (definitions below).

Whenever k is clear from the context it will be omitted in the above notation For a bialgebra B, a B-comodule V, and a subset X in V the right B-subcomodule

generated by X is denoted by X^{B}. In case V is a B-bicomodule, the left B-subcomodule generated by X is denoted by ^{B}X, and the subbicomodule generated by X is denoted ^{B}X^{B}.

The socle of a B-module V is denoted by σ(V)

If V is a vector space over a field F, X a subset of V, we adopt the notation ≺ X ≻_{F} or simply ≺ X ≻, if the field is clear from the context.

For an n ∈ Z^{+}

(n)_{q} = \frac{q^{n} − 1}{q − 1}  [n]_{q} = \frac{q^{n} − q^{−n}}{q − q^{−1}}
(n)_{q}! = ∏_{m=1}^{n}(m)_{q}  [n]_{q}! = ∏_{m=1}^{n}[m]_{q}
\binom{n}{m}_{q} = \frac{(n)_{q}!}{(m)_{q}!(n − m)_{q}!}  \left[{n \atop m}\right] = \frac{[n]_{q}!}{[m]_{q}![n − m]_{q}!}

The above defined rational functions lie in k[q, q^{−1}]([Lu ]). We write (n)_{ζ}, etc. for their values under the homomorphism q ↦ ζ.

We define a partial ordering ≺ on Z^{+} by declaring m ≺ n if and only if \binom{n}{m}_{ζ} ≠ 0. For an r ∈ Z^{+}, its l-adic decomposition is r = r_{0} + r_{1}l with 0 ≤ r_{0} ≤ l − 1. It is well known that m ≺ n ⟺ m_{0} ≤ n_{0} and m_{1} ≤ n_{1}

1 The k- Form of the Generic Algebras

Let k denote a field containing ζ. Let ν_{ζ} : k(q) ⟶ Z be the valuation on k(q) in which ν_{ζ}(f) is the order of f ∈ k(q) at ζ. Let R = {f ∈ k(q) | ν_{ζ}(f) ≥ 0} be the valuation ring. It is well known ([Kas]) that M_{q}(2) has a basis of x^{A} = x_{11}^{a_{11}}x_{12}^{a_{12}}x_{21}^{a_{21}}x_{22}^{a_{22}}, a_{ij} ∈ Z^{+}. The defining relations show that

Δ(x^{A}) = ∑ x^{A_{i}} ⊗ u_{i}  (∗)

with u_{i} lying in Q[q, q^{−1}]– span of the basic monomials x^{A}. This means M_{q}(2) contains an admissible R– lattice M_{R} := R − span{x^{A}} in the sense that M_{R} is closed to the comultiplication defined on M_{q}(2). f acts on k via f · 1k = f(ζ) · 1k. This action is identical with the natural one of R on R/M ≃ k, where M is the unique maximal ideal of R. In the spirit of ([Gr 1]) we pass from M_{q} to Mk via R defining

Mk = M_{R} ⊗_{R} k

Let R_{0} = Q[q, q^{−1}]. Equation (∗) shows that M_{q} has an admissible R_{0}– lattice, viz. the R_{0}– span of the x^{A} which we denote M_{R_{0}}. This lattice gives rise to the family of algebras Mk (cf. [Gr 2]). Namely k can be viewed as R_{0}-module via f ↦ f(ζ), f ∈ R_{0}. Now for every k Mk is isomorphic to M_{R_{0}} ⊗_{R_{0}} k under the correspondence sending x^{A} ⊗ 1_{k(ζ)} to x^{A} ⊗ 1_{k(ζ)}. Similar considerations can be used to pass from SL_{q} = k_{q}[SL(2)] and U_{q} to SL_{Q(ζ)} and U_{Q(ζ)}. The latter two algebras coincide with U_{ζ} and F_{ζ} defined in ([Lu ]) and ([DeC- Lu]) in the rank 1 case.

2 Weights and Weight Spaces

2.1. Let T = k[t_{1}, t_{2}] be a free commutative algebra generated by t_{1} and t_{2}. We give T a comodule structure by setting Δ_{T} (t_{i}) = t_{i} ⊗ t_{i}. Let δ : M → T be the algebra map sending x_{11} to t_{1}, x_{22} to t_{2}, and x_{i,j} to 0 for i ≠ j. δ is also a coalgebra map, thus a bialgebra one. This suffices to check on the generators x_{i,j} on account of both δ and Δ being algebra maps, which is easy. Next we make

Definition. We say that an element v in a right(left) M-comodule V has weight λ = (m, n) provided (1 ⊗ δ)Δ_{V} (v) = v ⊗ t_{1}^{m}t_{2}^{n} ((δ ⊗ 1)Δ_{V} (v) = t_{1}^{m}t_{2}^{n} ⊗ v).

We denote by Λ ≃ Z^{+} × Z^{+} the set of all weights. The subset of dominant weights, i.e. pairs (m, n) with m ≥ n, is denoted by Λ^{+}. The norm of a weight λ = (m, n) is set as | λ |= m + n. We indicate that λ is a dominant weight of norm d by writing λ ⊢ d. We put the lexicographical ordering ‘≤ ’on Λ. We say that v in V is a highest weight vector if the weight of v is maximal in that ordering. We say that V is a highest weight comodule if V is generated by some of its highest weight vectors. We denote by ^{λ}V (V^{λ}) the k-span of the elements of weight λ and refer to it as the λth weight subspace. In case V is an Mbicomodule we distinguish between the right and the left weight of a v ∈ V by writing wt_{r}(v) and wt_{l}(v), respectively. One can easily check that ^{λ}V (V^{λ}) is a right(left) M-subcomodule of V. Turning to the M-bicomodule M we notice

Lemma 1. (1) If α = (m, n) then ^{α}M =≺ x^{A} | a_{11} + a_{12} = m, a_{21} + a_{22} = n ≻ and M^{α} =≺ x^{A} | a_{11} + a_{21} = m, a_{12} + a_{22} = n ≻

(2) For every α ∈ Λ ^{α}M is a right M-comodule and M^{α} is a left M-comodule.

Proof. We compute the left weight of an x^{A}. Since (δ ⊗ 1)Δ is an algebra map, one has (δ ⊗ 1)Δ(x^{A}) = ∏ [(δ ⊗ 1)Δ(x_{i,j})]^{a_{i,j}} = ∏(δ(x_{i,i}) ⊗ x_{i,j})^{a_{i,j}} = t_{1}^{a_{11}+a_{12}}t_{2}^{a_{21}+a_{22}} ⊗ x^{A}.It follows that every element x^{A} of the first set in (1) lies in ^{α}M. Conversely, if x ∈ M has left weight α, then from x = ∑ c_{A}x^{A} and (δ ⊗ 1)Δ(x) = ∑ t^{λ} ⊗ c_{A}x^{A} = t^{α} ⊗ x it follows that λ = α for all x^{A} with c_{A} ≠ 0. The case of M^{α} is treated similarly.

(2) This follows from the fact that the right and left comodule structure maps commute_{△} M is clearly a direct sum of all ^{α}M as well as M^{α}. Hence Corollary. For every α ∈ Λ ^{α}M is an injective M-comodule _{△} We define the total degree of x^{A} to be deg(x^{A}) = ∑ a_{i,j} and put M^{d} for the span of all

monomials of degree d. We notice that M^{d} = ⊕{^{α}M | |α| = d} = ⊕{M^{α} | |α| = d}

For a general right M-comodule V the sum of V^{λ} is always direct, provided they are nonzero. In fact, V is always the sum of its weight subspaces

Lemma 2. Each right M-comodule is spanned by its weight elements.

Proof. It suffices to show that every cyclic submodule v^{M} has this property. Let ω be the structure map, and write ω(v) = ∑ v_{i} ⊗ x^{A_{i}}. By ([Gr 1]) the set {v_{i}} spans v^{M}. Now let us apply 1 ⊗ δ ⊗ 1 to the identity ∑ ω(v_{i}) ⊗ x^{A_{i}} = ∑ v_{i} ⊗ Δ(x^{A_{i}}). We get ∑(1 ⊗ δ)ω(v_{i}) ⊗ x^{A_{i}} = ∑ v_{i} ⊗ (δ ⊗ 1)Δ(x^{A_{i}}). As (δ ⊗ 1)Δ(x^{A_{i}}) = t^{β_{i}} ⊗ x^{A_{i}} with β_{i} = wt_{l}(x_{i}^{A}) we conclude (1 ⊗ δ)ω(v_{i}) = v_{i} ⊗ t^{β_{i}} or wt(v_{i}) = β_{i}.

Remark. The lemma follows directly from the fact that the bialgebra T is pointed cosemisimple.

2.2. In order to arrive at a representation theoretic structure of ^{α}M we bring in the quantum symmetric algebra S_{q}(2). We let S_{q}(2) stand for the quotient of the free algebra on two generators e_{1}, e_{2}, say, modulo the relation e_{2}e_{1} = qe_{1}e_{2} (see [Kas]). S_{q} is Z^{+} graded with the m-th component S_{q}^{m} =≺ e_{1}^{i}e_{2}^{j} | i + j = m ≻_{k(q)}. The R-lattice generated by the basis {e_{1}^{i}e_{2}^{j}} gives rise to the algebra Sk often abbreviated to S. The latter can be described as the quotient of the free k-algebra on {e_{1}, e_{2}} modulo the relation e_{2}e_{1} = ζe_{1}e_{2}. S is also graded by Z^{+} with the S^{m} =≺ e_{1}^{i}e_{2}^{j} | i + j = m ≻k.

We turn S into a right M-comodule via ω : S → S ⊗ M which is specified on the generators by

ω(e_{j}) = ∑ e_{k} ⊗ x_{kj}

It’s routine to check that ω respects the defining relation of S, thus extending uniquely to an algebra homomorphism. It’s easy to see that ω defines M-comodule structure on S, making S an M-comodule - algebra ([Kas]). A slightly more general family of M-comodules is constructed this. Let D^{b}, b ≥ 0 be the 1-dimensional M-bicomodule given by the grouplike det_{q}^{b}. For every λ = (m, n) ∈ Λ^{+} we put

M(λ) = S^{m−n} ⊗ D^{n}

.

Before passing on to the structure of these comodules we fix an important notation

Definition 1. For a dominant λ we denote by L(λ) the isomorphism class of the socle of M(λ). We shall write L(r) for L(λ) when λ = (r, 0).

Definition 2. We let ρ denote the mapping of Z^{+} × Z^{+} onto itself defined by ρ(m, n) = ρ(m − n) + (n, n) where ρ(r) is defined as follows. Write r = r_{0} + r_{1}l, 0 ≤ r_{0} ≤ l − 1, r_{1} ∈ Z^{+}. Then

ρ(r) = \{ \begin{matrix}r & if r_{0} = l − 1 \\ (r_{1}l − 1, r_{0} + 1) & else \end{matrix}

Note : Realizing Λ^{+} as a subset of Z × Z, geometrically ρ(m, n) is the reflection of (m, n) across x − y = sl − 1, where s is the largest positive integer such that sl − 1 < m − n.

Lemma 1. M(m, n) is a simple right M-comodule iff m−n = sl−1 for some s = 1, 2, …. Otherwise, M(m, n) is a nonsplit extension of L(m, n) by L(ρ(m, n)).

Proof. It suffices to deal with S^{d}. For an arbitrary e_{1}^{i}e_{2}^{j} with i+j = d one has i_{0} +j_{0} ≡ d_{0} (mod l). Hence either i_{0} + j_{0} = d_{0} or i_{0} + j_{0} = d_{0} + l. This gives rise to the division of the natural basis of S^{d} into two subsets according to the above dichotomy. Let L(d) be the k - span of the first set. It’s straightforward to show that L(d) is a simple subcomodule generated by e_{1}^{d}. Another verification shows that every monomial from the second set generates the entire S^{d}. Further, notice that i_{0} + j_{0} = d_{0} + l implies j_{0} = d_{0} + l − i_{0} ≥ d_{0} + 1. Now , select from the second set of monomials the one with the smallest value of j, i.e. j = d_{0} + 1, thus with i = d − j = d_{1}l − 1. The monomial e_{1}^{d_{1}l−1}e_{2}^{d_{0}+1} is the highest weight vector in S^{d}/L(d). It is of weight μ = (d_{1}l − 1, d_{0} + 1) for the weight of every e_{1}^{i}e_{2}^{j} is (i, j). It remains to show that S^{d}/L(d) ≃ L(μ). To this end we define φ : S^{d}/L(d) ⟶ L(μ) by φ(e_{1}^{i}e_{2}^{j} + L(d)) = e_{1}^{i′}e_{2}^{j′} ⊗ D^{d_{0}+1}, where i′ = i − (d_{0} + 1), j′ = j − (d_{0} + 1). A direct inspection shows that φ is a comodule morphism and by the dimension consideration it is an isomorphism _{△}

We now place M(λ) in M. A routine calculation shows that≺ x_{11}^{i}x_{12}^{j} | i + j = r ≻ is isomorphic to S^{r}, and further ≺ x_{11}^{i}x_{12}^{j}det_{q}^{b} : i + j = r ≻ is isomorphic to S^{r} ⊗ D^{b}. The isomorphism carries x_{11}^{i}x_{12}^{j} to e_{1}^{i}e_{2}^{j} and therefore x_{11}^{r}det_{q}^{b} to e_{1}^{r} ⊗ D^{b}. As the latter generates the socle of S^{r} ⊗ D^{b} we see that the former generates the simple comodule of highest weight (r + b, b).

Definition 3. For a dominant weight α = (a, b) we put g_{α} = x_{11}^{a−b}det_{q}^{b}.

Lemma 2. (1) g_{a,b}^{M} is a simple M-comodule of highest weight (a, b).

(2) The family {g_{a,b}^{M}} exhausts all simple right M-comodules.

Proof.(1) Has been proven above.

(2) Recall that M_{q} is Z^{+} graded by total degree, viz. M_{q} = ⨆_{d≥0} M_{q}^{d}. For every (a, b) with a + b = d, we have g_{a,b}^{M_{q}} ⊂ M_{q}^{d}. A direct verification based on ([Gr 1, 1.2b]) shows that g_{a,b}^{M_{q}} =≺ x_{11}^{i}x_{12}^{j}·det_{q}^{b} | i+j = a−b ≻. Put differently g_{a,b}^{M_{q}} ≃ S_{q}^{a−b}⊗D_{q}^{b} – tensor product of Mcomodules. Let n_{a,b} := dim_{k(q)}g_{a,b}^{M_{q}}. Then by general principles M_{q} contains n_{a,b} copies of g_{a,b}^{M_{q}}, because the latter is a simple M_{q}-comodule for every (a, b). We claim that ⨁_{(a,b)⊢d}(g_{a,b}^{M_{q}})^{n_{a,b}} equals to M_{q}^{d}. To this end we notice that putting \overline{M}(2) := k[\overline{x}_{i,j} | i, j = 1, 2] we have at once dim_{k(q)}M_{q}^{d} = dimk\overline{M}^{d}. It is well known ([Gr 2]) that \overline{M}^{d} = ⨁_{(a,b)⊢d} (\overline{S}^{a−b} ⊕ \overline{D}^{b})^{n_{a,b}} where

the \overline{S}^{r} and \overline{D}^{b} are the classical symmetric power of the defining representation and the b-th power of the determinant representation, respectively. The claim follows.

We can complete the proof by computing the simple factors of Mk^{d}. The principal fact (see [Gr 1, (2.5a)]) is that those coincide with the simple factors obtained through a reduction of the g_{a,b}^{M_{q}} ≃ S_{q}^{a−b} ⊗ D^{b} as described at the beginning of the §. By Lemma 1 its composition factors are of the form L(a, b) for some (a, b) ⊢ d. Therefore every composition factor of Mk is also of this form. As every simple Mk - comodule embeds in Mk the proof is completed _{△}

We point out a form of the Lusztig’s factorization theorem ([Lu ])

Lemma 3. Let λ = (r + n, n) and write r = r_{0} + r_{1}l. Then L(λ) ≃ L(r_{0}) ⊗ L(r_{1}l) ⊗ D^{n}.

Proof. By its definition L(r) =≺ e_{1}^{i}e_{2}^{j} | i_{0}+j_{0} = r_{0} ≻. The mapping e_{1}^{i}e_{2}^{j} ↦ e_{1}^{i_{0}}e_{2}^{j_{0}}⊗e_{1}^{i_{1}l}e_{2}^{j_{1}l} realizes an isomorphism of L(r) with L(r_{0}) ⊗ L(r_{1}l) _{△}

Corollary 1. In the notation of the Lemma, dim L(λ) = (r_{0} + 1)(r_{1} + 1).

Proof. From the Lemma dim L(λ) = dim L(r) and the latter equals dim L(r_{0})·dim L(r_{1}l). Clearly dim L(r_{0}) = r_{0} + 1. As for the second factor, it is a comodule over the subbialgebra Mk^{(l)} of Mk generated by the x_{i,j}^{l}. It’s easy to see that Mk^{(l)} is isomorphic with \overline{M} under the mapping x_{i,j}^{l} ↦ \overline{x}_{i,j}. But as an \overline{M}-comodule L(r_{1}l) is isomorphic to \overline{S}^{r_{1}} _{△}

2.3. S ⊗ S can be endowed with the tensor product comodule structure that turns it into a Z^{+} × Z^{+} graded comodule with the (m, n)-graded component S^{m} ⊗ S^{n}.

Lemma 1. For every λ = (m, n) ^{λ}M ≃ S^{m} ⊗ S^{n} as right M-comodules.

Proof. Associate to x^{A} = x_{11}^{a_{11}}x_{12}^{a_{12}}x_{21}^{a_{21}}x_{22}^{a_{22}} the decomposable tensor e_{1}^{a_{11}}e_{2}^{a_{12}} ⊗ e_{1}^{a_{21}}e_{2}^{a_{22}}. A verification that this map is a morphism of comodules is straightforward _{△}

3 Duality

We want to define a right comodule structure on the linear dual of an M- comodule V. Our construction follows closely that of ([Gr 2]).

We begin with definition of the mapping tr : Mk → Mk specified on generators by x_{ij}^{tr} = x_{ji}. tr respects the defining relations, hence extends to an automorphism on Mk. Further tr is a coalgebra antimorphism on account of identity (tr ⊗ tr)Δ = Δ^{op} ∘ tr. It suffices to check the latter on generators, because both tr and Δ are algebra morphisms, which presents no difficulties. Now, let ω : V → V ⊗ Mk be specified on a basis of {v_{i} | i ∈ I} of V through ω(v_{j}) = ∑ v_{i}⊗f_{ij}. The induced left coaction of Mk on V^{∗} is given by v_{j}^{∗} ↦ ∑ f_{ji}⊗v_{i}^{∗}.

This, in turn, induces a right Mk^{cop}- coaction v_{j}^{∗} ↦ ∑ v_{i}^{∗} ⊗ f_{ji}. Combining the latter with tr : Mk → Mk^{cop} we arrive at the right Mk- comodule structure on V^{∗}. We denote by V^{∘} the space V^{∗} endowed with this comodule structure. Explictly, ω^{∘}(v_{j}^{∗}) = ∑ v_{i}^{∗} ⊗ f_{ji}^{tr}.

Lemma 1. Every simple M- comodule is self-dual.

Proof. This is based on two properties, each well known in case of modules for algebraic groups. First, the annihilator in V of a subcomodule of V^{∘} in the natural pairing between V and V^{∘} is a subcomodule of V. This guarantees that the dual of a simple comodule is also simple. Second, dim V^{α} = dim(V^{∘})^{α} for every weight α. Indeed, its a straightforward matter to verify the identity

⟨(1 ⊗ δ)ω(v), ϕ ⊗ 1⟩ = ⟨v ⊗ 1, (1 ⊗ δ)ω^{∘}(ϕ)⟩ for every v ∈ V, and ϕ ∈ V^{∘} and deduce from it that V^{α} is orthogonal to (V^{∘})^{β} for every α ≠ β. These remarks show that V^{∘} is simple of highest weight α, if V is. As the simples are determined by their highest weights, the proof is completed_{△}

4 The Quantum Jeyakumar Comodules

Our first step will be a computation of the socle of ^{α}M.

Lemma 1. For a dominant α = (m, n)

σ(^{α}M) = ⨁ { L(m + i, n − i) | i ⪯ m − n + 2i & 0 ≤ i ≤ n}

Proof. By general coalgebraic theory M contains a copy of every simple right and left comodule. Further, σ(M) = ⊕^{M}L, where L runs over the set of representatives of simple right subcomodules of M. By Lemma 2.2.2 we may assume L = g_{a,b}^{M}. As σ(^{α}M) = ^{α}M ∩ σ(M) = ⊕_{L}(^{α}M ∩ ^{M}L), it remains to select the L with the property that ^{M}L contains a right subcomodule of the left weight α. We sketch an explicit computation of these L’s.

To begin with, ^{M}L = (^{M}g_{a,b})^{M} because the left and the right action commute with each other. Secondly, ^{M}g_{a,b} is the span of v_{i} = x_{11}^{a−b−i}x_{21}^{i} ·det_{q}^{b} where i ⪯ a−b. The above element has left weight (a − i, b + i). Therefore every element of v_{i}^{M} has this left weight. It follows αM ∩^{M}g_{a,b}^{M} = v_{i}^{M}, provided (a − i, b + i) = (m, n) for some i ⪯ a − b. The last condition is equivalent to a = m+i, b = n−i with i ⪯ m−n+2i. Since v_{i}^{M} is a simple M-subcomodule, and the right weight of v_{i} is (a, b), as all elements of ^{M}g_{a,b} have this weight, it follows that v_{i}^{M} ≃ L(m + i, n − i) _{△}

Definition. For a dominant α = (m, n) we let I(m, n) denote the injective hull of L(m, n).

A construction of I(m, n) begins with the l basic injectives corresponding to the weights (l − 1, k) for 0 ≤ k ≤ l − 1. These injectives are analogous to certain modules constructed by Jeyakumar ([Je]) and Humphreys ([Hu])

Theorem 1. For every k = 0, 1 …, l − 1 the M- comodules I(l − 1, k) have the following properties :

(i) I(l − 1, k) is a self-dual comodule (ii) I(l − 1, 0) = S^{l−1} while I(l − 1, k), k ≥ 1 resolves as follows

0 → L(l − 1, k) → I(l − 1, k) → S^{l−1+k} → 0

(iii) ^{(l−1,k)}M ≃ S^{l−1} ⊗ S^{k} ≃ I(l − 1, k) ⊕ I(l − 1, k − 2) ⊕ ⋯ ⊕ I(l − 1, k − 2[\frac{k}{2}]) ⊗ D^{[ \frac{k}{2} ]}

Proof. We take up (ii) and (iii) first. Recall that ^{α}M is injective for every α. Thus L(l − 1) = S^{l−1} ≃ ^{(l−1,0)}M is injective, whence I(l − 1, 0) = S^{l−1}. Assume (ii) and (iii) holds for every 0 ≤ k < n ≤ l − 1. Let’s consider S^{l−1} ⊗ S^{n}. From Lemma 1 we know that σ(S^{l−1)} ⊗ S^{n}) = ⊕L(l − 1 + i, n − i), where the sum is extended over nonnegative integers such that i ⪯ l − 1 − n + 2i and i ≤ n. Our first observation is that the above set coincides with the set {i | 0 ≤ 2i ≤ n}. For, if 2i ≤ n, then i ≤ 2i ≤ l − 1 − n + 2i on account of n ≤ l − 1. If 2i > n, then l − 1 − n + 2i = l + s for some s ≤ l − 1. Now i = l + s − (l − 1) + n − i = s + (n + 1 − i) > s contradicting i ⪯ l − 1 − n + 2i.

We notice next that L(l − 1 + i, n − i) ≃ L(l − 1, n − 2i) ⊗ D^{i}. Now for every i ≥ 1, L(l − 1, n − 2i) occurs in the socle of S^{l} ⊗ S^{n−2i}, and by the inductive hypothesis is the socle of I(l−1, n−2i). From this we conclude that I(l−1, n−2i)⊗D^{i} has L(l−1+i, n−i) as its socle. We claim that, in fact, I(l−1, n−2i)⊗D^{i} is the injective cover of L(l−1+i, n−i). In the first place, it follows from the inductive hypothesis that dimk I(l − 1, r) = 2l for every r ≥ 1. Secondly, I(l − 1, n − 2i) ⊗ D^{i} embeds into I(l − 1 + i, n − i). It remains to show that dimension of the latter is 2l. This is done in the following

Lemma 2. Let Δ be the set of weights given by Δ = {(m, n) | l − 1 ≤ m, m + n ≤ 2(l − 1) and m − n < l − 1}. Then for every (m, n) ∈ Δ dimk I(m, n) = 2l

Proof. The reflection of (m, n) ∈ Δ across the line x − y = l − 1, say, (m′, n′) gives by Lemma 2.2.1 a unique dominant weight connected to (m, n) in the sense that M(m, n) and M(m′, n′) share a common simple factor. In this case this factor is M(m, n) itself on account of its simplicity. Pick α := (l − 1 + i, n − i), 2i < n in Δ and let β be its reflection. Then the block of the decomposition matrix corresponding to {α, β} has the form

D = ( \begin{matrix}& L(α) & L(β) \\ M(α) & 1 & 0 \\ M(β) & 1 & 1 \end{matrix} )

where the (α, β) entry is the multiplicity of L(β) in M(α). By a theorem of Green ([Gr 1]) (see also Lemma 5.2) the matrix

C = D^{t} · D = ( \begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} )

records the multiplicities of L(α), L(β) in I(α), I(β). It follows I(α) ≈ 2L(α) + L(β), and M(β) ≈ L(α) + L(β), where we use ≈ to mean that two modules have the same composition factors. By Corollary 2.2.1 in view of the fact β = (l − 1 + n − i, i), the dimension of I(α) is 2(l − n + 2i) + 2(n − 2i) = 2l _{△}

We can complete now the proof of the Theorem. According to a theorem of ([Gr 1]) there is an injective indecomposable summand I(l − 1, n) of S^{l−1} ⊗ S^{n} with socle L(l − 1, n). The above discussion yields

S^{l−1} ⊗ S^{n} ≃ ⨁ \{I(l − 1, n − 2i) ⊗ D^{i} | 0 ≤ i ≤ [\frac{n}{2}]\}  (#)

One can easily deduce from this that dim I(l−1, n) = 2l. Now consider the multiplication map μ : S^{l−1} ⊗ S^{n} → S^{l−1+n}. As we know S is a comodule - algebra which implies μ is a comodule morphism. Since the composition factors of every I(l − 1, n − 2i) ⊗ D^{i}, i ≥ 1 are distinct from those of S^{l−1+n}, each of these summands is sent to zero by μ. Hence μ(I(l − 1, n)) = S^{l−1+n}, and a dimension count yields Kerμ |_{I(l−1,n)}= l − n. But the latter is exactly the dimension of the socle of I(l − 1, n), and the proof of (i) and (ii) is completed.

(i) By Lemma 3.1 S^{i} is self dual for every i ≤ l − 1. The usual isomorphism (V ⊗ W)^{∘} ≃ V^{∘} ⊗ W^{∘} implies self duality of S^{l−1} ⊗ S^{n}. Furthermore the decomposition (#) is unique and every summand has its unique highest weight. Applying Lemma 3.1 we see that I^{∘} has the same highest weight as I does for every indecomposable I occuring in (#) _{△}

Theorem 2. The multiplication table of I(l − 1, k) by S^{1} is given by:

(1) For k = 0

I(l − 1) ⊗ S^{1} = I(l − 1, 1)

(2) For k = 1

I(l − 1, 1) ⊗ S^{1} = I(l − 1, 2) ⊕ 2I(l − 1) ⊗ D

(3) For every k, 2 ≤ k ≤ l − 2

I(l − 1, k) ⊗ S^{1} = I(l − 1, k + 1) ⊕ I(l − 1, k − 1) ⊗ D

(4) For k = l − 1

I(l − 1, l − 1) ⊗ S^{1} = I(l − 1) ⊗ L(l) ⊕ I(l − 1, l − 2) ⊗ D

Proof. (1) This is a particular case of the previous theorem.

Every indecomposable I under consideration occurs in S^{l−1} ⊗ S^{k} for some k ≤ l − 1. Therefore I ⊗ S^{1} is a summand of (S^{l−1} ⊗ S^{k}) ⊗ S^{1}. We claim that the latter comodule is injective. Indeed, by Lemma 2.3.1 S^{a} ⊗ S^{b} ≃ ^{(a,b)}M is injective for every weight (a, b). Combine this with a series of isomorphisms (S^{l−1} ⊗ S^{k}) ⊗ S^{1} ≃ S^{l−1} ⊗ (S^{k} ⊗ S^{1}), and S^{k} ⊗ S^{1} ≃ S^{k+1} ⊕ S^{k−1} ⊗ D, the last one by Lemma 1. We conclude that I ⊗ S^{1} is injective.

Recall that the socle of I(l − 1, k) is L(l − 1, k) ≃ S^{l−1−k} ⊗ D^{k}. Therefore

L(l − 1, k) ⊗ S^{1} ≃ S^{l−1−k} ⊗ S^{1} ⊗ D^{k} ≃ S^{l−k} ⊗ D^{k} ⊕ S^{l−1−(k+1)} ⊗ D^{2}
≃ L(l, k) ⊕ L(l − 1, k + 1).

This, of course, shows that L(l − 1, k) ⊗ S^{1} is semisimple, and therefore lies in the socle of I(l − 1, k) ⊗ S^{1}.

Suppose k is as in (3). Then by Theorem 1 the injective hulls of those simples are I(l − 1, k − 1) ⊗ D and I(l − 1, k + 1), so that their sum embeds in I(l − 1, k) ⊗ S^{1}. As k ≥ 2, part (ii) of that theorem shows that their dimensions are 2l. As dimension of I(l − 1, k) ⊗ S^{1} is 4l, (3) is established.

The cases (2) and (4) need separate treatment because in these cases the socle of the product is larger than the product of the socles.

(2) Recall that I(l − 1, 1) ≃ S^{l−1} ⊗ S^{1}. But

(S^{l−1} ⊗ S^{1}) ⊗ S^{1} ≃ S^{l−1} ⊗ (S^{1} ⊗ S^{1}) ≃ S^{l−1} ⊗ S^{2} ⊕ S^{l−1} ⊗ D

and the assertion follows from Theorem 1.

(4) The socle of I(l − 1, l − 1) is D^{l−1}. Therefore

D^{l−1} ⊗ S^{1} ≃ S^{1} ⊗ D^{l−1} = L(l, l − 1) = L(l − 1, l − 2) ⊗ D

is in the socle of I(l−1, l−1)⊗S^{1}. Therefore I(l−1, l−2)⊗D is a summand of I(l−1, l−1)⊗S^{1}. Further I(l − 1, l − 1) contains unique line of elements of weight 2(l − 1). Therefore the (2l − 1, 0)- weight space of I(l − 1, l − 1) ⊗ S^{1} is one dimensional. For every z in this weight space the subcomodule z^{M} is indecomposable (because its highest weight subspace is 1-dimensional) and has L(2l − 1) as its epimorphic image. But L(2l − 1) ≃ S^{2l−1} by Lemma 2.2.1, and S^{2l−1} is injective by Lemma 2.3.1 and Corollary 2.1. Therefore z^{M} ≃ S^{2l−1}, for otherwise (z^{M})^{∘} would have (S^{2l−1})^{∘} ≃ S^{2l−1} as a submodule, hence splits up, which in turn implies a splitting of z^{M}. We see that I(l − 1, l − 1) ⊗ S^{1} contains S^{2l−1} ⊕ I(l − 1, l − 2) ⊗ D. By dimensions, they are equal.

In conclusion we notice that the comodule S^{l−1} ⊗ L(l) has 1- dimensional (2l − 1, 0)- weight subspace, so that the argument just used leads up to an isomorphism of it with S^{2l−1}_{△}

5 Determination of I(m, n)

We will show that the I(n, m) fall into two classes. One of these consists of the comodules M(m, n). The larger second class is built from the basic injectives I(l − 1, k), 1 ≤ k ≤ l − 1 via tensoring them with a suitable simple comodule of the form L(sl). The L(sl) corresponds to simple representation of \overline{M}(2) of highest weight s.

We define sets of dominant weights for s ≥ 0

L_{s} = {(m, n) ∈ Λ^{+} | m − n = sl − 1} for s ≥ 1
L_{0} = {(m, n) ∈ Λ^{+} | m = n}

the strips of weights

Γ_{s} = {(m, n) ∈ Λ^{+} | sl − 1 < m − n < (s + 1)l − 1} for s ≥ 1
Γ_{0} = {(m, n) ∈ Λ | 0 < m − n < l − 1}, and
Γ̌_{s} = {(m, n) ∈ Γ_{s} | (s + 1)sl − 1 ≤ m} for s ≥ 0

and finally, the set difference

Δ_{s} = Γ_{s} \ Γ̌_{s}, s ≥ 0

. Clearly Λ^{+} is a disjoint union of the Γ̌_{s}, Δ_{s}, and L_{s} for s running over Z^{+}. For λ = (m, n) ∈ Γ̌_{s} we put k = m − (s + 1)l + 1 and q = n − k. In these notation we state the main theorem

Theorem 1 (1) If λ ∈ Γ̌_{s}, then

I(λ) = I(l − 1, q) ⊗ L(sl) ⊗ D^{k}

(2) If λ ∈ Δ_{s} or λ ∈ L_{s}, then

I(λ) = M(λ)

We need a few lemmas. The main first requires some preparations.

Definition. An Mk- comodule V is said to be homogeneous if the coefficient space of V is a subspace of Mk^{d} for some d.

Let π : Mk → SLk be the canonical map. Recall that ker π = J is generated by det_{ζ} −1. For every Mk-comodule V we denote by V_{S} the SLk-comodule induced by V via pushing the structure map ω along π. That is to say, ω_{S}(v) = ∑ v_{i} ⊗ π(x_{i}), provided ω(v) = ∑ v_{i} ⊗ x_{i}.

Lemma 1. Let I be a homogeneous Mk-comodule. Suppose I_{S} is an SLk- injective comodule. Then I is Mk- injective.

Proof. We may assume I embedded into a ”free” injective comodule (X) ⊗ Mk for a suitable vector space X ([Gr 1]). Extend π to the mapping π̂ : (X) ⊗ Mk → (X) ⊗ SLk. Clearly ker π̂ = (X) ⊗ J, and therefore ker π̂ does not contain homogeneous subcomodules. Further, by our assumption, π(I) is SLk- injective. Therefore there is a subcomodule C of (X) ⊗ SLk for which π(I) ⊕ C = (X) ⊗ SLk. Let Ĉ be the preimage of C. Were I ∩ Ĉ ≠ (0), I ∩ ker π̂ would also be nonzero, a contradiction _{△}

Lemma 2. (1) For every λ ∈ Γ̌_{s}, dim I(λ) = 2(s + 1)l

(2) For every λ ∈ Δ_{s} ∪ L_{s}, dim I(λ) = dim M(λ)

Proof. (1) The proof is based on a theorem of Green ([Gr 1]) relating the decomposition numbers to the Cartan invariants. We recall the necessary definitions. Let d_{αβ} and c_{αβ} be the composition multiplicities of L(β) in M(α) and I(α), respectively. Let D = (d_{αβ}) and C = (c_{αβ}). Then the aforementioned result states

c_{αβ} = ∑ d_{γα} · d_{γβ}

or, in the matrix form,

C = D^{t} · D

Let’s consider the sequence bl.(λ) of all dominant weights containing λ, taken in the decreasing order with every two consecutive weights λ_{i} and λ_{i+1} satisfying λ_{i} = ρ(λ_{i+1}), where ρ is defined in Lemma 2.2.1. By this same lemma the matrix D breaks up into disjoint submatrices corresponding to the various bl.(λ). Every such submatrix is the m × m matrix, m = the number of weights in bl.(λ), of the form

D = ( \begin{matrix}1 & 0 … 0 & 0 \\ 1 & 1 … 0 & 0 \\ . & . . . . . & . \\ . & . . . . . & . \\ . & . . . . . & . \\ 0 & 0 … 1 & 1 \end{matrix} )

where the dots stand for zero entries. The corresponding block of C is

C = ( \begin{matrix}2 & 1 & 0 … 0 & 0 \\ 1 & 2 & 1 … 0 & 0 \\ . & . & …. 0 & 0 \\ . & . & …. 0 & 0 \\ . & . & …. 0 & 0 \\ 0 & 0 & 0 … 1 & 1 \end{matrix} )

From the matrix C we deduce that for every λ which is neither the first, nor the last element of bl.(λ), I(λ) ≃ 2L(λ)+L(′λ)+L(λ′), where′λ and λ′ are the immediate predecessor and successor of λ, respectively. It remains to notice that by Lemma 2.2.1 dim I(λ) = dim M(λ) + dim M(λ′). The same lemma shows that λ′ is the reflection of λ in x − y = (s + 1)l − 1, hence λ′ = (n + (s + 1)l − 1, m − (s + 1)l + 1). The assertion follows.

(2) If λ ∈ Δ_{s} for some s ≥ 0, then the last row of C shows that I(λ) ≈ L(λ) + L(′λ) ≈ M(λ).

If λ ∈ L_{s}, then bl.(λ) = {λ} is a singleton, forcing the equality I(λ) = L(λ) = M(λ)_{△}

Proof of the Theorem. (1) We claim that the restriction of J := I(l − 1, q) ⊗ L(sl) ⊗ D^{k} to SLk is an injective SLk- module. By The Fundamental Theorem of Hopf Modules ([Mo]) it suffices to show that I(l − 1, q)_{S} is such. Now by Theorem 4.1 I(l − 1, q) is a summand of S^{l−1} ⊗ S^{q}, so everything follows from injectivity of S_{S}^{l−1}. A more general version of this fact (under a more stringent conditions) appeared in the literature ([APW]). An independent proof fitting the framework of this paper will be given below (Lemma 6.2.1). Thus by Lemma 1 J is Mk- injective. It remains to notice that J contains L(l − 1, q) ⊗ L(sl) ⊗ D^{k} = L(λ) in its socle. Hence I(λ) is a summand of J, and a reference to Lemma 2(1) completes the proof.

(2) Since M(λ) has socle L(λ), it embedds into I(λ). Now use Lemma 2(2) _{△} The last result contains the subcomodule structure of I(λ)

Theorem 2. (1) For a λ ∈ Γ̌_{s} the lattice of subcomodules of I(λ) is given by the diagram

[figure: I(λ); J; M(λ); W (λ′); L(λ)]

That is to say, I(λ) has a unique maximal subcomodule J satisfying J/σ(I(λ)) ≃ L(′λ)⊕ L(λ′).

(2) For a λ ∈ Γ̌_{s} I(λ) is a nonsplit extention of M(λ) by M(λ′).

Note 1: When s = 0 ′λ is not defined and L(′λ) should be omitted in part (1).

Proof. (1) If s = 0, then by Theorem 1 I(λ) = I(l − 1, q) ⊗ D^{k} and the assertion follows from Theorem 4.1(ii). In general, i.e. s ≥ 1, I(λ) = I(l − 1, q) ⊗ L(sl) ⊗ D^{k}. The comodule on the right is self-dual by the combination of Lemma 3.1 and Theorem 4.1(i). Since I(λ) has a simple socle, it therefore also has a simple head, and a unique maximal subcomodule, call it J. Dualizing the exact sequence of Theorem 4.1(ii) we arrive at a subcomodule W (l −1+q) := (S^{l−1+q})^{∘} of I(l −1, q) with the property I(l −1, q)/W (l −1+q) ≃ L(l −1, q). This leads to conclusion that W (l − 1 + q) ⊗ L(sl) ⊗ D^{k} is a maximal subcomodule, hence equal to J. Further the socle and head of W (l − 1 + q) are L(l − 1, q) and L(l − 1 + q), respectively on account of Lemma 2.2.1. Since σ(I(λ)) = L(l − 1, q) ⊗ L(sl) ⊗ D^{k}, we deduce J/σ(I(λ)) ≃ L(l − 1 + q) ⊗ L(sl) ⊗ D^{k}. By Lemma 2.2.3 L(l − 1 + q) ≃ L(q − 1) ⊗ L(l), so that applying the classical Clebsch-Gordon formula to L(l) ⊗ L(sl) we complete the proof _{△}

(2) Since the socle of M(λ) is L(λ) M(λ) embedds into I(λ). The quotient is a nonsplit extension of L(λ′) by L(λ) isomorphic to M(λ′) _{△}

6 The Case of GLk and SLk

6.1. The algebra GLk is the result of localization of Mk with respect to the multiplicative set {det_{ζ}^{m} | m ≥ 0}. Every Mk- comodule is clearly a GLk one.

Lemma . S^{l−1} is an injective GLk- comodule.

Proof. Let X = ⨁_{m≥0} S^{l−1} · det_{ζ}^{m}. By Theorem 5.1(2) X is Mk- injective, and therefore

Mk = X ⊕ Y where Y is the sum of the injectives isomorphic to I(λ) for every λ ≠ (l − 1 + m, m), m ∈ Z^{+}. The key observation is that Y · det_{ζ} ⊂ Y. This follows from the combination of Lemma 5.2 and part (2) of Theorem 5.1 which does not depend on injectivity of S^{l−1} as SLk- comodule. Now define X̂ as the subspace of GLk spanned by all x · det_{ζ}^{s} for x ∈ X and s ∈ Z, and similarly for Y. It is routine to check that X̂ ⊕ Ŷ = GLk. But X̂ is a direct sum of S^{l−1} · det_{ζ}^{s}, s ∈ Z, so that S^{l−1} is a direct summand of GLk _{△}

Let Π^{+} be the set of dominant weights λ = (m, n), m ≥ n. We denote by Π_{s}^{+}, s ≥ 0 the subset of weights satisfying sl − 1 < m − n < (s + 1)l − 1. Put k = m − (s + 1)L + 1 and q = n − k. Notice that q satisfies 0 ≤ q ≤ l − 1. Let L̂_{s} be the set of all dominant weights on the line x − y = sl − 1.

Theorem. (1) For a λ ∈ L̂_{s}

I(λ) = M(λ)

(2) For a λ ∈ Π_{s}^{+}

I(λ) = I(l − 1, q) ⊗ L(sl) ⊗ D^{k}

Proof. (1) M(λ) = S^{sl−1} ⊗ D^{n} by Lemma 2.2.1. Further S^{sl−1} ≃ S^{l−1} ⊗ L(sl) on account of Lemma 2.2.3. Now the assertion follows from the Fundamental Theorem of Hopf modules_{△}

(2) Denote by J the right hand side of equality in (2). By lemma 1 I(l − 1, q) is GLkinjective, and then so is J. By Theorem 5.1(1) the socle of J is L(λ) _{△}

6.2. We begin the case of SLk with a few remarks. Recall that given an Mk- comodule V via the structure map ω : V → V ⊗ Mk, the restriction V_{S} of V to SLk is obtained through ω_{S} := πω. Our first observation is that for every homogeneous Mk- comodule V every SLk- subcomodule of V_{S} is of the form U_{S} for some Mk- subcomodule U. It follows readily that the restriction of every simple Mk- comodule L(m, n) to SLk remains simple. As L(m, n) = σ(S^{m−n} ⊗ D^{n}), and D^{n} restricts to the trivial SLk- comodule, L(m, n)_{S} = L(m − n)_{S}. In what follows we delete the subscript and write this simple comodule as L(r).Defining weights for SLk through the algebra \overline{T} = k[t, t^{−1}] in a fashion similar to that in §2 one can see easily that L(r) i s the highest weight r simple comodule for SLk. In fact the family {L(r)} exhausts all simple SLk- comodules. One way to see this is to utilize the fact that the algebra Uk = Uk(sl(2)) constructed in ([Lu]) is in the perfect duality with SLk ([DeC-Lu]) and the fact that {L(r)} exhaust all simple type 1 modules for Uk. The next remark is analogous to Lemma 6.1

Lemma 1. S_{S}^{l−1} is an injective SLk- module.

Proof. Let X and Y be as in the proof of Lemma 1.1. Then π(X) + π(Y) = SLk and the assertion follows from the equality π(X) ∩ π(Y) = (0) in view of the fact that π(X) = S_{S}^{l−1}. The last equality is equivalent to (X + J) ∩ Y ⊂ J where J = ker π. As J = X(det_{ζ} −1) + Y (det_{ζ} −1) and Y det_{ζ} ⊂ Y the necessary inclusion is obvious _{△}

Definition. We let I(r) denote the injective cover of L(r)

Theorem 1. (1) For r = sl − 1

I(r) = S^{r}

(2) For r = r_{0} + r_{1}l, r_{0} ≠ l − 1

I(r) = I(l − 1, l − 1 − r_{0})_{S} ⊗ L(r_{1}l)_{S}

Proof. (1) S^{sl−1} is a simple Mk- module by Lemma 2.2.1, while by Lemma 2.2.3 S^{sl−1} ≃ S^{l−1} ⊗ L((s − 1)l). Restricting to SLk and using Lemma 1 yields the claim.

(2) Let J = I(l − 1, l − 1 − r_{0}) ⊗ L(r_{1}l). Then J_{S} is an injective SLk- comodule. The socle σ(J) is L(l − 1, l − 1 − r_{0}) ⊗ L(r_{1}l) and since L(l − 1, l − 1 − r_{0}) restricts to L(r_{0}) we see that σ(J)_{S} = L(r) _{△}

7 Grothendieck Rings of SL

7.1. Let [V] denote the class of V in G_{0}(SL). We put λ for [L(1)].

Theorem 1. G_{0}(SL) ≃ Z[λ].

Proof. By definition G_{0} is a free span of the L(a), a ≥ 0. We claim that the set {λ^{a} | a ≥ 0} forms another basis for G_{0}. It suffices to show that these two sets are connected by an unitriangular matrix. To this end we prove (⋆) λ^{a} = L(a) + lesser terms, where L(b) is termed lesser than L(a) provided b < a.

To prove the formula we induct on a. The case a = 1 holds by definition and we assume the formula for all m < a. Then λ^{a} = L(a) ⊗ L(1) + ∑_{b<a} c_{b}L(b) ⊗ L(1) which coupled with

commutativity of G_{0} ([Mo, 10.2.11]) gives L(a) ⊗ L(1) = (L(a_{0}) ⊗ L(1)) ⊗ L(a_{1}l). Now L(a_{0}) ⊗ L(1) = L(a_{0} + 1) + L(a_{0} − 1) for a_{0} < l − 1. This is quite easy to see and is also a particular case of a lemma below. When a_{0} = l − 1, L(l − 1) ⊗ L(1) can be viewed as the restriction to SL of S^{l−1} ⊗ S^{1} which is I(l − 1, 1) by Theorem 4.1(iii). Using part (ii) of this theorem we arrive at [I(l − 1, 1] = L(l) + 2L(l − 2) which proves (⋆) for all a_{0}, hence for all a _{△}

7.2. Our next goal is to give a description of K_{0}(SL). The latter ring is by definition the free Z- span of I(r), r ≥ 0.

We begin, though, with a subring R of the Green ring for SL which is, additively, the free Z- span of L(s), 0 ≤ s ≤ l − 1, and I(r), r ≥ 0. We pause to make

Definition. We define the ‘l- reflections’ ρ : Z → Z by ρ(r) = ρ(r_{0} + r_{1}l) = 2x_{r} − r, where x_{r} = (r_{1} + 1)l − 1 if r_{0} ≠ l − 1, and x_{r} = r otherwise. Notice that ρ is an invertible mapping. It’s inverse ρ^{−1} acts on r by ρ^{−1}(r) = 2y_{r} − r with y_{r} = r_{1}l − 1 if r_{0} ≠ l − 1, and ρ^{−1}(r) = r otherwise. We claim that R is a commutative ring under the tensor product multiplication. The commutativity of R follows from the almost commutativity of SL ([Mo, loc cit]). The second assertion, viz. R is closed to multiplication, is a consequence of the fact that the product of an injective with an arbitrary comodule remains injective and the following precise statement

Lemma. (Krull - Schmidt Decomposition of L(a) ⊗ L(b))

Suppose 0 ≤ a, b ≤ l − 1, and a ≥ b. Then

1. If a + b ≤ l − 1

L(a) ⊗ L(b) = ⨁_{0≤i≤b} L(a − b + 2i)

2. If a + b > l − 1, letting s = a + b − l + 1 we have

L(a) ⊗ L(b) = ⨁_{0≤i≤l−2−a} L(a − b + 2i) ⨁( ⨁_{0≤2t≤s} I(ρ^{−1}(a + b − 2t))

Proof. L(a)⊗L(b) is the restriction of S^{a} ⊗S^{b} to SL. We shall take advantage of the fact that S^{a} ⊗S^{b} is M- injective. Our computation of the socle in §4 leads up to S^{a} ⊗S^{b} = ⊕I(λ) where λ runs over the set of weights {(a + i, b − i) | i ⪯ a − b + 2i and i ≤ b}. An easy calculation (cf. proof of Theorem 4.1) shows that the i above are exactly the ones satisfying a − b + 2i ≤ l − 1. Therefore all those λ lie in Γ_{0}. If λ ∈ Γ̌_{0}, then the corresponding I(λ) restricts to the SL- injective of the form I(ρ^{−1}(a+b−2t)) for some t by Theorems 5.1(1) and 6.2.1(2). Otherwise λ ∈ Δ_{0}, and then I(λ) = M(λ), and the latter restricts to L(a − b + 2i). This yields the assertion regarding R. A more thorough analysis of the λ’s leads up to the formulas of the lemma _{△}

Remark. An argument similar to the one given above produces the Krull - Schmidt decomposition of S^{a} ⊗ S^{b} for arbitrary a and b.

Proposition. Let x = [L(1)] be the class of L(1) in R. Then R ≃ Z[x].

Proof. We begin by relabiling our spanning set. We shall use the highest weights of the generators as their subscripts. This is sensible as distinct generators have distinct highest weights. In fact by Theorem 6.2.1 a generator is of the form L(r) or I(s) = I(l − 1, l − 1 − s_{0})_{S} ⊗ L(s_{1}l). In the first case the highest weight is r, while in the second it equals to 2(l − 1) − s_{0} + s_{1}l on account of Theorem 4.1(ii) and an easy observation that the highest weight of a product equals sum of the highest weights of factors. We write g_{r} for the basic element of the highest weight r. Thus g_{a} = [L(a)], if 0 ≤ a ≤ l − 1, or a = sl − 1, and g_{r} = I(ρ^{−1}(r)) for r ≥ l, r_{0} ≠ l − 1. We aim at showing that {x^{n} | n ≥ 0} is a basis of R. To this end we prove that (⋆) x^{r} = g_{r}+ lesser terms. By induction on r, it suffices to show that (♯) g_{r} · x = g_{r+1}+ lesser terms. Let s = ρ^{−1}(r) = s_{0} + s_{1}l. Then g_{r} = I(s_{0}) ⊗ L(s_{1}l), and h ence we have g_{r} · x = I(s_{0}) ⊗ L(1) ⊗ L(s_{1}l). But I(s_{0}) = I(l − 1, l − 1 − s_{0})_{S}, and a reference to Theorem 4.2 shows that I(l − 1, k) ⊗ L(1) is a sum of indecomposables with exactly one summand having the highest weight equal to the sum of the highest weights of I(l − 1, k) and L(1). Restricting to SL we obtain (♯) and hence (⋆) _{△}

We now can give a characterization of K_{0}(SL)

Theorem. K_{0}(SL) is isomorphic to the principal ideal of R generated by L(l − 1).

Proof. In the notation of the Proposition K_{0} is the span of all g_{r}, r ≥ l − 1. We know that K_{0} is an ideal of R, hence L(l − 1) · R ⊆ K_{0}. The opposite inclusion follows from the equality L(l − 1) · x^{n} = g_{l−1+n}+ lesser terms. The latter is a an immediate consequence of the observation that by the Proposition L(l − 1) is a monic polynomial of x of degree l − 1 and property (⋆) established there _{△}

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