From representations of quivers via Hall and Loewy algebras to quantum groups

Claus Michael Ringel

Let Δ be a symmetric generalized Cartan matrix, and g = g(Δ) the corresponding Kac–Moody Lie–algebra with triangular decomposition g = n_{−} ⊕ h ⊕ n_{+} (see [K]. We denote by b_{+} = b_{+}(Δ) = h ⊕ n_{+} the Borel subalgebra. Let U_{q}(b_{+}) be the quantization of the universal enveloping algebra of b_{+}, it is defined by generators and relations as we will recall below. For Δ of finite or affine type, we want to survey a construction of U_{q}(b_{+}) using the representation theory of quivers, following [R2], [R3], [R4] and [R5].

1. Representations of quivers

A (finite) quiver Q = (Q_{0}, Q_{1}, s, e) is given by two finite sets Q_{0}, Q_{1} and two maps s, e : Q_{1} ⟶ Q_{0}, the elements of Q_{0} will be called vertices or points, those of Q_{1} arrows; if α is an arrow, then s(α) is called its start vertex, e(α) its end vertex, and α is said to go from s(α) to e(α), written α : s(α) ⟶ e(α). (Thus, a quiver is nothing else than a directed graph with possibly multiple arrows and loops, or a diagram scheme in the sense of Grothendieck; so the concept is old, the denomination “quiver” being due to Gabriel.) Given a quiver Q = (Q_{0}, Q_{1}, s, e), there is the opposite quiver Q^{∗} = (Q_{0}, Q_{1}, e, s) with the same set of vertices but with the reversed orientation for all the arrows. Recall that an n×n–matrix (a_{ij})_{ij} with a_{ii} = 2 and a_{ij} = a_{ji} ≤ 0 for all i ≠ j is called a symmetric generalized Cartan matrix [K]. Given a symmetric generalized Cartan n × n–matrix Δ = (a_{ij})_{ij}, we associate the following quiver Q(Δ); its set of vertices is Q(Δ)_{0} = {1, 2, …, n}, and for 1 ≤ i < j ≤ n, we draw −a_{ij} arrows from i to j. The quivers of the form Q(Δ) have the following property: their vertices may be labelled by 1, …, n such that for any arrow α, we have s(α) < e(α) : conversely, any quiver with this property is of the form Q(Δ). The reader should be aware that the construction of Q(Δ) takes into account the order of the rows and columns of Δ.

Given a quiver Q, a path in Q of length ℓ ≥ 1 is of the form (x|α_{1}, …, α_{ℓ}|y), where α_{i} are arrows with x = s(α_{1}), e(α_{i}) = s(α_{i+1}) for 1 ≤ i < ℓ, and e(α_{ℓ}) = y; a path in Q of length 0 is of the from (x|x) with x ∈ Q_{0}. A path of the form (x|α_{1}, …, α_{ℓ}|x) with ℓ ≥ 1 is called a cyclic path. Note that the quivers of the form Q(Δ) with Δ a symmetric generalized Cartan matrix are precisely the quivers without a cyclic path.

Let k be a field. The path algebra kQ of Q over k is the free vectorspace over k with basis the set of paths in Q, with distributive multiplication given on the basis by

(x|α_{1}, …, α_{ℓ}|y) · (x′|α′_{1}, …, α′_{ℓ′}|y′) = \{ \begin{matrix}(x|α_{1}, … α_{ℓ}, α′_{1}, …, α′_{ℓ′}|y′) & if y = x′ \\ 0 & if y ≠ x′ . \end{matrix}

The elements (x|x) with x ∈ Q_{0} are primitive and orthogonal idempotents, and 1 = ∑_{x∈Q_{0}}(x|x) is the unit element of kQ. Note that kQ is finite–dimensional if and

only if Q has no cyclic path.

Recall that a ring of global dimension ≤ 1 is said to be hereditary, a finitedimensional k–algebra A with radical N is said to be split basic provided A/N is a product of copies of k. It is easy to see that the algebras kQ(Δ) with Δ a symmetric generalized Cartan matrix are precisely the finite–dimensional k–algebras which are hereditary and split basic. Again, we stress that the algebra kQ(Δ) depends on the given ordering of the rows and columns of Δ; different orderings of the rows and columns usually will lead to algebras which are not isomorphic. For example,

the Cartan matrix Δ = [ \begin{matrix}2 & −1 & 0 \\ −1 & 2 & −1 \\ 0 & −1 & 2 \end{matrix} ] yields the quiver ∘^{1} ⟶ ∘^{2} ⟶ ∘^{3}, and the

path algebra of Q(Δ) is given by kQ(Δ) = [ \begin{matrix}k & k & k \\ 0 & k & k \\ 0 & 0 & k \end{matrix} ] , the ring of upper triangular

matrices, whereas for Δ′ = [ \begin{matrix}2 & 0 & −1 \\ 0 & 2 & −1 \\ −1 & −1 & 2 \end{matrix} ] , the corresponding quiver is ∘^{1} ⟶ ∘^{3} ⟵ ∘^{2},

and its path algebra is the algebra kQ(Δ′) = [ \begin{matrix}k & k & k \\ 0 & k & 0 \\ 0 & 0 & k \end{matrix} ]; note that dim_{k} kQ(Δ) = 6,

whereas dim_{k} kQ(Δ′) = 5.

A ring A is called an artin algebra provided its center C is artinian, and A is a finitely generated C–module. Given an artin algebra A, the A–modules we are interested in usually will be right A–modules of finite length, and we denote the category of these modules by mod A. The composition of two A–module maps f : M ⟶ M′, g : M′ ⟶ M′′ will be denoted by gf : M ⟶ M′′ (applying such maps on the opposite side of the scalars).

Given an artin algebra A, we denote by K(A) the Grothendieck group of all (finite–dimensional) A–modules modulo exact sequences. Similarly, let K(mod A) be the Grothendieck group of all (finite–dimensional) A–modules modulo split exact sequences. The theorem of Jordan–Hölder asserts that K(A) is the free abelian group on the set of isomorphism classes of simple A–modules. Given an A–module M, we denote by dim M its equivalence class in K(A). The theorem of Krull–Schmidt asserts that K(mod A) is the free abelian group on the set of isomorphism classes of indecomposable A–modules. If M is an indecomposable A– module, we denote by u_{[M]} = u_{M} the corresponding element in K(mod A); always,

[M] will denote the isomorphism class of a module M. Also note that K(mod A) has a canonical K(A)–grading: given x ∈ K(A), let K(mod A)_{x} be the subgroup generated by all u_{[M]} where M is indecomposable with dim M = x. It follows that for Δ a symmetric generalized Cartan n × n–matrix, and A = kQ(Δ), we have K(A) ≅ ℤ^{n}, whereas K(mod A) does not have to be finitely generated. An algebra A is said to be representation–finite provided there are only finitely many isomorphism classes of indecomposable A–modules.

Let Q be a quiver and k a field. A representation V = (V_{x}, V_{α}) of Q over k is given by a family of (finite–dimensional) vectorspaces V_{x} (x ∈ Q_{0}), and a family of linear maps V_{α} : V_{s(α)} ⟶ V_{e(α)} (α ∈ Q_{1}). Given two representations V, V′, a map f = (f_{x}) : V ⟶ V′ is given by linear maps f_{x} : V_{x} ⟶ V′_{x} such that for any α ∈ Q_{1}, we have V′_{α}f_{s(α)} = f_{e(α)}V_{α}. Note that the category mod kQ is equivalent to the category of representations of the opposite quiver Q^{∗} of Q, and we may (and will) identify these categories. Given a representation V, its dimension vector dim V has, by definition, coordinates (dim V)_{x} = dim_{k} V_{x}, for x ∈ Q_{0}; it belongs to ℤ^{Q_{0}}, and ∑_{x∈Q_{0}}(dim V)_{x} is called the dimension of V. For any vertex x ∈ Q_{0}, we

will consider the one–dimensional representation S(x) of Q^{∗} defined by S(x)_{x} = k, S(x)_{y} = 0 for y ≠ x ∈ Q_{0} and S(x)_{α} = 0 for α ∈ Q_{1}. Assume now that Q has no cyclic path, say let Q = Q(Δ) with Δ = (a_{ij})_{ij} a symmetric generalized Cartan n × n–matrix. Then S(1), …, S(n) are the simple kQ(Δ)–modules, and we have

Ext^{1}(S(i), S(j)) = 0 for i ≥ j,

and

dim_{k} Ext^{1}(S(i), S(j)) = −a_{ij} for i < j.
(For,  let  I(i)  be  the  subspace  of  kQ(Δ)

generated by all paths in Q(Δ) different from (i|i), and u(i) the inclusion map of I(i) into kQ(Δ). Clearly, I(i) is a twosided ideal, and kQ(Δ)/I(i) and S(i) are isomorphic right kQ(Δ)–modules. It follows that Ext^{1}(S(i), S(j)) is isomorphic as a k–space to the cokernel of Hom_{kQ(Δ)}(u(i), S(j)), and therefore to the k– dual of the subspace of kQ(Δ) generated by the paths of the form (i|α|j).) In particular, we see how to recover Δ from kQ(Δ). Also, we note the following: Given a representation V of Q, and x ∈ Q_{0}, the k–dimension of V_{x} is just the Jordan–Hölder multiplicity of S(x) in V, thus the two definitions of dim V given above coincide; in particular, M ↦ dim M yields the canonical identification of K(kQ(Δ)) with ℤ^{n}.

Let us write down at least a few representations explicitly. First, consider the quiver Q

∘^{5}

↓

∘_{1} → ∘_{2} → ∘_{6} ← ∘_{4} ← ∘_{3}

of type E_{6}, and its representation V

U
↓
k0^{2} → k^{2}0 → k^{3} ← 0k^{2} ← 0^{2}k

with U = V_{5} the 2–dimensional subspace of k^{3} generated by (110) and (011). It is an indecomposable representation and its dimension vector is

\begin{aligned}& 2 \\ 1 2 & 3 & 2 1, \end{aligned}

where we arrange its coordinates according to the shape of the quiver. The reader should observe that dealing with this particular quiver Q, we obtain a lot of representations by considering (as in the example above) a vectorspace V_{6} endowed with five subspaces V_{1}, …, V_{5} such that V_{1} ≤ V_{2}, and V_{3} ≤ V_{4}. Actually, one may observe without difficulties that all but seven isomorphism classes of indecomposable representations of Q are obtained in this way. Second, for the quiver

∘
↓
∘
↓
∘ → ∘ → ∘ ← ∘ ← ∘

of type Ẽ_{6}, we obtain a 1–parameter family of indecomposable representations

\begin{aligned}& U_{λ} \\ & ↓ \\ & U \\ & ↓ \\ k0^{2} → k^{2}0 → & k^{3} & ← ok^{2} ← 0^{2}k \end{aligned} ,

where U is as above, and U_{λ} is the 1–dimensional subspace of k^{3} generated by (1, 1 + λ, λ), with λ a fixed element of k.

We recall that in 1972, Gabriel has shown that the path algebra of a quiver Q is representation–finite if and only if Q is the disjoint union of quivers of the form Q(Δ), with Δ of finite type (thus of type A_{n}, D_{n}, E_{6}, E_{7}, or E_{8}), and that for Q = Q(Δ), with Δ of finite type, dim furnishes a bijection between the set B_{0} of isomorphism classes of indecomposable representations of Q and the set Φ^{+} of positive roots for Δ. We can reformulate this result as follows: K(mod kQ(Δ))⊗ℂ is the free ℂ–space with basis (u_{[M]})_{[M]∈B_{0}}, whereas for the semisimple complex Lie algebra g = g(Δ) with triangular decomposition g = n_{−} ⊕h⊕n_{+}, the ℂ–space n_{+} has a canonical decomposition n_{+} = ⊕_{α∈Φ^{+}}g_{α} into weight spaces g_{α}, and all g_{α}

are 1–dimensional. Thus,

K(mod kQ(Δ)) ⊗ ℂ ≅ n_{+}

as ℂ–spaces, and in fact as ℤ^{n}–graded ℂ–spaces: here, K(mod kQ(Δ)) is graded by ℤ^{n} = K(kQ(Δ)), whereas n_{+} is graded by the root lattice.

We may ask whether the representation theory of quivers may be used to endow K(mod kQ(Δ))⊗ℂ, or even K(mod kQ(Δ)) itself with a Lie structure so that this isomorphism becomes an isomorphism of Lie algebras.

2. Hall algebras

Let R be a finite ring. We denote by mod R the category of finitely generated R– modules. Note that an R–module is finitely generated if and only if it is of finite length if and only if it has finitely many elements, so we call these R–modules just the finite R–modules. Given finite R–modules M, N_{1}, …, N_{t} let F_{N_{1}...N_{t}}^{M} be the number of filtrations M = M_{0} ⊇ M_{1} ⊇ … ⊇ M_{t} = 0 of M such that M_{i−1}/M_{i} ≅ N_{i}, for 1 ≤ i ≤ t. The Hall algebra H(R) is, by definition, the free abelian group with basis (u_{[M]})_{[M]∈B} indexed by the set of B of isomorphism classes [M] of finite R–modules M, with multiplication given by

u_{[N_{1}]}u_{[N_{2}]} = ∑_{[M]} F_{N_{1}N_{2}}^{M}u_{[M]};

note that we have F_{N_{1}N_{2}}^{M} ≠ 0 only in case dim M = dim N_{1} + dim N_{2}, and that the number of isomorphism classes of modules M with given dim M is finite, thus the sum considered above always is a finite sum. We note that the multiplication is associative, in fact we have

u_{[N_{1}]}u_{[N_{2}]}u_{[N_{3}]} = ∑_{[M]} F_{N_{1}N_{2}N_{3}}^{M}u_{[M]},

and u_{[0]} is the unit element for this multiplication, thus H(R) is a ring.

More generally, we may start with a ring R having the property that given two simple R–modules S_{1}, S_{2} with only finitely many elements, then also Ext_{R}^{1}(S_{1}, S_{2}) has only finitely many elements, and take for B the set of isomorphism classes of R–modules with finitely many elements. If R = ℤ_{(p)} then B is just the set of isomorphism classes of finite abelian p–groups. It is this case which has been considered in detail in the literature, first in 1900, by Steinitz [S], then in 1959 by Ph. Hall [H]: he called H(ℤ_{(p)}) the “algebra of partitions”, since the isomorphism classes of finite abelian p–groups correspond bijectively to the partitions. The main reference for this particular Hall algebra is a book by Macdonald [M]. (The Hall algebra H(ℤ_{(p)}) can also be interpreted as a Hecke algebra: Let G_{n}^{+} be the set of n × n–matrices over ℤ_{(p)} which are invertible over ℚ, and K_{n} the subset of those elements which are invertible over ℤ_{(p)}. The canonical embedding of G_{n}^{+} into G_{n+1}^{+} yields a semigroup G^{+} = ⋃_{n} G_{n}^{+}, with a subgroup K = ⋃_{n} K_{n}, and H(ℤ_{(p)}) may be identified with the Hecke algebra for the inclusion of K into G^{+} (note that the double cosets of K in G^{+} correspond bijectively again to the partitions), see [M].)

If R_{1}, R_{2} are finite rings, then H(R_{1} × R_{2}) = H(R_{1}) ⊗ H(R_{2}), thus for studying Hall algebras H(R), we may assume that R is connected, or, equivalently, that the center C = C(R) of R is a local ring, and we denote by q = q(R) the cardinality of C/ rad C, and by l_{C}(M) the length of the C–module M. Since C/ rad C is the only simple C–module, we have |M| = q^{l_{C}(M)} for any (finite) C–module M. Note that if X is an R–module, then X, End(X), and rad End(X) all are C–modules.

The Hall algebra H(R) usually is non–commutative, in contrast to the classical case of R = ℤ_{(p)}. For example, let R = [ \begin{matrix}k & k \\ 0 & k \end{matrix} ] , the ring of upper triangular 2 × 2– matrices over the finite field k. Then there are two simple R–modules S_{1}, S_{2}, one of them, say S_{1}, is injective, the other one, S_{2}, is projective. It follows that in H(R), we have

u_{S_{2}}u_{S_{1}} = u_{S_{1}⊕S_{2}},
u_{S_{1}}u_{S_{2}} = u_{S_{1}⊕S_{2}} + u_{P} ,

where P is the unique indecomposable R–module of length 2. In particular,

u_{P} = u_{S_{1}}u_{S_{2}} − u_{S_{2}}u_{S_{1}} = [u_{S_{1}}, u_{S_{2}}],

so the basis element u_{P} corresponding to the indecomposable R–module P is expressed as a commutator of other basis elements (corresponding to other indecomposable modules). This is a rather general feature and it demonstrates the usefulness of H(R) : we obtain large indecomposable modules (or better, their counterparts in H(R)) by forming iterated commutators or related algebraic operations, starting with small indecomposable modules, say simple, or at least serial ones. Note that by, definition, K(mod R) is a subgroup of H(R), namely the subgroup generated by the elements u_{[M]}, with M indecomposable. We claim that up to multiples of q − 1, the subgroup K(mod R) is closed under commutators:

Proposition 1. Let R be a finite connected ring, and q = q(R). Let N_{1}, N_{2} be indecomposable R–modules, let M be a decomposable R–module. Then q − 1 divides F_{N_{1}N_{2}}^{M} − F_{N_{2}N_{1}}^{M}.

For example, consider the “3–subspace quiver”

∘^{2}

↓

∘_{1} → ∘_{0} ← ∘_{3}

We denote by P (i) the indecomposable projective, by Q(i) the indecomposable injective module corresponding to the vertex i. For 1 ≤ i ≤ 3, let M(i) be the maximal submodule of Q(0) with Q(0)/M(i) ≅ Q(i), and let M be the unique indecomposable module with dimension vector _{1 2 1}^{1} . Then

u_{P (0)}u_{Q(0)} = u_{P (0)⊕Q(0)},

u_{Q(0)}u_{P (0)} = qu_{P (0)⊕Q(0)} + (q − 1) ∑_{i=1}^{3} u_{P (i)⊕M(i)} + (q − 2)u_{M},

thus

[u_{P (0)}, u_{Q(0)}] ≡ u_{M} mod(q − 1).

Proof of Proposition 1. We can assume that N_{1} ≉ N_{2}. Let M = M′ ⊕ M′′ with M′ indecomposable and M′′ ≠ 0. Let U be the set of submodules U of M with U ≠ M′, U ≠ M′′, U ≅ N_{2} and M/U ≅ N_{1}. We claim that q − 1 divides |U|. Note that for U ∈ U, we have that U ∩M′ is a proper submodule of M′. Otherwise, M′ ⊆ U, thus U = M′ ⊕ (U ∩ M′′); but U is indecomposable, therefore U = M′, contrary to our assumption. Let G = Aut M′, the automorphism group of M′.

Then |G| = | End M′| − | rad End M′| = q^{r}(q^{e} − 1), where r = ℓ_{C}(rad End M′) and e = ℓ_{C}(End M′/ rad End M′). We identify G with a subgroup of Aut M via g ↦ g⊕1_{M′′}, thus G also acts on U via g∗U = {(gu′, u′′)|(u′, u′′) ∈ U} where g ∈ G and U ∈ U. To compute the stabilizer G_{U} of U in G, note first that we may consider Hom(M′, U ∩ M′) as a subset of rad End M′. Now, if f ∈ Hom(M′, U ∩ M′), then (1 + f) ∗ U ⊆ U, since for (u′, u′′) ∈ U, also ((1 + f)u′, u′′) = (u′, u′′) + (fu′, 0) ∈ U, thus (1 + f) ∗ U = U. Conversely, if g ∗ U = U, then for all (u′, u′′) ∈ U also (gu′, u′′) ∈ U, thus (gu′ − u′, 0) ∈ U, consequently g − 1 ∈ Hom(M′, U ∩ M′), thus g ∈ 1 + Hom(M′, U ∩ M′). It follows that |G_{U}| = | Hom(M′, U ∩ M′)| = q^{r′} for some r′ ≤ r, and therefore |G/G_{U}| is divisible by q^{e} − 1. Since |G/G_{U}| is the orbit length of U, and U was arbitrary, we see that q^{e} − 1 divides |U|, so q − 1 divides |U|.

In order to finish the proof, we have to distinguish to cases. If let M ≉ N_{1} ⊕N_{2}, then any submodule U of M with U ≅ N_{2}, M/U ≅ N_{1} satisfies in addition U ≠ M′, U ≠ M′′, thus F_{N_{1}N_{2}}^{M} = |U|, and therefore q − 1|F_{N_{1}N_{2}}^{M}. The same argument shows q − 1|F_{N_{2}N_{1}}^{M}.

Otherwise, we can assume M′ = N_{1} and M′′ = N_{2}. Then, besides the elements of U, there is just one additional submodule U of M with U ≅ N_{2}, M/U ≅ N_{1}, namely U = M′′. Therefore F_{N_{1}N_{2}}^{M} ≡ 1 mod(q − 1), and similarly, F_{N_{2}N_{1}}^{M} ≡ 1 mod(q − 1).

Given a specific R–module M, it usually is not easy to decide whether M is indecomposable or not. Proposition 1 yields an effective procedure for certain cases. For example, consider the E_{6}–quiver and its representation V with dim V = _{1 2 3 2 1}^{2} , exhibited in section 1. There is a unique submodule V′ with dimension vector _{1 1 2 1 1}^{1} , namely

\begin{aligned}& \{(x0x)\} \\ & ↓ \\ k0^{2} → k0^{2} → & k0k & ← 0^{2}k ← 0^{2}k \end{aligned} ,

V′ is indecomposable, and V/V′ is isomorphic to the indecomposable representation

k
↓
0 → k → k ← k ← 0.

Thus F_{V/V′,V′}^{V} = 1. On the other hand, we clearly have Hom(V/V′, V) = 0, therefore F_{V′,V/V′}^{V} = 0. So Proposition 1 asserts that V is indecomposable. (Of course, we can apply Proposition 1 only in case k is a finite field, and there are other methods which may be used for the example above in case k is arbitrary: We have Hom(V′, V/V′) = 0, therefore V has to be indecomposable!)

We will see in the next section that for special rings R we can calculate F_{N_{1}N_{2}}^{M} modulo q − 1 for all indecomposable R–modules N_{1}, N_{2}, M.

In order to deal with presentations of Hall algebras, we mention the following fundamental relations, where \left[{n \atop t}\right]_{T} = \frac{φ_{n}(T)}{φ_{t}(T)φ_{n−t}(T)} is the Gauss polynomial, with φ_{n}(T) = (1 − T) ⋯ (1 − T^{n}).

Proposition 2. Let R be a finite ring, let S_{i}, S_{j} be simple R–modules with Ext^{1}(S_{i}, S_{j}) = 0. Let q_{i} = | End(S_{i})|, q_{j} = | End(S_{j})|, and let

a_{ij} = − dim _{End(S_{i})}Ext^{1}(S_{j}, S_{i}), a_{ji} = − dim Ext^{1}(S_{j}, S_{i})_{End(S_{j})}.

Let u_{i} = u_{[S_{i}]}, u_{j} = u_{[S_{j}]}. If Ext^{1}(S_{i}, S_{i}) = 0, then

∑_{t=0}^{n}(−1)^{t}\left[{n \atop t}\right]_{q_{i}}q\binom{t}{2}_{i}u_{i}^{t}u_{j}u_{i}^{n−t} = 0 with n = 1 − a_{ij},

if Ext^{1}(S_{j}, S_{j}) = 0, then

∑_{t=0}^{n}(−1)^{t}\left[{n \atop t}\right]_{q_{j}}q\binom{t}{2}_{j}u_{j}^{n−1}u_{i}u_{j}^{t} = 0 with n = 1 − a_{ji}.

These relations look rather similar to those used by Drinfeld and Jimbo in order to define quantizations of Lie algebras, this similarity will be discussed in the next section. But the reader may observe already here that the polynomials

ρ_{n}^{+}(T, X, Y) = ∑_{t=0}^{n}(−1)^{t}\left[{n \atop t}\right]_{T} T^{\binom{t}{2}}X^{t}Y X^{n−t}

+ρ_{n}(T, X, Y) = ∑_{t=0}^{n}(−1)^{t}\left[{n \atop t}\right]_{T} T^{\binom{t}{2}}X^{n−t}Y X^{t}

both yield for T = 1 an iterated commutator:

ρ_{n}^{+}(1, X, Y) = (ad X)^{n}Y,
+ρ_{n}(1, X, Y) = (−1)^{n}(ad X)^{n}Y.

The proof of Proposition 2 (see [R4]) is not difficult:

We assume that Ext^{1}(S_{i}, S_{i}) = 0. By the definition of the multiplication in H(R), the elements u_{i}^{t}u_{j}u_{i}^{n−t} are linear combinations of elemens u_{[M]} where M is an R–module with dim M = dim S_{j} + n dim S_{i}. Since Ext^{1}(S_{i}, S_{i}) = 0, Ext^{1}(S_{i}, S_{j}) = 0, such a module M is the direct sum of an indecomposable module M′ with M′/ rad M′ = S_{j}, and several copies of S_{i}, say M = M′ ⊕ dS_{i}. Since dim_{End(S_{i})} Ext^{1}(S_{j}, S_{i}) < n, it follows that d ≥ 1. The coefficient of u_{[M]} in u_{i}^{t}u_{j}u_{i}^{n−t} just counts the number of composition series of M of the form

M = M_{0} ⊃ M_{1} ⊃ … ⊃ M_{n+1} = 0

with M_{t}/M_{t+1} ≅ S_{j}. If t > d, then there is no such composition series, if t ≤ d, then the number of such composition series is α_{i}(q_{i}), where

α_{t}(T) = \frac{φ_{d}(T)φ_{n−t}(T)}{(1 − T)^{n}φ_{d−t}(T)}

and

∑_{t=0}^{d}(−1)^{t}\left[{n \atop t}\right]_{T} T^{\binom{t}{2}}α_{t} = 0.

The ring H(R) is K(R)–graded: for x ∈ K(R), let H(R)_{x} be the subgroup generated by the elements u_{[M]} with dim M = x; clearly, for x, y ∈ K(R), we have H(R)_{x} · H(R)_{y} ⊆ H(R)_{x+y}. As a consequence, any element d ∈ K(R)^{∗} = Hom(K(R), ℤ) gives rise to a derivation δ_{d} : H(R) ⟶ H(R), defined by δ_{d}(u_{[M]}) = d(dim M) · u_{[M]}. In particular, let S_{1}, …, S_{n} be the simple R–modules, thus dim S_{1}, …, dim S_{n} is a basis of K(R), and we denote by d_{1}, …, d_{n} the dual basis of K(R)^{∗}; thus d_{i}(dim M) = (dim M)_{i} is the Jordan Hölder multiplicity of S_{i} in M. In this way, we obtain derivations δ_{i} = δ_{d_{i}} of H(R), and we may form the skew polynomial ring

H′(R) = H(R)[T_{i}, δ_{i}]_{i},

where T_{1}, …, T_{n} are indeterminates satisfying the commutation rules

[T_{i}, T_{j}] = 0
[T_{i}, u_{[M]}] = δ_{i}(u_{[M]}) = (dim M)_{i}u_{[M]}.

We call H′(R) the extended Hall algebra of R.

For e ∈ ℕ_{0}, let ψ_{e}(T) = \frac{(1−T)···(1−T^{e})}{(1−T)^{e}}.

Proposition 3. Let X_{1}, …, X_{m} be indecomposable modules with End(X_{i}) the field with q_{i} elements. Assume that Ext^{1}(X_{i}, X_{j}) = 0 for i ≤ j and that Hom(X_{i}, X_{j}) = 0 for i > j. Let a ∈ ℕ_{0}^{m}. Then

u_{[X_{1}]}^{a(1)} ⋯ u_{[X_{m}]}^{a(m)} = (∏_{i=1}^{m} ψ_{a(i)}(q_{i})) · u_{[⊕a(i)X_{i}]},

and

u_{[a(1)X_{1}]} ⋯ u_{[a(m)X_{m}]} = u_{[⊕a(i)X_{i}]}.

Proof: First, we show the second formula. Given M with a filtration

M = M_{0} ⊇ M_{1} ⊇ … ⊇ M_{m} = 0

such that M_{i−1}/M_{i} ≅ a(i)X_{i}, then all M_{i} are direct summands and M ≅ ⊕^{m}_{i=1}a(i)X_{i},

since Ext^{1}(X_{i}, X_{j}) = 0 for i ≤ j. Thus, let M = ⊕^{m}_{i=1}a(i)X_{i}. Using the condition

Hom(X_{i}, X_{j}) = 0 for i > j, we conclude that M_{i} = ⊕_{j>i}a(j)X_{j} is uniquely

determined, and therefore M has precisely one filtration with the prescribed factors a(i)X_{i}.

It remains to show that

u_{[X_{i}]}^{a(i)} = ψ_{a(i)}(q_{i}) · u_{[a(i)X_{i}]}.

However, since End(X_{i}) is a field, the filtrations of a(i)X_{i} with factors X_{i} correspond bijectively to the complete flags in the a(i)–dimensional End(X_{i})–space, but the number of such flags is ψ_{a(i)}(q_{i}).

We call R representation–directed provided R has only finitely many indecomposable modules and they can be ordered as X_{1}, …, X_{m} such that Hom(X_{i}, X_{j}) = 0 for i > j. Clearly, such an ordering satisfies in addition Ext^{1}(X_{i}, X_{j}) = 0 for i ≤ j.

3. Generic Hall algebras

We return to the path algebra of a quiver, and consider the representation–finite case. Thus, let Δ be a symmetric Cartan matrix (therefore of type A_{n}, D_{n}, E_{6}, E_{7}, or E_{8}), and let R = kQ(Δ). Recall that Φ^{+} denotes the set of positive roots for Δ. Given α ∈ Φ^{+}, let M(α) = M(α, R) be the indecomposable R–module with dim M(α) = α. Given a function a : Φ^{+} ⟶ ℕ_{0}, let M(a) = M(a, R) = ⊕_{α∈Φ^{+}}a(α)M(α). In this way, we obtain a bijection between the set of functions

Φ^{+} ⟶ ℕ_{0} and B.

Proposition 4. Given a, b, c : Φ^{+} ⟶ ℕ_{0}, there exists a polynomial φ_{ac}^{b} ∈ ℤ[T] with the following property: if k is a finite field, q = |k|, then

F_{M(a,R),M(c,R)}^{M(b,R)} = φ_{ac}^{b}(q).

For the proof, we refer to [R2]. The polynomials φ_{ac}^{b} occurring in this way will be called Hall polynomials. This proposition allows us to introduce generic Hall algebras. Before we do this, let us write down several of these ploynomials explicitly. If α ∈ Φ^{+}, we will denote the corresponding characteristic function Φ^{+} ⟶ ℕ_{0} also by α. If α, γ ∈ Φ^{+}, we have to distinguish the two possible additions: the addition inside the root lattice will be denoted by +, the addition of functions Φ^{+} ⟶ ℕ_{0} may be denoted by ⊕, but we will not need the notation.

The symmetric generalized Cartan matrix Δ (with its fixed ordering of rows and columns) determines a usually non–symmetric bilinear form ⟨−, −⟩ on the root lattice ℤ^{n} as follows: Let \overline{Δ} be the (uniquely determined) lower triangular matrix with Δ = \overline{Δ} + \overline{Δ}^{t}, thus \overline{Δ} = (\overline{a}_{ij})_{ij}, with \overline{a}_{ii} = 1, \overline{a}_{ij} = a_{ij} for i > j, and \overline{a}_{ij} = 0 for i < j, and let ⟨α, β⟩ = α\overline{Δ}β^{t}. The importance of this bilinear form for the representation theory of Q(Δ)^{∗} is well–known: given representations V, V′ of Q(Δ)^{∗}, we have

⟨dim V, dim V′⟩ = dim_{k} Hom(V, V′) − dim_{k} Ext^{1}(V, V′).

Proposition 5. Let α, γ ∈ Φ^{+} with ⟨α, γ⟩ < 0. Let β = α + γ. Then β ∈ Φ^{+}, and φ_{αγ}^{β} is as follows:

\begin{matrix}⟨γ, α⟩ & φ_{αγ}^{β} \\ 0 & 1 \\ 1 & T − 2 \\ 2 & (T − 2)^{2} \\ 3 & (T − 2)^{3} or T^{3} − 5T^{2} + 10T − 7 \\ 4 & (T − 2)(T^{3} − 4T^{2} + 8T − 6) \\ 5 & T^{5} − 6T^{4} + 15T^{3} − 23T^{2} + 25T − 13 \end{matrix}

The proof is rather tedious, see [R3].

Corollary. Let α, γ ∈ Φ^{+} with ⟨α, γ⟩ < 0. Let β = α + γ. Then φ_{αγ}^{β}(1) = (−1)^{⟨γ,α⟩}, whereas φ_{γα}^{β} = 0.

The first assertion is a direct consequence of Proposition 5. The second assertion is shown as follows: From ⟨α, γ⟩ < 0 we conclude that Ext^{1}(M(α, R), M(γ, R)) ≠ 0, and therefore Ext^{1}(M(γ, R), M(α, R)) = 0, for R = kQ(Δ). But this implies that F_{M(γ,R)M(α,R)}^{M(β,R)} = 0, and therefore φ_{γα}^{β} = 0, by Proposition 3.

Let us change the notation as follows: we will denote by q instead of T the indeterminate we are dealing with, thus ℤ[q] is the polynomial ring over ℤ in one variable, and given a, b, c : Φ^{+} ⟶ ℕ_{0}, then φ_{ac}^{b} ∈ ℤ[q]. Let Λ be the completion of the polynomial ring ℂ[q] at the maximal ideal generated by q − 1. In Λ we may consider h = ln q, thus Λ is a complete discrete valuation ring and h (or also q − 1) generates its radical.

As before, Δ is a symmetric Cartan n × n–matrix. The generic Hall algebra H(Δ) is the free ℤ[q]–module with basis (u_{a})_{a} indexed by the set of functions a : Φ^{+} ⟶ ℕ_{0}, with multiplication

u_{a}u_{c} = ∑_{b} φ_{ac}^{b}u_{b}.

Again, H(Δ) is a ring, its identity element is the zero function. Also. H(Δ) is generated by the root lattice ℤ^{n}, thus, as before, we can form the extended generic Hall algebra H′(Δ) = H(Δ)[T_{i}, δ_{i}]_{i}, where T_{1}, …, T_{n} are indeterminates satisfying the commutation rules

[T_{i}, T_{j}] = 0,
[T_{i}, u_{a}] = δ_{i}(u_{a}) = (dim a)_{i}u_{a},

where (dim a)_{i} = ∑_{α∈Φ^{+}}a(α)α_{i} with α = (α_{1}, …, α_{n}) ∈ Φ^{+} ⊆ ℤ^{n}. The ring we are

interested in is

Ĥ′(Δ) := lim_{←}_{m}H′(Δ) ⊗_{ℤ[q]} ℂ[q]/(q − 1)^{m},

it is a complete Λ–algebra containing H′(Δ) as a subring. Instead of dealing with the elements T_{i}, u_{i}, 1 ≤ i ≤ n, let us consider the elements

H_{i} := ∑_{j=1}^{n} a_{ij}T_{j}

X_{i} := exp(−\frac{1}{2} ∑_{j=1}^{i−1} a_{ij}T_{j} ln q) · u_{i}.

Theorem 1. Let Δ be a symmetric Cartan matrix. The algebra Ĥ′(Δ) is, as a complete Λ–algebra, generated by H_{1}, …, H_{n}, X_{1}, …, X_{n} with the relations

[H_{1}, H_{j}] = 0, [H_{i}, X_{j}] = a_{ij}X_{j}, for all i, j,

∑_{t=0}^{n}(−1)^{t}\left[{n \atop t}\right]_{q}q^{− \frac{t(n−t)}{2}} X_{i}^{t}X_{j}X_{i}^{n−t} = 0 with n = 1 − a_{ij}, and i ≠ j.

This shows that Ĥ′(Δ) ≅ U_{q}(b_{+}(Δ)), the quantization of U(b_{+}(Δ)) in the sense of Drinfeld and Jimbo [D].

It is not difficult to derive the relations mentioned in the theorem from the known relations for T_{i}, u_{i}: since [T_{i}, T_{j}] = 0 for all i, j, also [H_{i}, H_{j}] = 0 for all i, j; since [T_{i}, u_{i}] = u_{i} and [T_{i}, u_{j}] = 0 for i ≠ j, it follows that [H_{i}, u_{j}] = a_{ij}u_{j}, and therefore also [T_{i}, X_{j}] = a_{ij}X_{j}. Finally, one easily verifies that for c ∈ ℂ

exp(cT_{i} ln q)u_{i} = q^{c} · u_{i} exp(cT_{i} ln q),

whereas for i ≠ j, the elements exp(cT_{i} ln q) and u_{j} do commute. Using this information one observes without difficulties (see [R4]) that the last relation just rewrites the relations

ρ_{n}^{+}(q, u_{i}, u_{j}) = 0, ^{+}ρ_{n}(q, u_{j}, u_{i}) = 0

which are valid for i < j. Note that the only difference between^{+}ρ_{n}, ρ_{n}^{+} and the new relation are the factors q^{\binom{n}{t}} and q^{− \frac{t(n−1)}{2}} . Whereas the relations ^{+}ρ_{n}, ρ_{n}^{+} depend on the ordering of rows and columns of Δ, thus on the orientation of Q(Δ), the Drinfeld–Jimbo relation is independent of this ordering. Of course, the definition of X_{i} in terms of u_{i} also takes into account the ordering of Δ.

Starting with the isomorphism Ĥ′(Δ) ≅ U_{q}(b_{+}(Δ)), and factoring out on both sides the ideals generated by q − 1, we obtain the following:

Corollary.

H′(Δ) ⊗_{ℤ[q]} ℂ[q]/(q − 1) ≅ U(b_{+}(Δ)), and
H(Δ) ⊗_{ℤ[q]} ℂ[q]/(q − 1) ≅ U(n_{+}(Δ)).

Note that H(Δ)_{1} = H(Δ) ⊗_{ℤ[q]} ℤ[q]/(q − 1) is the free abelian group with basis

(u_{a})_{a} indexed by the functions a : Φ^{+} ⟶ ℕ_{0}, with multiplication

u_{a}u_{c} = ∑_{b} φ_{ac}^{b}(1)u_{b},

we may call it the degenerate Hall algebra. Proposition 1 implies that the subgroup generated by all u_{α}, α ∈ Φ^{+}, actually is a Lie subalgebra of H(Δ)_{1}. Of course, this subgroup is canonically identified with K(mod kQ(Δ)), with u_{α} corresponding to u_{M(α,kQ(Δ))}. Thus, we have found a Lie structure on K(mod kQ(Δ)) which is derived from the representation theory of the quiver Q(Δ), namely

[u_{M(α)}, u_{M(γ)}] = ∑_{β}(φ_{αγ}^{β}(1) − φ_{γα}^{β}(1))u_{M(β)},

and

K(mod kQ(Δ)) ⊗ ℂ and n_{+}

are isomorphic as Lie algebras. More is true: K(mod kQ(Δ)) is a ℤ–form of K(mod kQ(Δ)) ⊗ ℂ ≅ n_{+}, and in fact it is a Chevalley ℤ–form of n_{+}, as the corollary to Proposition 5 shows. Recall that a finite–dimensional semisimple complex Lie algebra has several Chevalley ℤ–forms; in order to write down the corresponding structure constants, the subtle point are the signs. In [T], Tits gave a complete description of all possible sign choices. The signs which we obtain using the representation theory of quivers were first exhibited by Frenkel and Kac [FK].

Finally, let us mention that H(Δ)_{1} itself may be considered as a corresponding Kostant ℤ–form of U(n_{+}): we can order the positive roots α_{1}, …, α_{m} in such a way that for R = kQ(Δ) we have Hom(M(α_{i}, R), M(α_{i}, R)) only in case i ≤ j. Thus we can apply Proposition 3 in order to conclude that for u_{i} = u_{α_{i}} and a(i) ∈ ℕ_{0} we have

u_{1}^{a(1)} … u^{a(m)} = ∏_{i=1}^{m} ψ_{a(i)} · u_{a}

in H(Δ). Now ψ_{a(i)}(1) = a(i)!, thus, in H(Δ)_{1}, we have

u_{1}^{a(1)} … u_{m}^{a(m)} = ∏_{i=1}^{m} a(i)! · u_{a},

and we can rewrite this as

u_{a} = \frac{u_{1}^{a(1)}}{a(1)!} … \frac{u_{m}^{a(m)}}{a(m)!}.

The right side is a typical generator in the Kostant ℤ–form, on the left we just have the basis element of H(Δ)_{1} corresponding to the module M(a, R); note that all R–modules are of the form M(a, R), so the rather technical looking generators in the Kostant ℤ–form do correspond just to modules.

For a full proof of Theorem 1 we refer to [R3] and [R4]. This proof first establishes the corollary and derives from this the theorem. The proof of the corollary presented in [R3] relies on the actual calculation of the Hall polynomials as stated in Proposition 5. A more direct proof will be given in [R5].

4. Composition algebras and Loewy algebras

In order to deal with symmetric generalized Cartan matrices which are not of finite type we have to change our view point slightly. For general symmetric generalized Cartan matrices, there does not exist a generic Hall algebra: the idea of constructing a generic Hall algebra relies on the fact that the set of finite– dimensional kQ(Δ)–modules, for Δ of finite type, is indexed by a fixed set (namely the set of function Φ^{+} ⟶ ℕ_{0}) independent of k, and that on this set we have various multiplications using different finite fields k. However, if Δ is not of finite type, then already the set of finite–dimensional kQ(Δ)–modules will depend on k. Of course, there are classes of kQ(Δ)–modules which can be indexed without reference to k, for example the simple, or more generally, the semisimple modules.

Let R be a finite ring. The subring C(R) of H(R) generated by the elements u_{[S]} with S simple is called the composition algebra of R, the subring L(R) of H(R) generated by the elements u_{[M]} with M semisimple will be called the Loewy algebra of R, since it encodes the possible Loewy series of modules (a Loewy series of a module is by definition a filtration with semisimple factors).

The rings C(R) and L(R) may be constructed also as follows: Let S_{1}, …, S_{n} be a complete set of simple R–modules, we denote the isomorphism class of S_{i} by s_{i} = [S_{i}], and we define S(s_{i}) = S_{i}. We use the set S = {s_{1}, …, s_{n}} as a set of letters in order to form words. Thus, let W = W(S) be the set of words in S (there is the empty word 1, the remaining words are of the form w = a_{1}a_{2} … a_{t} with t ≥ 1 and all a_{i} ∈ S). Of course, words are multiplied using juxtaposition, thus W is a semigroup. Given w = a_{1}a_{2} … a_{t} ∈ W, and M and R–module, let F_{w}^{M} = F_{S(a_{1})...S(a_{t})}^{M}, and, of course, F_{1}^{M} = 0 for M ≠ 0, and F_{1}^{0} = 1. Note that F_{w}^{M} counts the number of composition series of M of tye w, the type of the composition series M = M_{0} ⊃ M_{1} ⊃ … ⊃ M_{t} = 0 of M being the word [M_{0}/M_{1}] … [M_{t−1}/M_{t}] ∈ W. Note that the free ℤ–algebra A(S) generated by S is just the semigroup algebra ℤ(W(S)). In A(S), we consider the set I(R) of all finite linear combinations ∑_{i}λ_{i}w_{i} with λ_{i} ∈ ℤ, w_{i} ∈ W such that ∑_{i}λ_{i}F_{w_{i}}^{M} = 0 for

all R–modules M. Clearly, I(R) is an ideal, it is just the kernel of the canonical surjection A(S) ⟶ C(R) which sends s_{i} onto u_{[S(s_{i})]}. Thus,

C(R) ≅ A(S)/I(R).

Similarly, given d ∈ ℕ_{0}^{n}, let s_{d} = [⊕d(i)S_{i}], and S(s_{d}) = ⊕d(i)S_{i}. Again, we may consider the set W̃ of words in S̃ = {s_{d}|d ∈ ℕ_{0}^{n}}. Given a word a_{1} … a_{t} ∈ W̃ with a_{i} ∈ S̃, and M an R–module, let F_{a_{1}...a_{t}}^{M} = F_{S(a_{1})...S(a_{t})}^{M}, this counts the number of filtrations M = M_{0} ⊇ M_{1} ⊇ … ⊇ M_{t} = 0 with a_{i} = [M_{i−1}/M_{i}], for 1 ≤ i ≤ t. Thus, here we deal with Loewy series. As before, we denote by A(S̃) the free ℤ–algebra generated by S̃, and denote by Ĩ(R) the set of finite linear combinations ∑ λ_{i}w_{i} with λ_{i} ∈ ℤ, w_{i} ∈ W̃ such that ∑ λ_{i}F_{w_{i}}^{M} = 0 for all R– modules M. Then the canonical map A(S̃) ⟶ L(R), which sends s_{d} onto u_{[S(s_{d})]} yields an isomorphism

L(R) ≅ A(S̃)/Ĩ(R).

Proposition 6. If S_{1}, …, S_{n} are the simple R–modules and Ext_{R}^{1}(S_{i}, S_{j}) = 0 for i ≥ j, then C(R) ⊗_{ℤ} ℚ = L(R) ⊗_{ℤ} ℚ. If R is representation–directed, then

L(R) = H(R).

Proof: We assume Ext_{R}^{1}(S_{i}, S_{j}) = 0 for i ≥ j, and let u_{i} = u_{[S_{i}]}. For d ∈ ℕ_{0}^{n}, let S(d) = ⊕d(i)S_{i}, and u_{d} = u_{[S(d)]}. We can apply Proposition 4 and conclude that

u_{d} = u_{[S(d)]} = ∏ \frac{1}{ψ_{d(i)}(q_{i})} · u_{1}^{d(1)} ⋯ u_{n}^{d(n)},

where q_{i} = | End(S_{i})|. This yields the first assertion.

Assume now that R is representation–directed, with indecomposable modules X_{1}, …, X_{m} such that Hom(X_{i}, X_{j}) = 0 for i > j. Note that this implies that End(X_{i}) is a division ring, thus a field, and that Ext^{1}(X_{i}, X_{j}) = 0 for i ≤ j. Thus, again we can use Proposition 4 and conclude that for a ∈ ℕ_{,}^{m}

(*)  u_{[⊕a(i)X_{1}]} = u_{[a(1)X_{1}]} ⋯ u_{[a(m)X_{m}]}.

It remains to see that for X indecomposable, b ∈ ℕ_{0}, the element u_{[bX]} belongs to L(R). We use induction on the length of bX. Given d ∈ ℕ_{0}^{n}, we have

u_{[d(n)S_{n}]} ⋯ u_{[d(1)S_{1}]} = ∑_{dim M=d} u_{[M]}.

For, since S_{n} is projective, any module M with dim M = d has a unique submodule M_{1} with M/M_{1} ≅ d(n)S_{n}, thus for any module M with dim M = d, we have F_{d(n)S_{n},... ,d(1)S_{1}}^{M} = 1. Therefore for d = dim bX, we have

u_{[bX]} = u_{[d(n)S_{n}]} ⋯ u_{[d(1)S_{1}]} − ∑_{\begin{matrix}dim M=d \\ [M]≠[bX] \end{matrix}} u_{[M]}.

Consider a module M with dim M = dim bX and [M] ≠ [bX]. It is well– known that such a module M has at least two non–isomorphic indecomposable direct summands, thus M ≅ ⊕_{i=1}^{m}a(i)X_{i}, and all a(i)X_{i} are proper submodules. By

induction, u_{[a(i)X_{i}]} ∈ L(R), and therefore u_{[M]} ∈ L(R), according to (*). It follows that u_{[bX]} ∈ L(R). This completes the proof.

Generic Loewy algebras

Let Δ be a symmetric generalized Cartan matrix. For 1 ≤ i ≤ n, and k a field, we denote by S(i, kQ(Δ)) the simple kQ(Δ)–module corresponding to the vertex i of Q(Δ). For d ∈ ℕ_{0}^{n}, let S(d, kQ(Δ)) = ⊕_{i}d(i)S(i, kQ(Δ)). We consider the

product

∏_{k} H(kQ(Δ))

where k runs through all (isomorphism classes of) finite fields, its elements are of the form x = (x^{(k)})_{k}, where x^{(k)} ∈ H(kQ(Δ)). In particular, we will have to consider the element q = (q^{(k)})_{k}, where q^{(k)} = |k| = |k|u_{[0]}, for 1 ≤ i ≤ n the elements u_{i} = (u_{i}^{(k)})_{k}, and for d ∈ ℕ_{0}^{n} the elements u_{d} = (u_{d}^{(k)})_{k}, where u_{i}^{(k)} = u_{[S(i,kQ(Δ))]}, u_{d}^{(k)} = u_{[S(d,kQ(Δ))]}. Note that q is a central element of ∏ H(kQ(Δ)).

The generic composition algebra C(Δ) is the subring of ∏ H(kQ(Δ)) generated by q and u_{i}, where 1 ≤ i ≤ n. The generic Loewy algebra L(Δ) is the subring generated by q and u_{d}, where d ∈ ℕ_{0}^{n}. Note that we have C(Δ) ⊗ ℚ = L(Δ) ⊗ ℚ, by propositon 4.

In order to define the corresponding extended algebras, we have to be aware that the generalized Cartan matrix Δ = (a_{ij})_{1≤i,j≤n} may be singular, so that we have to deal with a realisation of Δ (see [K]). Let l be the rank of Δ, let n′ = 2n−l, and let Δ′ = (a′_{ij})_{1≤i,j≤n′} be a non–singular n′ × n′–matrix with integer coefficients such that a′_{ij} = a_{ij} for all 1 ≤ i, j ≤ n. For 1 ≤ i ≤ n′, let γ_{i} = ∑_{j=1}^{n} a′_{ij}δ_{j}, this is a derivation of H(kQ(Δ)). We form the skew polynomial ring

H′(kQ(Δ)) = H(kQ(Δ))[H_{i}, γ_{i}]_{1≤i≤n′},

thus H_{1}, …, H_{n′} are indeterminates satisfying the commutation rules

[H_{i}, H_{j}] = 0, for 1 ≤ i, j ≤ n′,
[H_{i}, u_{j}] = a′_{ij}u_{j}, for 1 ≤ i ≤ n′, 1 ≤ j ≤ n.

In ∏ H′(kQ(Δ)) we denote the element (H_{i}^{(k)})_{k} with H_{i}^{(k)} = H_{i} ∈ H′(kQ(Δ)) again by H_{i}. Let C′(Δ) be the subring of ∏ H′(kQ(Δ)) generated by q and all u_{i}, H_{j}, where 1 ≤ i ≤ n, 1 ≤ j ≤ n′, and let L′(Δ) be its subring generated by q, and all u_{d}, H_{j}, where d ∈ ℕ_{0}^{n}, and 1 ≤ j ≤ n′. There is the following alternative description. The derivations γ_{i} of the various H(kQ(Δ)) yield a derivation, again denoted by γ_{i}, of ∏ H(kQ(Δ)), and both C(Δ) and L(Δ) are mapped into itself under γ_{i}, thus we can form the corresponding skew polynomial rings and we have

C′(Δ) = C(Δ)[H_{i}, γ_{i}]_{1≤i≤n′}, L′(Δ) = L(Δ)[T_{i}, γ_{i}]_{1≤i≤n′}.

Note that instead of starting with ∏ H(kQ(Δ)), we may consider the ℤ^{n}–graded subring

⨁_{d}∏_{k}H(kQ(Δ))_{d},

since both C(Δ) and L(Δ) lie inside this subring.

Since q is a central element, we will consider C(Δ), L(Δ), C′(Δ) and L′(Δ) as ℤ[q]–algebras, and we form

{ \atop L′(Δ)} := lim_{←}_{m} L′(Δ) ⊗_{ℤ[q]} ℂ[q]/(q − 1)^{m}

(since C′(Δ) ⊗_{ℤ} ℚ = L′(d) ⊗_{ℤ} ℚ, there is no need to introduce also the notation

{ \atop C′(Δ)}). Let (b_{ij})_{1≤i,j≤n′} = (Δ′)^{−1}, and define

T_{i} = ∑_{j=1}^{n′} b_{ij}H_{j},

for 1 ≤ i ≤ n. Then, clearly,

[T_{i}, u_{i}] = u_{i}, and [T_{i}, u_{j}] = 0 for i ≠ j,

since ∑_{t=1}^{n′} b_{it}a′_{tj} = 1 for i = j, and zero otherwise. As before, we consider instead of u_{i} the element

X_{i} := exp(−\frac{1}{2} ∑_{j=1}^{i−1} a_{ij}T_{j} ln q)u_{i}.

The elements X_{i}, H_{j} with 1 ≤ i ≤ n, 1 ≤ j ≤ n′ generate { \atop L′(Δ)} as a complete Λ–algebra. By the definition of H_{i}, and X_{i} the following relations are satisfied

(*)  [H_{i}, H_{j}] = 0, for 1 ≤ i, j ≤ n′,

(**)  [H_{i}, X_{j}] = a′_{ij}X_{j}, for 1 ≤ i ≤ n′, 1 ≤ j ≤ n.

According to Proposition 2, we have

ρ_{n}^{+}(q, u_{i}, u_{j}) = 0, ^{+}ρ_{n}(q, u_{j}, u_{i}) = 0 for i < j, where n = 1 − a_{ij},

and, as before, this translates to

(***)  ∑_{t=0}^{n}(−1)^{t}\left[{n \atop t}\right]_{q}q^{− \frac{t(n−t)}{2}} X_{i}^{t}X_{j}X_{i}^{n−t} = 0 with n = 1 − a_{ij}, and i ≠ j.

Altogether, we see:

Theorem 2. Let Δ be a symmetric generalized Cartan matrix. Then { \atop L′(Δ)} is generated, as a complete Λ–algebra, by H_{1}, …, H_{2n−l}, X_{1}, …, X_{n}, and these elements satisfy the relations (*), (**), and (***).

It follows that there is a surjective ring homomorphism

η_{Δ} : U_{q}(b_{+}(Δ)) ⟶ { \atop L′(Δ)},

since, by definition, U_{q}(b_{+}(Δ)) is the free complete Λ–algebra, with generators X_{1}, …, X_{n}, H_{1}, …, H_{2n−l} and relations (*), (**) and (***).

We have seen in Theorem 1 that η_{Δ} is an isomorphism for Δ of finite type. We want to announce the following result:

Theorem 3. If Δ is of affine type, then η_{Δ} is an isomorphism.

The proof uses the structure theory of mod kQ(Δ), it will be given in [R5].

6. Symmetrizable generalized Cartan matrices

The results presented above all generalize to the case of a symmetrizable generalized Cartan matrix. In this case, we have to deal with species instead of quivers, see [R2], [R3], [R4] and [R5]. The actual calculation of the corresponding Hall polynomials as in Proposition 5 above yields direct sign choices for the Chevalley ℤ–forms of the Lie algebras of type B_{n}, C_{n}, F_{4}, G_{2} which seem to be new.

References

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[FK] Frenkel, I.B., and Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math 62 (1980), 23–66.

[G] Gabriel. P.: Unzerlegbare Darstellungen I. Manuscripta Math. 6 (1972), 71–103.

[H] Hall, P.: The algebra of partitions. Proc. 4th Canadian Math. Congress. Banff (1959), 147–159.

[K] Kac, V.G.: Infinite dimensional Lie algebras. Second edition Cambridge University Press 1985.

[Ko] Kostant, B.: Groups over ℤ. In: Algebraic Groups and Discrete subgroups. Proc. Symp. Pure Math. IX. Am. Math. Soc. (1966).

[M] Macdonald, I.G.: Symmetric functions and Hall polynomials. Clarendon Press (1979).

[R1] Ringel, C.M.: Tame algebras and integral quadratic forms. Springer LNM 1099 (1984).

[R2] Ringel, C.M.: Hall algebras. In: Topics in Algebra. Banach Centre. Publ. 26. Warszawa. To appear.

[R3] Ringel, C.M.: Hall polynomials for the representation–finite hereditary algebras. Adv. Math. To appear.

[R4] Ringel, C.M.: Hall algebras and quantum groups. Invent. Math.

[R5] Ringel, C.M.: Loewy algebras and quantum groups. In preparation.

[S] Steinitz, E.: Zur Theorie der Abel’schen Gruppen. Jahresber. DMV 9 (1901), 80–85.

Claus Michael Ringel Fakultät für Mathematik Universität D–4800 Bielefeld 1 West Germany.