Existence of Richardson elements for seaweed Lie algebras of finite type

Seaweed Lie algebras are a natural generalisation of parabolic subalgebras of reductive Lie algebras. A well‐known theorem of Richardson says that the adjoint action of a parabolic group has a dense open orbit in the nilpotent radical of its Lie algebra (Richardson, Bull. Lond. Math. Soc. 6 (1974) 21–24.). Elements in the open orbit are called Richardson elements. In (Jensen, Su and Yu, Bull. Lond. Math. Soc. 42 (2009) 1–15.) together with Yu, we generalised Richardson's Theorem and showed that Richardson elements exist for seaweed Lie algebras of type A . Using GAP, we checked that Richardson elements exist for all exceptional simple Lie algebras except E8 , where we found a counterexample.


Introduction
Throughout, we assume that k is the field of complex numbers, g is a reductive Lie algebra and G is a connected reductive algebraic group with Lie algebra g. Definition 1.1 [5,12]. A Lie subalgebra q of g is called a seaweed Lie algebra (or simply a seaweed) if there exists a pair (p, p ) of parabolic subalgebras of g such that q = p ∩ p and p + p = g. Two parabolic subalgebras p and p of g such that p + p = g are said to be opposite.
Seaweed Lie algebras (later also called biparabolic algebras, see, for example, [9,10]) were defined by Dergachev and Kirillov in their study of indices of Lie algebras of type A [5] and were generalised by Panyushev to arbitrary reductive Lie algebras [12]. By definition, parabolic subalgebras of g are seaweed Lie algebras. Substantial work on seaweeds has been done on generalising results on parabolic subalgebras and beyond. Among others, there is work on indices by Joseph [9,10], on affine slices for the coadjoint action by Yu and Tauvel [17] and Joseph [10,11]. Also, Panyushev and Yakimova study meander graphs in [13,14]. Along the line of generalising results on parabolic subalgebras, Baur and Moreau study quasi-reductive biparabolic algebras in [2].
We are interested in the adjoint action of a seaweed Lie algebra on its nilpotent radical and the density of the action, with a view from quiver representation theory. In the case where q is parabolic, a well-known theorem of Richardson [15] (see also [16,Chapter 33]) says that Richardson elements exist. In this case, Brüstle, Hille, Ringel and Röhrle gave a categorical construction of Richardson elements, when g is of type A, using representations of a double quiver with relations of a linear quiver [4]. The following natural question was raised independently by Duflo and Panyushev. Question 1.3 [8]. Does a seaweed Lie algebra have a Richardson element?
Surprisingly, a quiver model can also be constructed from a given seaweed Lie algebra to understand the adjoint action on the nilpotent radical. In [8] together with Yu, we made use of the construction in [4] to build rigid modules for the quiver model and obtained a positive answer to Question 1.3 for all seaweeds in Lie algebras of type A. An example (Example 3.6) is given in Section 3 to illustrate the construction. This quiver model is the double quiver with relations of a quiver of type A. The path algebra of the double quiver with relations is quasi-hereditary and had been studied by Hille and Vossieck in work on the radical bimodule of a hereditary algebra [6].
In this paper, we prove the existence of Richardson elements for seaweed Lie algebras of type B, C and D.
Theorem 1.4. Let g be a Lie algebra of type B, C or D. Then any seaweed Lie algebra in g has a Richardson element.
A natural approach to answer Question 1.3 would be to adapt Richardson's proof for the case of parabolic subalgebras. However, the fact that the corresponding parabolic group is the normaliser of the nilpotent radical plays an important role in Richardson's proof, but fails for seaweed Lie algebras. At this point, we emphasise the advantages of techniques from quiver representation theory, which do not require this fact, as can be seen in Brüstle, Hille, Ringel and Röhrle's construction for parabolic subalgebras [4] and our construction for seaweeds [8]. The key ingredient of the categorical approach is the interplay between Lie algebras and quiver representation theory. For instance, when g is of type A, seaweed Lie algebras are endomorphism algebras of projective representations of quivers of type A. Furthermore, endomorphism algebras of representations that give us Richardson elements correspond to stabilisers of Richardson elements. In this paper, we exclusively analyse local properties of endomorphisms at a vertex of the quiver and apply these properties to prove the main theorem.
As a consequence of Theorem 1.4 and the results in [8], Richardson elements exist for all seaweed Lie algebras of finite type except for E 8 . We remark that the existence of Richardson elements in type E 6 and E 7 , and the counterexample in E 8 (see [8]) were verified using GAP [18, version 4. 8. 7].
The remainder of this paper is organised as follows. In Section 2, we recall the notion of a standard seaweed Lie algebra and prove some useful lemmas. We decompose standard seaweeds, including standard parabolic subalgebras, as sums of subalgebras and analyse how their stabilisers act. Furthermore, we show that a local property of stabilisers of Richardson elements for seaweeds of type A is sufficient for the existence of Richardson elements for all seaweeds of other types, except in two special cases. This condition can be verified using the categorical construction of Richardson elements [4,7,8] in type A. So in Section 3, we recall the construction in type A and explain the link between seaweed Lie algebras and representations of quivers. We prove essential results on stabilisers in Section 4. In Section 5, we prove the main results. Techniques from quiver representation theory play an important role in the proofs.

Standard seaweeds and parabolic subalgebras
We fix a Borel subalgebra b of g and a Cartan subalgebra h contained in b. Denote by Φ, Φ + , Φ − and Π, respectively, the root system, the set of positive roots, the set of negative roots and the set of positive simple roots, determined by h, b and g. For α ∈ Φ, denote by g α the root space corresponding to α. Write We say that α is supported at a positive simple root α i if x i = 0 and call the set of all such simple roots the support of α.
For S, T ⊂ Π, let Φ S be the set of roots with support in S, Note that p ± S are parabolic subalgebras and q S,T is a seaweed Lie algebra. Parabolic subalgebras and seaweeds constructed in this way are said to be standard with respect to h and b and we call p + S (respectively, p − S ) a positive parabolic subalgebra (respectively, a negative parabolic subalgebra).
Proposition 2.1 [12,16]. Any seaweed Lie algebra in g is G-conjugate to a standard seaweed Lie algebra.
As a consequence, it suffices to consider standard seaweeds when proving the existence of Richardson elements. Let g α and l S,T = h ⊕ α∈ΦS∩T g α . Then l S,T is the Levi-subalgebra of q S,T and n S,T = n + S,T ⊕ n − S,T is the nilpotent radical of q S,T . In the sequel, we will assume that neither S or T is equal to ∅ or Π. Note that two Lie algebras of the same type may have different rank and we view By the type of a seaweed Lie algebra q ⊆ g, we mean the type of g and thus the type of q is not well-defined without an embedding q ⊆ g.
We note the following symmetry with respect to the choice of S and T . Proof. The lemma follows from the involution g → g mapping g α onto g −α .

A decomposition of seaweed Lie algebras and Richardson elements
Let q S,T be a standard seaweed in a simple Lie algebra g of type B, C or D. Note that Denote the positive simple roots of g by α 1 , . . . , α n , with the corresponding Dynkin graph numbered as follows.
Let C = (c ij ) be the Cartan matrix of g. For instance, when g is of type B 3 , the Cartan matrix is The Cartan matrix of type C n is the transpose of that of type B n .
Let e i , f i , h i be the Chevalley generators of g. That is, g is generated by these generators subject to the relations We choose a new basis h 1 , . . . , h n of the Cartan subalgebra of g as follows. First, let Next, for i 2, let Lemma 2.3. When i = 2 and g is of type D, When g is of type D and i = 2, or g is of type B or C, Proof. Direct computation gives the following, otherwise, except when i = 2 and g is of type D, where we have otherwise.
So the lemma follows.
Let h i be the subspace spanned by h i . Let By Lemma 2.2, we may assume that η 1. Let We define two subspaces of g, Lemma 2.4. (1) If > 2, when g is of type D or > 1 for other types, then g 1 is a Lie subalgebra of the same type as g.
Proof. By the definition of h i , we have So the lemma follows.
Lemma 2.5. If = ω, then q S,T has a Richardson element.
Proof. If ω = = 1, then each root space g α in q S,T is supported on a subgraph of type A, that is, the support of α contains only simple roots corresponding to vertices in the subgraph.
So q S,T is isomorphic to a seaweed of type A. Similarly, if ω = = 2 and g is of type D, then q S,T is also isomorphic to a seaweed of type A. If ω = = n, then q S,T is a reductive Lie algebra of the same type as g with the nilpotent radical 0 and thus obviously 0 is the Richardson element. In the first two cases, by [8,Theorem 1.2] or Richardson's Theorem [15], q S,T has a Richardson element.
In all the other cases, where q 1 ⊆ g 1 is a parabolic subalgebra of the same type as g, and q 2 ⊆ g 2 is a seaweed of type A. Furthermore, by equation (a), [g 1 , g 2 ] = 0 and so [q 1 , q 2 ] = 0.
Since both q 1 and q 2 have Richardson elements, we can conclude that q S,T has a Richardson element.
Consequently, we may assume that > ω for the remainder of the paper. When g is of type D, we also assume ( , ω) = (2, 1), in which case we prove separately the existence of Richardson elements in Theorem 5.4. Let Note that by the definition of , S contains all the simple roots α i with i < . These subsets determine two subalgebras of q S,T , namely the positive parabolic subalgebra c S,T of g 1 determined by T and the seaweed Lie subalgebra a S,T of g 2 determined by S and T .
Example 2.6. Let g be a Lie algebra of type D 6 , S = {α 5 , α 3 , α 2 , α 1 } and T = {α 6 , α 4 , α 2 , α 1 }. Then = 4, ω = η = 3. The subalgebras a S,T and c S,T can for instance be described using matrices as follows, where a S,T is marked by * and †, and c S,T is marked by and †. The one-dimensional intersection is marked by †, and there is an anti-symmetry to the anti-diagonal. ⎛ Let n a and n c be the nilpotent radicals of a S,T and c S,T , respectively. (1) q S,T = a S,T + c S,T ; (2) a S,T ∩ c S,T = l is a block in the Levi subalgebra of q S,T ; (3) n S,T = n a ⊕ n c . Proof.
(1) Let α be a positive root such that g α ⊆ q S,T . Then α is not supported at α , since α ∈ S. By the assumption on the root system, α must be supported on simple roots α i with all i < or all i > . So g α ⊆ a S,T or g α ⊆ c S,T . Similarly, a negative root β with g β ⊆ q S,T is supported on simple roots α i with all i < ω or all i > ω, and so g β ⊆ a S,T or g β ⊆ c S,T . By construction, [g αi , g −αi ] ⊆ a S,T + c S,T for all simple roots α i , and so q S,T = a S,T + c S,T ; (2) follows from the construction; (3) follows from (1) and (2).
By [8, Theorem 1.2], Richardson elements exist in a S,T . Let r 2 ∈ a S,T be a Richardson element and denote by the stabiliser of r 2 in a S,T . For a subalgebra u ⊆ g given as a direct sum of root spaces and subspaces h i , let x |u be the canonical projection of x ∈ g onto u. Let Proof. Assume [c r2 , r 1 ] = n c . Take any (x a , x c ) ∈ n a ⊕ n c . There exists y a ∈ a S,T such that Write y a = y a + (y a ) |l . Note that r 1 | gα = 0 can only occur for positive roots α with support contained in {α −1 , . . . , α 1 }, and y a | g β = 0 can occur only for positive roots β with support contained in {α n , . . . , α +1 }, or negative roots with support contained in {α n , . . . , α ω+1 } and containing at least one α j for some j . So by equation (a) in Subsection 2.2 and the fact from Lemma 2.3 that Let y c ∈ c r2 be such that Let z ∈ stab aS,T (r 2 ) with z |l = (y c ) |l and z = z − z |l . Then similar to equation (b), Therefore, This completes the proof of the lemma.

A decomposition of parabolic subalgebras and Richardson elements
The main goal in this subsection is to present a key sufficient condition for the existence of Richardson elements in general, except in the two special cases, namely when = ω as in Lemma 2.5 and when ( , ω) = (2, 1) in type D. We transfer the problem of the existence of Richardson elements to a local relationship (see Lemma 2.12) between a parabolic subalgebra constructed from a given seaweed Lie algebra and the seaweed Lie algebra itself. We first give a decomposition of parabolic subalgebras, discuss properties of subalgebras in the decomposition and then state and prove the sufficient condition at the end of the section. Let S, T, , ω, g 1 , g 2 and l be defined as in Subsection 2.2 with ω < . In the remainder of this section, for type D, besides the assumption that we also assume that Assumption (ii) is purely a technical issue, to avoid a complication in the description of the decomposition discussed in this subsection and it does not compromise the completeness of the existence of Richardson elements for q S,T with ( , ω) = (2, 1), due to the symmetry between α 1 and α 2 when g is of type D.
Let g be a Lie algebra of the same type as g, with rank at least and root system denoted by Φ . We may assume that both g and g are subalgebras of a Lie algebra of the same type as g such that g ⊆ g or g ⊆ g. Here all inclusions are induced by inclusions of Dynkin diagrams. This, in particular, implies that g and g have the simple roots, α i for 1 i , in common. Let p + U ⊆ g be the standard parabolic subalgebra determined by U with α ∈ U and We choose a basis {h i } i for the Cartan subalgebra h of g in the same manner as we did for the basis {h i } i of the Cartan subalgebra of g. Let g 1 = g 1 and let g 2 ⊆ g be defined similarly to These two sets determine the following standard parabolic subalgebras of g 2 and g 1 , Note that c U = c S,T . Let d U ⊆ p + U be the direct sum of all root spaces g α with α a positive root such that g α is neither contained in a U nor in c U . Let n a be the nilpotent radical of a U . Recall that n c is the nilpotent radical of c U = c S,T .
The subalgebras a S,T and c S,T are as below, marked by * , † and †, , respectively, where the intersection is marked by †, and (2) Let g = g and U = {α 6 , α 4 , α 2 }, which satisfies the conditions The subalgebras a U marked by * and †, c U by † and , and d U by − are as below. (1) and (2) there is an anti-symmetry to the anti-diagonal.
Lemma 2.10. The following are true.
Proof. By the construction, d U is the direct sum of the root spaces g α with α positive and supported at both simple roots α and α ω . For any g −α ⊆ p + U with −α a negative root, the root −α is not supported at α and α ω . So (1) follows. Similarly, (2) holds.
By Richardson's theorem, there exists (1) and (2), we may assume that r 1 is the Richardson element for c U . Again by Lemma 2.10, we can identify where y + y |l ∈ a U and y |l + y ∈ c U . Since n a ∩ n c = 0 and [y, r 2 + r 1 ] = x ∈ n c , we have [y, r 2 ] = [y + y |l , r 2 ] = 0 and [y, It follows that y |l + y ∈ c r 2 and so [c r 2 , r 1 ] = n c .
Recall that r 2 is a Richardson element for a S,T . We have the following key observation, which gives a sufficient condition for the existence of Richardson elements.
Lemma 2.12. If stab aU (r 2 ) |l = stab aS,T (r 2 ) |l , then q S,T has a Richardson element.
Proof. Assume stab aU (r 2 ) |l = stab aS,T (r 2 ) |l . Then c r 2 = c r2 and so [c r2 , r 1 ] = n c by Lemma 2.11. Then q S,T has a Richardson element, by Lemma 2.8.
Verifying the condition in Lemma 2.12 is a key step in the proof of the existence of Richardson elements in the seaweed q S,T . We will make use of the categorical construction of Richardson elements [8], using quiver representation theory. That is, we will analyse the properties of local endomorphisms (that is, restrictions of endomorphisms to a vertex) of rigid modules constructed in [8]. So we recall the construction in the next section.

Rigid D-modules and Richardson elements in type A
In this section, we first recall a quasi-hereditary algebra D, which is the path algebras of a double quiver with relations of a quiver Q of type A, and the construction of rigid good D-modules [8]. We then explain in examples how to construct Richardson elements for the corresponding seaweed Lie algebras from rigid modules. We remark that the quiver Q can be constructed from a given seaweed and the relations defining the algebra D can be read off from the seaweed as well [8].

The path algebra D of a double quiver with relations
Let Q be a quiver of type A m with vertices Q 0 = {1, . . . , m} and arrows Let A = kQ, the path algebra of Q. We denote the projective indecomposable A-module associated to vertex i by P i . Let is a seaweed in a Lie algebra of type A, and a Richardson element in radEnd A P (d) can be constructed from a certain representation X(d) (to be made precise in Subsections 3.2 and 3.3) of a double quiver of Q with relations [7,8]. We recall the double quiver with relations from [3,6] and some related basic definitions.
LetQ be the double quiver of Q, that is,

Any D-module is an A-module via the inclusion A ⊆ D and any A-module is a D-module via the surjection D
A mapping all arrows in Q * 1 to zero. We use the notation A X to indicate the A-module structure of a D-module X and note that be the Δ-support of M . This definition is similar to the support of a module, which is defined using the usual dimension vector.
We identify modules with the corresponding quiver representations. So a D-module M is a collection of vector spaces M i , i ∈ Q 0 and linear maps M β , β ∈ Q 1 ∪ Q * 1 , satisfying the relations I, and a homomorphism f : M → N of D-modules is a collection of linear maps (f i ) i∈Q0 commuting with the module structure on M and N .
Note that a D-module M is rigid if it has no self-extensions, that is, In the remainder of this section, we briefly recall the construction of rigid D-modules and their corresponding Richardson elements [8].

Construction of rigid D-modules: the linear case [4]
Let Q be a linear quiver with m the unique sink vertex. Then I is generated by commutative relations at 2, . . . , m − 1, and a zero relation at 1. In this case, the indecomposable projective Thus, there is a natural bijection between subsets I ⊆ Q 0 and submodules of R m . More precisely, under this bijection a subset I corresponds to the unique submodule X(I) ⊆ R m with Δ-support I. For any vector d ∈ Z m 0 , define with dim Δ X(d) = d and I 1 ⊆ I 2 ⊆ · · · ⊆ I t . Then X(d) is a rigid D-module. We give an example to illustrate the construction. See [4] for more details.

Construction of rigid D-modules: the general case
Now suppose that Q has an arbitrary orientation. Recall that a vertex is admissible if it is a source or a sink. Let  When u = i v+1 is admissible, u is contained in both the vth and the (v + 1)th intervals. We can also define u based on the (v + 1)th interval and obtain an order that is opposite to the one defined in Definition 3.2. This explains for instance in Example 3.4, why the summands M 1 versus M 2 and N 1 versus N 2 are ordered the way they are. Now using the order u , we can construct rigid good D-modules as follows. Let M and N be two good rigid D-modules satisfying With respect to ij , we glue the ith biggest summand of M to the ith biggest summand of N and then take the direct sum of all the glued modules. In this way, we obtain a rigid good D-module X(d) for any Δ-dimension vector d. We illustrate the construction by an example. See [7,8] for more details. We have M 1 3 M 2 and N 1 3 N 2 . So X(d) is the direct sum of the gluings of M 1 , M 2 with N 1 and N 2 , respectively, that is, By the construction of rigid good modules, we have the following lemma.

Construction of Richardson elements
Observe that a standard parabolic subalgebra in gl n can be naturally identified with the endomorphism algebra of a projective representation of a linear quiver. For instance, p + U gl 5 with U = Π\{α 2 } can viewed as End(P 2 1 ⊕ P 3 2 ) for the projective representation P 2 1 ⊕ P 3 2 of the quiver where the number of vertices is the number of Levi-blocks of p + U and the multiplicities 2 and 3 of P 1 and P 2 are the sizes of the Levi-blocks. This identification is due to the following, Hom(P 1 , P 1 ) = Hom(P 2 , P 2 ) = Hom(P 2 , P 1 ) = k and Hom(P 1 , P 2 ) = 0.
Similarly, a seaweed Lie algebra q S,T can be viewed as an endomorphism algebra of a projective module of a quiver of type A. In both cases, the nilpotent radical can then be identified with the Jacobson radical of the endomorphism algebra.
Let V = ⊕ i V i with spaces V i determined by the Levi-blocks and of dimension 2, 1, 2, 1, 2, respectively. We have the following embeddings: This is the projective representation P = P 2 1 ⊕ P 2 ⊕ P 2 3 ⊕ P 4 ⊕ P We now describe how to construct a Richardson element r(d) for q S,T ⊆ gl n from the rigid module X(d). Note that X(d) is constructed based on data contained in q S,T . Let such that x ij = x lj for two different summands X i and X l . If X i is Δ-supported at both s and t with s < t, but not at s + 1, . . . , t − 1, then the matrix r(d) has a 1 at either (x is , x it ) or (x it , x is ), depending on which root space belongs to q S,T = End A (P (d)). All other entries in r(d) are equal to 0. Note that End D (X(d)) ∼ = stab qS,T (r(d)).
The matrix r(d) is also the adjacency matrix of an oriented graph with components corresponding to indecomposable summands of X(d). See the example below for an illustration and [1,4,8] for more detail.
Example 3.7. The rigid modules X(d) in Example 3.1 and 3.4 correspond, respectively, to the parabolic subalgebra of gl 5 with Richardson element r 1 and the seaweed Lie algebra in gl 8 , which is exactly the seaweed given in Example 3.6, with Richardson element r 2 as follows The corresponding oriented graphs are

Stabilisers of Richardson elements in type A
Consider the quiver Q of type A m of arbitrary orientation and the algebras A and D as in Section 3. For D-modules M and N , let be the space of homomorphisms from N to M restricted to vertex i. Let End D (M ) i and Aut D (M ) i be defined similarly. We study the structure of the endomorphism algebra of a good rigid D-module and its restriction to a vertex. Let We order the summands such that X i < m X i+1 for all indecomposable summands X i .

Restriction to m, when m is a source
Proof. By the construction of X(d), Hom D (X i , X j ) m is one dimensional for i j and zero otherwise. Furthermore, we may choose basis elements r ji ∈ Hom D (X i , X j ) m for all i and j, such that r ji r kl = r jl if i = k and zero otherwise. The lemma follows.

Restriction to m, when m is a sink
Recall that α m−1 : m − 1 → m is the arrow that ends at m. For D-modules M and N , let Proof. The proof is similar to the proof of Lemma 4.1.
We remark that gl(V ) is in fact isomorphic to End A (P dm m ).

Stabilisers of indecomposable rigid modules
There is an obvious embedding of endomorphism rings i End D (X i ) ni ⊆ End D (X(d)).
Let v be a sink in Q and let α be an arrow ending at v. Then αα * induces a nilpotent endomorphism x of X i . We have such an endomorphism x s : X i → X i for each interval [i s−1 , i s ] with s = 1, . . . , t + 1, where x s can be zero, depending on the intersection of supp Δ (X i ) and Lemma 4.3. The map y s → x s induces an isomorphism k[y 1 , . . . , y t+1 ] y s y l = 0 for s = l, y ms Proof. Let x s and x l be two arbitrary endomorphisms, induced by αα * and ββ * , respectively, where α and β are two different arrows. Clearly, x s x l = 0 if α and β end at different sinks.
Otherwise, x s x l = 0 follows from the relation α * β = 0. By definition, x ms s = 0. So the map is well defined.
By the construction of X i , End D (X i ) is generated by the x s and thus the map is surjective. The intersection of the images of x s and x l is zero if s = l, and so the injectivity follows.
By the embedding, End D (X(d)) ⊆ End A (P (d)) ⊆ gl n and the construction of Richardson elements discussed in Subsection 3.4, each element of i End D (X i ) ni can be explicitly described in terms of matrices. They can also be described in terms of the oriented graphs constructed from X i . Example 4.4. We use the two Richardson elements from Example 3.7. In both cases, there are two indecomposable summands X 1 and X 2 in X(d), and we use black entries to describe End D (X 1 ) and blue entries to describe End D (X 2 ) as follows, To illustrate, for instance in the first matrix, the a-entries are a · 1 X 1 , the b-entries are b · x 1 1 with x 1 1 : X 1 → X 1 , the c-entries are c · (x 1 1 ) 2 and the e-entry is e · x 2 1 with x 2 1 : X 2 → X 2 . The non-zero off-diagonal entries in the matrices correspond to non-trivial paths in the oriented graphs as follows.

The main result
Theorem 5.1. Let g be a simple Lie algebra of type B, C or D. Then any seaweed in g has a Richardson element.
We continue to use the notation from Section 2. The proof of the theorem is split into two cases. The first case (see Theorem 5.2) deals with type B, C and D, under the assumptions (i) and (ii) in Subsection 2.3 for type D. We will use quiver representations and results from Section 4 to verify that the sufficient condition in Lemma 2.12 holds, and so Richardson elements exist. The second case deals with the special situation, where g is of type D and ( , ω) = (2, 1). Unlike the first case, there can be root spaces that are not contained in a S,T + c S,T (cf Lemma 2.7). That is, q S,T is not necessarily equal to a S,T + c S,T .
Theorem 5.2. Let q S,T ⊆ g be a seaweed, where g is of type B or C, or of type D with ( , ω) = (2, 1). Then q S,T has a Richardson element.
Proof. Note that we only need to consider the situation, where > ω and neither S nor T is equal to Π or ∅. We also assume (ii) in Subsection 2.3.
Let P = P (d) be a projective A-module such that Let B be the path algebra of the linearly oriented quiver A l with the unique sink l and let P (ĉ) = Pĉ 1 1 ⊕ · · · ⊕ Pĉ l l , a projective representation of this linear quiver. Denote by F the algebra of the associated double quiver with relations, defined in the same way as the algebra D in Subsection 3.1, and let X(ĉ) be the rigid good F -module.
Note that End B (P (ĉ)) = End F (P (ĉ)) and by abuse of notation we let End B (P (ĉ)) 0 l = End F (P (ĉ)) 0 l , which is in fact isomorphic to End B (Pĉ l l ). Choose g and U (see Subsection 2.3), and an embedding End B (P (ĉ)) ⊆ g such that End B (P (ĉ)) = a U and End B (P (ĉ)) 0 l = l. Let r ∈ a U be a Richardson element corresponding to X(ĉ), where the summands of X(ĉ) are ordered from small to big with respect to the order l , so that End F (X(ĉ)) 0 l = stab aU (r ) |l is standard upper triangular. Both End F (X(ĉ)) 0 l ⊆ l and End D (X(d)) m ⊆ l are standard upper triangular with Levi blocks of equal sizes, and so stab aS,T (r) |l = stab aU (r ) |l .
Then q S,T has a Richardson element by Lemma 2.12.
The following example illustrates the construction of X(ĉ) in the proof above. where the summands are ordered from big to small with respect to 4 . In the base interval [3,4], the last summand has the smallest Δ-support {4} and so is the smallest one, the other summands have the same Δ-support {3, 4} and so they are compared at next interval [1,3], in which case representations with smaller supports are actually bigger with respect to 4 . We have , a standard upper triangular parabolic in gl 4 ; (2) the number of blocks is l = 3 with the sizes 2, 1, 1 andĉ = (1, 2, 4); (3) the representation P (ĉ) = P 1 ⊕ P 2 2 ⊕ P 4 3 is the projective representation of the quiver 1 − − → 2 − − → 3; (4) the rigid F -module X(ĉ) is as follows The summands are ordered from small to big with respect to 3 . The space End(X(ĉ)) 0 3 is the space of induced homomorphisms between the top 3 in the summands and indeed For the proof of the following theorem, we denote by E ij the elementary matrix with 1 at (i, j)-entry and 0 elsewhere.
Theorem 5.4. If g has type D and ( , ω) = (2, 1), then the seaweed q S,T ⊆ g has a Richardson element.
Proof. Let W = S\{1} and Then q S,T = q W,T ⊕ q 1 S,T and n S,T = n W,T ⊕ q 1 S,T . Note that q W,T is a seaweed of type A and so it has a Richardson element r ∈ n W,T . Since S,T and let y ∈ q W,T and y ∈ stab qW,T (r) such that Then [y + y , r + r ] = x + x and so r + r is a Richardson element. Hence, to prove the theorem, it suffices to show [stab qW,T (r), r ] = q 1 S,T . We choose the representation of g given by 2n × 2n-matrices anti-symmetric to the antidiagonal, g α1 = k · (E n−1,n+1 − E n,n+2 ) and g α2 = k · (E n−1,n − E n+1,n+3 ).
Fix an embedding gl n ⊆ g induced by We may assume the Richardson element r is r(d), constructed from a rigid good module X(d) = ⊕ i X i . Recall that the underlying quiver Q is of type A m , where m is the number of Levi-blocks in q W,T . Choose an embedding End A (P (d)) ⊆ g such that End A (P (d)) = q W,T and End D (X(d)) = stab qW,T (r).

Then
End D (X i ) ⊆ stab qW,T (r), where the x ij are constructed from the summand X i as in Subsection 3.4. As ( , ω) = (2, 1), the vertex m is a source and the mth Levi-block in q W,T is of rank 1, that is, d m = 1. Furthermore, the mth Levi-block is spanned by E n,n − E n+1,n+1 . Observe first that for any x ∈ g α ⊆ q 1 S,T , [E n,n − E n+1,n+1 , x] = x and when i < n, Note also that there is a unique summand of X(d), say X i0 , such that A X i0 contains P m as a summand and X i0 = P m , that is, A X i0 has at least two summands and so We will decompose q 1 S,T as direct sum of subspaces V i , determined by the summands of X(d), and then construct a 'Richardson element' r i in each subspace, in the sense that The orientation of the arrows at vertex m − 1 determines the construction of V i . There are two cases to be considered. Case 1. The vertex m − 1 is non-admissible and so dim q 1 S,T = d m−1 . Note that each summand P m−1 is contained in a different indecomposable summand of X(d) and we may assume Then q 1 S,T = ⊕ i V i . Let δ ij be Kronecker numbers. For any r j ∈ V j , we have Now choose a non-zero element r i ∈ V i and let Case 2. The vertex m − 1 is admissible. Then it is a sink and Note that V i can be 0 and we have Then for any non-zero r i ∈ g βi , where x is the endomorphism of X i induced by αα * with α the arrow from vertex m − 2 to vertex m − 1 (see Lemma 4.3). By convention, x 0 denotes the identity endomorphism 1 X i . Choose a non-zero element r i ∈ g βi and let In both cases, Therefore q S,T has a Richardson element.
Remark 5.5. (a) The Richardson elements and their stabilisers can be explicitly constructed using results from [8], the proofs of Theorem 5.2 and Theorem 5.4. The work of Baur [1] on parabolic subalgebras is also needed in the case of Theorem 5.2.
(b) The method of Theorem 5.4 could be generalised to Lie algebras of exceptional types and therefore provide an explanation to why Richardson elements do not exist for some seaweed Lie algebras of type E 8 .
We end this paper with an example of constructing Richardson elements, using the method discussed in Theorem 5.4.
Example 5.6. Let g = so 10 , a Lie algebra of type D 5 .
(3) The Richardson elements of q S\{1},T and q K\{1},L are as below. Entries of the same colour come from the same indecomposable summand. Note that for X(d), one of the indecomposable summands is a Verma module, so both the fourth column and row are zero. (4) The stabilisers of the two Richardson elements are as follows. ⎛ There are two indecomposable direct summands in each of X(c) and X(d). The different colours indicate homomorphisms between different pairs of summands. The action of stab q K\{1},L (r K\{1},L ) on q 1 K,L is equivalent to the natural action of ( a + e 0 0 2e ) on k 2 , and r = ( 1 1 ) ∈ q 1 K,L . So we have the following Richardson elements for the two seaweeds q S,T and q K,L , with the red entries coming from the contributions of r in q 1 S,T and q 1 K,L , respectively. ⎛