Meanders and Frobenius Seaweed Lie Algebras

The index of a seaweed Lie algebra can be computed from its associated meander graph. We examine this graph in several ways with a goal of determining families of Frobenius (index zero) seaweed algebras. Our analysis gives two new families of Frobenius seaweed algebras as well as elementary proofs of known families of such Lie algebras. MSC 2010: 17B05, 17B08


Introduction
Let L be a Lie algebra over a field of characteristic zero. For any functional F ∈ L * there is an associated skew bilinear form B F on L defined by B F (x, y) = F ([x, y]) for x, y ∈ L. The index of L is defined to be The Lie algebra L is Frobenius if dim = 0; equivalently, if there is a functional F ∈ L * such that B F (−, −) is non-degenerate. Frobenius Lie algebras were first studied by Ooms in [9], where he proved that the universal enveloping algebra U L is primitive (i.e. admits a faithful simple module) provided that L is Frobenius and that the converse holds when L is algebraic. The relevance of Frobenius Lie algebras to deformation and quantum group theory stems from their relation to the classical Yang-Baxter equation (CYBE). Suppose B F (−, −) is non-degenerate and let M be the matrix of B F (−.−) relative to some basis {x 1 , . . . , xn} of L. Belavin and Drinfel'd showed that r = i,j (M −1 ) ij x i ∧ x j is a (constant) solution of the CYBE; see [1]. Thus, each pair consisting of a Lie algebra L together with functional F ∈ L * such that B F is non-degenerate provides a solution to the CYBE; see [5,6] for examples.
The index of a semisimple Lie algebra g is equal to its rank and thus such algebras can never be Frobenius. However, there always exist subalgebras of g which are Frobenius. In particular, many amongst the class of biparabolic subalgebras of g are Frobenius. A biparabolic subalgebra is the intersection of two parabolic subalgebras whose sum is g. They were first introduced in the case g = sl(n) by Dergachev and Kirillov in [2] where they were called Lie algebras of seaweed type. Associated to each seaweed algebra is a certain graph called the meander. One of the main results of [2] is that the algebra's index is determined by graph-theoretical properties of its meander; see Section 3 for details.
Using different methods, Panyushev developed an inductive procedure for computing the index of seaweed subalgebras; see [10]. In the same paper, he exhibits a closed form for the index of a biparabolic subalgebra of sp(n). One may see also [8,11].
Tauvel and Yu found in [12] an upper bound for the index of a biparabolic subalgebra of an arbitrary semisimple Lie algebra, and they conjectured that this was an equality. Joseph proved the Tauvel-Yu conjecture in [7].
The methods of [2,7,10,12] are all combinatorial in nature. Yet even with this theory available, it is difficult in practice to implement this theory to find families of Frobenius biparabolic Lie algebras. In contrast, for many cases it is known explicitly which biparabolic algebras have the maximum possible index. For example, the only biparabolics in sl(n) and sp(n) which have maximal index are the Levi subalgebras. In contrast, the problem of determining the biparabolics of minimal index is an open question in all cases.
Our focus in this note is on the seaweed Lie algebras-these are the biparabolic subalgebras of sl(n). The only known families of Frobenius seaweed Lie algebras that seem to be in the literature will be outlined in Section 4, although the unpublished preprint [4] may offer more examples. We will examine these families using the meander graphs of Dergachev and Kirillov. Our methodology provides new proofs that these algebras are indeed Frobenius. We also exhibit a new infinite family of Frobenius seaweed Lie algebras in Section 4.4.

Seaweed Lie algebras
In this section, we introduce the seaweed Lie algebras of [2]. Recall that a composition of a positive integer n is an unordered partition x = (a 1 , . . . , am). That is, each a i ≥ 0 and a i = n. Definition 1. Let V be an n-dimensional vector space with a basis e 1 , . . . , en. Let x = (a 1 , . . . , am) and y = (b 1 , . . . , b t ) be two compositions of n and consider the flags where V i = span{e 1 , . . . , e a1+···+ai } and W j = span{e b1+···+e b j +1 , . . . , en}. The subalgebra of sl(n) preserving these flags is called a seaweed Lie algebra and is denoted by p(x | y).
• The Borel subalgebra is p(1 | n) and has index (n + 1)/2 . • A maximal parabolic subalgebra is of the form p(a, b | n).Èlashvili proved in [3] that its index is gcd(a, n) − 1.
A tantalizing question is how to classify which seaweed algebras are Frobenius, especially given their importance in the general theory of Lie algebras and applications to deformations and quantum groups.

Meanders
As stated earlier, Dergachev and Kirillov have developed a combinatorial algorithm to compute the index of an arbitrary p(x | y) from its associated meander graph M (x | y) determined by the compositions x and y. The vertices of M (x | y) consist of n ordered points on a horizontal line, which can be called 1, 2, . . . , n. The edges are arcs above and below the line connecting pairs of different vertices.
More specifically, the composition x = (a 1 , . . . , am) determines arcs above the line which we will call the top edges. The component a 1 of x determines a 1 /2 arcs above vertices 1, . . . , a 1 . The arcs are obtained by connecting vertex 1 to vertex a 1 , vertex 2 to vertex a 1 − 1, and so on. If a 1 is odd then vertex a a1/2 has no arc above it. For the component a 2 of a, we do the same procedure over vertices a 1 + 1, . . . , a 1 + a 2 , and continue with the higher a i .  The arcs corresponding to y = (b 1 , . . . , b t ) are drawn with the same rule but are under the line containing the vertices. These are called the bottom edges.
It is easy to see that every meander consists of a disjoint union of cycles, paths, and isolated points, but not all of these are necessarily present in any given meander.
Theorem 3 (Dergachev-Kirillov). The index of the Lie algebra of seaweed type p(a | b) is equal to the number of connected components in the meander plus the number of closed cycles minus 1.

Remark 4.
The presence of the minus one in the theorem is due to our use of seaweed subalgebras of sl(n) rather than of gl(n) as used by Dergachev and Kirillov [2]. The index drops by one by the restriction to sl(n) from gl(n).
We see that there is a single path and a single cycle. Using the theorem above, the index is 2 is not a Frobenius algebra.
It is easy to see that to obtain a Frobenius algebra, the only possibility for the meander is that it consist of a single path with no cycles and no isolated points. The following illustrates this point.

Question 7.
What are the conditions on the compositions x and y so that the meander M (x | y) consists of a single path with no cycles or isolated points?
As stated, this seems to be an elementary question involving nothing more that the basics of graph theory. However, the apparent simplicity of the question is misleading since an answer would provide a complete classification of Frobenius seaweed algebras-a difficult problem. Even so, it is easy to give some necessary conditions on x = (a 1 , . . . , am) and y = (b 1 , . . . , b t ) for M (x | y) to be a single path. For example, exactly two elements of the set (a 1 , . . . , am, b 1 , . . . , b t ) must be odd. This is because a path must have a starting point and an ending point, and these correspond to vertices of degree one. A vertex of degree one is either missing a top edge or bottom edge connecting to it, and this happens only if some a i or b j is odd.
Another necessary condition for M (x | y) to be a single path is that a 1 = b 1 . In this case, and thus p(x | y) is not Frobenius since the index is additive for direct sums of Lie algebras. More generally, if r i=1 a i = r j=1 b j for some r ≤ min{m, t}, then the meander is not a single path. Other necessary conditions can be given, but none seems to shed light on what is sufficient.
To analyze some other cases, it is convenient to modify the definition of the meander M (x | y).
Note that in M (x | y), each vertex is incident with exactly one top and one bottom edge or loop.

Theorem 12. Suppose x and y are compositions of n. Then the meander M (x | y) is a single path if and only if the meander permutation σx,y is an n-cycle in Sn.
Proof. Suppose the meander M (x | y) is the single path a 1 , a 2 , . . . , an. By switching x and y if necessary, we can assume that b(a 1 ) = a 2 . Then the meander permutation is the n-cycle (a 1 , a 3 , . . . , a n−1 , an, a n−2 , . . . , a 2 ) if n is even, and if n is odd it is the n-cycle if (a 1 , a 3 , . . . , an, a n−1 , a n−3 , . . . , a 2 ). Conversely, suppose σx,y is an n-cycle but M (x | y) is not a single path. Then M (x | y) contains either an isolated point, a path of length less than n, or a cycle. We will show that each of these possibilities leads to a contradiction.
If i is an isolated point of M (x | y), then it is a fixed point of σx,y which therefore cannot be an n-cycle. If a 1 , . . . , a k is a path in M (x | y) with k < n, then, depending on whether k is even or odd, either the (a 1 , a 3 , . . . , a k−1 , a k , a k−2 , . . . , a 2 ) or (a 1 , a 3 , . . . , a k , a k−1 , a k−3 , . . . , a 2 ) appears in the cycle decomposition of σx,y. Since k < n, we conclude that σx,y is not an n-cycle. Now if M (x | y) contains a cycle a 1 , a 2 , . . . , a k , a 1 , then the meander permutation contains either the k/2 cycle (a 1 , a 3 , . . . , a n−1 ) if n is even or the k-cycle (a 1 , a 3 , . . . , an, a 2 , a 4 , . . . , a n−1 ) if n is odd. If k < n, then σx,y is not an n-cycle. If k = n is even, then the same argument shows that σx,y is not an n-cycle. The remaining case is that k = n is odd. If this happens though, we must have M (x | y) = M (x | y), and consequently all components a i and b j are even. Since a i = n, we have a contradiction. Thus, in all cases when M (x | y) is not a single path, the meander permutation σx,y is not an n-cycle, which is a contradiction. The proof is complete. Lemma 13. Consider the compositions x = (a, b) and y = n. The meander permutation σx,y is the map sending i to i + a mod n for all i.
Proof. By definition of the top and bottom maps, we have and thus RecallÈlashvili's result asserting that the maximal parabolic algebra p(a, b | n) is Frobenius if and only if gcd(a, n) = 1. An immediate corollary of the previous lemma gives a new simple proof ofÈlashvili's result. Proof. By Theorem 12 it suffices to show that the meander permutation is an n-cycle. According to Lemma 13, σx,y(i) = i + a mod n for all i. Thus, the meander permutation is an n-cycle if and only if the sequence i, i + a, i + 2a, . . . , i + (n − 1)a forms a complete residue system modulo n. This occurs precisely when gcd(a, n) = 1. The proof is complete.

Opposite maximal parabolic subalgebras
We now use the same ideas to present another family of Frobenius seaweed algebras each of which is an intersection of a positive and a negative maximal parabolic algebra. Such algebras are of the form p(a, b | c, d) and are called opposite maximal parabolic subalgebras. Proof. The bottom and top maps are given by There are four possible compositions t(b(i)), depending on and whether i ≤ c or i > c and whether b(i) ≤ a or b(i) > a. It is an easy calculation to see that in each case t(b(i)) = a − c + i mod n.
An immediate consequence is the following result. Proof. The argument is exactly as that used in Corollary 14. Namely, that the meander permutation is an n-cycle if and only if the sequence i, i + (a − c), i + 2(a − c), . . . , i + (n − 1)(a − c) is a complete residue system modulo n, and this is the case if and only if gcd(a − c, n) = 1.
The result of the corollary was first proved using different methods by Stolin in [11]. For example, the Lie algebra p(2, 3 | 4, 1) is Frobenius since 2 − 4 = −2 is relatively prime to 7. At this time, the above line of reasoning does not easily extend to compositions x and y with more than two components. However, some calculations offer hope of producing more families of Frobenius Lie algebras using methods similar to those above.

Submaximal parabolic algebras
We conclude with another new family of Frobenius algebras. These are of the form p(a, b, c | n), so they are parabolic algebras omitting exactly two simple roots. We use a different technique than for maximal or opposite maximal algebras to analyze this family. Our result is the following classification theorem.
Theorem 17. The submaximal parabolic algebra p(a, b, c | n) is Frobenius if and only if gcd(a + b, b + c) = 1.
We first establish some conditions on the degrees of the vertices {v 1 , v 2  Proof. Suppose for a moment that there exists a vertex v of degree 0. This vertex must have no bottom edge, meaning that n is odd and v = v (n+1)/2 . We also know that v has no top edge so b is odd and v is halfway between v a+1 and v a+b . This implies that a = c so a + b = b + c, a contradiction. Hence, we get exactly one vertex of degree 1 for each integer in {a, b, c, n} which is odd.
If n is odd, the vertex v (n+1)/2 has degree 1. If all three of a, b and c are odd, then a + b and b + c are both even, meaning they have a common factor of 2, a contradiction. This implies that exactly one of a, b or c must be odd. Then, there is exactly one other vertex of degree one as desired.
If n is even, the bottom edges form a perfect matching. If all three of a, b and c are even, then a + b and b + c are again even, a contradiction. This implies that exactly two of a, b or c are odd, meaning there are two vertices of degree 1 as desired.
By Lemma 18, one component of M must be a path and there are possibly more components which are all cycles. Let P be this path and suppose P has a ≤ a vertices in the first part of the partition, and b ≤ b and c ≤ c vertices in the other parts, respectively. Note that one of a , b or c may be zero. Label the vertices of P with u 1 , u 2 , . . . , u n , where n = |P | following the inherited order (the order of the labels v i ) of the vertices. Notice that the path P forms a meander graph on its own. This means that, by the proof of Lemma 18, we know that exactly two of the integers in {a , b , c , n } are even and two are odd. Now suppose there exists at least one component of M that is a cycle. Let C be the set of all vertices in cycles of M . Suppose C has d vertices in the interval between u i and u i+1 . For the moment, let us suppose that i = n 2 . Following the bottom edges, this means that C must also have d vertices in the interval between u n −i and u n −i+1 . Using this argument, we will show that C has d vertices in almost every interval.
Define a dead end in M to be an interval u i to u i+1 such that M contains an edge joining u i and u i+1 . In particular, if n is even, then the interval between u n /2 and u n /2+1 is a dead end. Proof. A dead end is formed by two consecutive vertices of P which are adjacent. Each occurrence of a dead end coincides with one of a , b , c or n being even, and we know that exactly two of these are even. Thus, there are exactly two dead ends and the proof is complete.
Proof of Theorem 17. Call an interval a partition interval if it is the meeting point of two parts of our partition. Namely, the partition intervals are from u a to u a +1 and from u a +b to u a +b +1 . Now suppose C has d vertices in the interval from u i to u i+1 . For the moment, we suppose this interval is not a dead end. As mentioned before, this means that, by following bottom edges, C must also have d vertices in the interval from u n −i to u n −i+1 . Also, by following top edges, C must have d vertices in another interval (depending where the top edges go).
If the interval from u i to u i+1 happens to be one of the two partition intervals (e.g. suppose i = a ), then this means that C must have d 1 vertices in the interval outside u 1 and at least d 2 vertices in the interval from u a +b to u a +b +1 where d 1 + d 2 = d. This then implies that C has d 1 vertices in the interval beyond un (following bottom edges) and another d 1 vertices in the interval from u a +b to u a +b +1 (following top edges) for a total of d vertices in the interval u a +b to u a +b +1 . See Figure 4 for an example. In this figure, the dark lines represent the edges of P while light lines represent edges of C. The unlabeled light lines represent d edges each. Here n = 7, a = 2, b = 2 and c = 3.
Alternating following top and bottom edges, we see that the cycle C has exactly d vertices in every interval between vertices and possibly d 1 ≤ d vertices on each end beyond u 1 and beyond u n . Carefully counting, we see that the first part of our partition has a = a + 2d 1 + (a − 1)d vertices. Similarly, the second part has b =   b + 2d 2 + (b − 1)d and the third part has c = c + 2d 1 + (c − 1)d. This means that a + b = (a + b )(d + 1) and b + c = (b + c )(d + 1) and these have a common factor of d + 1, a contradiction. This shows that C must be empty, so G is simply the path P .
The following is an example to show that this argument does not work when we break n into more pieces. Consider the meander M = M (3, 2, 2, 2 | 9) pictured in Figure 5.
Here we have broken the top into 4 pieces while leaving the bottom in one piece. Notice that we can add a cycle to this structure which does not pass through all the intervals. This happens because, as the number of pieces we have increases, the number of dead ends also increases, allowing more flexibility in the placement of the cycles.
The above illustrates the complexity of the meander graphs M (x | y) as the number of parts of x and y grow. At the moment, the problem of classifying all Frobenius seaweed Lie algebras seems to be out of reach. Of late, there has been a great deal of interest in Frobenius Lie algebras. Perhaps these recent developments will be instrumental in the development of a classification theory.