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**Meanders and Frobenius Seaweed Lie Algebras ^{*}**

**Vincent Coll ^{1} Anthony Giaquinto^{2} and Colton Magnant^{3}
**

^{1}Department of Mathematics, College of Arts and Sciences, Lehigh University, Bethlehem, PA 18015, USA

^{2}Department of Mathematics and Statistics, College of Arts and Sciences, Loyola University Chicago, Chicago, IL 60626, USA

^{3}Department of Mathematical Sciences, College of Science and Technology, Georgia Southern University, Statesboro, GA 30460, USA Address correspondence to Anthony Giaquinto, [email protected]

^{*}This article is a part of a Special Issue on Deformation Theory and Applications (A. Makhlouf, E. Paal and A. Stolin, Eds.).

**Received Date:** September 03, 2010; **Accepted Date:** May 09, 2011

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

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

The Lie algebra L is Frobenius if dim = 0; equivalently, if there is a functional F ∈ L* such that BF (−,−) is non-degenerate.

Frobenius Lie algebras were first studied by Ooms in [9], where he proved that the universal enveloping algebra
UL 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 BF (−,−) is non-degenerate and let
M be the matrix of *B _{F}* (−.−) relative to some basis {x

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.

In this section, we introduce the seaweed Lie algebras of [2]. Recall that a composition of a positive integer n is an unordered partition That is, each

**Definition 1.** Let V be an n-dimensional vector space with a basis e1, . . . , en. Let and be two compositions of n and consider the flags

where The subalgebra of sl(n) preserving these flags is called a seaweed Lie algebra and is denoted by

A basis-free definition is available but is not necessary for the present discussion. The name seaweed Lie algebra was chosen due to their suggestive shape when exhibited in matrix form. For example, the algebra p(3, 1, 3, 2 | 4, 2, 3) consists of traceless matrices of the form

where the entries marked by the dots are zero.

Many important subalgebras of sl(n) are of seaweed type, as illustrated in the following example.

Example 2. • The entire algebra sl(n) = p(n | n) has index n − 1.

• The Cartan subalgebra of traceless diagonal matrices is where 1 = (1, 1, . . . , 1) and has index n − 1.

• The Borel subalgebra is and has index

• A maximal parabolic subalgebra is of the form p(a, b | n). Élashvili proved in [3] that its index is gcd(a, n) − 1.

The only explicitly known Frobenius examples in the above list are the maximal parabolic algebras p(a, b | n) with a and n relatively prime. Of course, another infinite family of Frobenius seaweed algebras occurs when a = (2, . . . , 2, 1), b = (1, 2, . . . , 2), and n is odd. A similar case is a = (1, 2, . . . , 2, 1), b = (2, . . . , 2), and n is even. These two families are detailed in [10].

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.

As stated earlier, Dergachev and Kirillov have developed a combinatorial algorithm to compute the index of an
arbitrary from its associated *meander* graph determined by the compositions The vertices
of 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 determines arcs above the line which we will call the top edges. The component determines arcs above vertices 1, . . . , a1. The arcs are obtained by connecting vertex 1 to vertex a1, vertex 2 to vertex a1 − 1, and so on. If a1 is odd then vertex has no arc above it. For the component we do the same procedure over vertices a1+1, . . . , a1+a2, and continue with the higher ai.

The arcs corresponding to 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 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).

**Example 5.** **Figure 1** shows the meander corresponding to the compositions We see that there is a single path and a single cycle. Using the theorem above, the index is 2+1−1 = 2. Hence,
p(5, 2, 2 | 2, 4, 3) 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.

**Example 6.** Consider the algebra p(3, 2, 2 | 2, 5). Its meander is given in **Figure 2**.

Labeling the vertices with {1, 2, . . . , n} from left to right, notice that M(3, 2, 2 | 2, 5) is the single path 2, 1, 3, 7, 6, 4, 5 (if we start with 2) or its reversal 5, 4, 6, 7, 3, 1, 2 if we start with 5. In particular, the index is 1−1 = 0 and so this is a Frobenius algebra.

**Question 7.** What are the conditions on the compositions so that the meander 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 to be a single path. For example, exactly two elements of the set (a1, . . . , am, b1, . . . , bt) 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 ai or bj is odd.

Another necessary condition for to be a single path is that a1 ≠ b1. In this case,

and thus is not Frobenius since the index is additive for direct sums of Lie algebras. More generally, if then the meander is not a single path. Other necessary conditions can be given, but none seems to shed light on what is sufficient.

In this section, we revisit some known families of Frobenius seaweed algebras in terms of meanders. At the end we also provide two new families.

First consider Panyushev’s example with and n is odd. Again, numbering as in Example 6, the top edges connect 2 to 4, 4 to 6, and so on. while the bottom edges connect 1 to 3, 3 to 5, and so on. Hence, the meander consists of the single path 1, 2, . . . , n. A similar argument verifies that the meander for with n even is also the path 1, 2, . . . , n.

To analyze some other cases, it is convenient to modify the definition of the meander

**Definition 8.** Suppose are compositions of n. The modified meander is the graph appended with a loop corresponding to each odd ai and bj . Specifically, for all odd ai, add a loop connecting to itself. Similarly, for all odd bj , add a bottom loop connecting to itself.

Note that in each vertex is incident with exactly one top and one bottom edge or loop.

**Example 9.** **Figure 3** shows the modified meander M’(5, 2, 2 | 2, 4, 3). Compare with the meander M(5, 2, 2 |
2, 4, 3) given in Example 5.

**The top and bottom bijections**

Each modified meander determines two bijections of S = {1, 2, . . . , n} to itself. Define a “top” bijection t of S by t(i) = i, where j is the unique vertex incident with the same top edge as i. If i is joined to itself by a top loop, then t(i) = 1. In a similar way, define a “bottom” bijection b of S by b(i) = j, where j is the unique vertex incident with the same bottom edge as j. If i is joined to itself by a bottom loop, then b(i) = 1. Clearly the maps t and b are well defined. For instance, in Example 9, we have t(3) = 3 and b(3) = 6.

**Definition 10.** be compositions of n. The meander permutation is the permutation t ◦ b of S.
That is,

**Example 11.** Consider the meander permutation as in Example 9.We can write as a product
of disjoint cycles in Sn: (1, 4)(2, 5)(3, 7, 8, 9, 6) (note the different use of the term “cycle”).

**Theorem 12.** Suppose are compositions of n. Then the meander is a single path if and only if the
meander permutation is an n-cycle in Sn.

Proof. Suppose the meander is the single path a1, a2, . . . , an. By switching if necessary, we can assume that b(a1) = a2. Then the meander permutation is the n-cycle (a1, a3, . . . , an−1, an, an−2, . . . , a2) if n is even, and if n is odd it is the n-cycle if (a1, a3, . . . , an, an−1, an−3, . . . , a2).

Conversely, suppose is an n-cycle but is not a single path. Then 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 then it is a fixed point of σx,y which therefore cannot be an n-cycle. If a1, . . . , ak is a path in with k < n, then, depending on whether k is even or odd, either the (a1, a3, . . . , ak−1, ak, ak−2, . . . , a2) or (a1, a3, . . . , ak, ak−1, ak−3, . . . , a2) appears in the cycle decomposition of Since k < n, we conclude that is not an n-cycle.

Now if contains a cycle a1, a2, . . . , ak, a1, then the meander permutation contains either the k/2 cycle (a1, a3, . . . , an−1) if n is even or the k-cycle (a1, a3, . . . , an, a2, a4, . . . , an−1) if n is odd. If k < n, then is not an n-cycle. If k = n is even, then the same argument shows that is not an n-cycle. The remaining case is that k = n is odd. If this happens though, we must have and consequently all components ai and bj are even. Since we have a contradiction. Thus, in all cases when is not a single path, the meander permutation is not an n-cycle, which is a contradiction. The proof is complete.

**Maximal parabolic subalgebras**

To generate more examples of Frobenius Lie algebras, we consider maximal parabolic seaweed subalgebras of sl(n) which are necessarily of the form p(a, b | n).

**Lemma 13.** Consider the compositions The meander permutation 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

Therefore,

Recall Èlashvili’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 of Èlashvili’s result.

**Corollary 14.** The maximal parabolic algebra p(a, b | n) is Frobenius if and only if gcd(a, n) = 1.

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.

**Lemma 15.** be compositions of n. The permutation meander is the map sending
i to a − c mod n for all i.

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 + imod n.

An immediate consequence is the following result.

**Corollary 16.** The opposite maximal parabolic seaweed algebra p(a, b | c, d) is Frobenius if and only if gcd(a −
c, n) = 1.

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 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 {v1, v2, . . . , vn} of the meanderM = M(a, b, c | n). Since the vertices of M are viewed as the numbers {1, 2, . . . , n} on a line, the interval between vertices vi and vi+1 makes sense.

**Lemma 18.** Suppose gcd(a + b, b + c) = 1. Then, there are exactly two vertices of degree 1 in M and all other
vertices have degree 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 va+1 and va+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 vertices in the first part of the partition, and and vertices in the other parts, respectively. Note that one of may be zero. Label the vertices of P with u1, u2, . . . , un’, where following the inherited order (the order of the labels vi) 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 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 ui and ui+1. For the moment, let us suppose that Following the bottom edges, this means that C must also have d vertices in the interval between 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 ui to ui+1 such that M contains an edge joining ui and ui+1. In particular, if is even, then the interval between is a dead end.

**Lemma 19.** Suppose gcd(a + b, b + c) = 1. Then there are exactly two dead ends in M.

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 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 Now suppose C has d vertices in
the interval from ui to ui+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 Also, by
following top edges, C must have d vertices in another interval (depending where the top edges go).

If the interval from ui to ui+1 happens to be one of the two partition intervals (e.g. suppose ), then this means that C must have d1 vertices in the interval outside u1 and at least d2 vertices in the interval from to This then implies that C has d1 vertices in the interval beyond un (following bottom
edges) and another d1 vertices in the interval from (following top edges) for a total of d vertices
in the interval 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

Alternating following top and bottom edges, we see that the cycle C has exactly d vertices in every interval between vertices and possibly d1 ≤ d vertices on each end beyond u1 and beyond Carefully counting, we see that the first part of our partition has vertices. Similarly, the second part has b = and the third part has This means that and 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 as the number of parts of 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.

- Belavin AA, Drinfel’d VG (1982) Solutions of the classical Yang-Baxter equations for simple Lie algebras. Funct Anal Appl 16: 159–180.
- Dergachev V, Kirillov A (2000) Index of Lie algebras of seaweed type. J Lie Theory 10: 331–343.
- Elashvili AG (1982) Frobenius Lie algebras. Funktsional Anal i Prilozhen 16: 94–95.
- Elashvili AG (1990) On the index of parabolic subalgebras of semisimple lie algebras.
- Gerstenhaber M, Giaquinto (1997) A Boundary solutions of the classical Yang-Baxter equation. Lett Math Phys 40: 337–353.
- Gerstenhaber M, Giaquinto (2008) A Graphs, Frobenius functionals, and the classical Yang-Baxter equation.
- Joseph A (2006) On semi-invariants and index for biparabolic (seaweed) algebras. I J Algebra 305: 487–515.
- Khoroshkin SM, Pop II, Samsonov ME, Stolin AA, Tolstoy VN (2008) On some Lie bialgebra structures on polynomial algebras and their quantization. Comm Math Phys 282: 625–662.
- Ooms AI (1974) On Lie algebras having a primitive universal enveloping algebra. J Algebra 32: 488–500.
- Panyushev DI (2001) Inductive formulas for the index of seaweed Lie algebras. Moscow Math J 2: 221–241.
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