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Journal of Generalized Lie Theory and Applications
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Meander Graphs and Frobenius Seaweed Lie Algebras III

Vincent Coll1*, Dougherty A1, Hyatt M2 and Mayers N1

1Department of Mathematics, Lehigh University, Bethlehem, PA, USA

2Department of Mathematics, Pace University, NY, USA

*Corresponding Author:
Vincent Coll
Department of Mathematics
Lehigh University, Bethlehem
PA, USA
Tel:
267-471-5320
E-mail: [email protected]

Received Date: February 28, 2017; Accepted Date: April 18, 2017; Published Date: April 28, 2017

Citation: Coll V, Dougherty A, Hyatt M, Mayers N (2017) Meander Graphs and Frobenius Seaweed Lie Algebras III. J Generalized Lie Theory Appl 11: 266. doi: 10.4172/1736-4337.1000266

Copyright: © 2017 Coll V, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

We investigate properties of a Type-A meander, here considered to be a certain planar graph associated to seaweed subalgebra of the special linear Lie algebra. Meanders are designed in such a way that the index of the seaweed may be computed by counting the number and type of connected components of the meander. Specifically, the simplicial homotopy types of Type-A meanders are determined in the cases where there exist linear greatest common divisor index formulas for the associate seaweed. For Type-A seaweeds, the homotopy type of the algebra, defined as the homotopy type of its associated meander, is recognized as a conjugation invariant which is more granular than the Lie algebra's index.

Keywords

Lie algebra; Seaweed; Biparabolic; Meander; Rank; Index; Frobenius

Introduction

The index of a Lie algebra is an important invariant of the Lie algebra and is bounded by the algebra's rank: ind Equation, with equality when Equation is reductive. More formally, the index of a Lie algebra Equation is given by

Equation

where f is a linear functional on Equation and Bf is the associated skew-symmetric Kirillov form defined by Bf (x, y) = f([x, y]) for all Equation. Of particular interest are those Lie algebras which have index zero. Such algebras are called Frobenius and have been studied extensively from the point of view of invariant theory [1] and are of special interest in deformation and quantum group theory stemming from their connection with the classical Yang-Baxter equation [2,3]. Simple Lie algebras can never be Frobenius but there always exits subalgebras that must be.

In [4] Dergachev and A. Kirillov introduced a combinatorial method for computing the index of certain seaweed (biparabolic) subalgebras of Equation, based on counting the number and type of connected components of a planar graph representation of the seaweed algebra, called a (Type-A) meander. A Type-C, or symplectic meander and an attendant combinatorial index formula was developed by Coll et al in [2] for seaweed subalgebras of Equation. (See also [5]).

Using a collection of deterministic graph theoretic moves, a given meander can be “wound down" to reveal its (simplicial) homotopy type. The sequence of moves used in this winding-down procedure is called the signature of the meander and may be regarding as a graph theoretic rendering of Panyushev's well-known reduction [6]. In [7,8], Coll et al used the signature to develop closed form formulas for the index of a seaweed algebra in terms of the block sizes of the defining flags in the Type-A and Type-C cases when the number of blocks in the flags is small. Subsequently, Karnauhova and Liebsher [10] used signature type moves and complexity arguments to establish that the index formulas developed in these papers are the only linear greatest common divisor formulas for the index based on the flags defining the seaweed. One finds that in the Type-A case, a seaweed is Frobenius precisely when its associated meander consists of a single path [4,7,8]. For a Type-C seaweed to be Frobenius, its associated meander must reduce to a certain collection of paths [5,10].

Since the homotopy type is not defined in terms of the algebraic structure of the original Lie algebra, it is not a priori clear that it is an algebraic invariant of the original algebra. In fact, this remains an open question for an arbitrary seaweed. However, it is implicit in the work of Moreau and Yakimova [11] that for Type-A seaweeds the homotopy type is a conjugation invariant which is more granular than the index. The example at the end of this paper provides two seaweed algebras which have the same dimension, rank, and index − but different homotopy types. So, are not conjugate.

In Theorems 5.1 and 5.2, we classify those homotopy types of seaweeds where there exist linear greatest common divisor index formulas developed in the prequels to this article.

Seaweed Algebras

Let q and q′ be two parabolic subalgebras of a reductive Lie algebra g. If q + q′ = g then q∩q′ is called a seaweed, or in the terminology of A. Joseph [13] biparabolic, subalgebra of Equation. In what follows, we further assume that Equation is simple and comes equipped with a triangular decomposition

Equation

where Equation is a Cartan subalgebra of Equation and Equation and Equation are the subalgebras consisting of the upper and lower triangular matrices, respectively. Let Π be the set of Equation's simple roots and for β∈Π, let Equation denote the root space corresponding to β. A seaweed subalgebra q∩q′ is called standard if Equation and Equation We tacitly assume that the ground field is an algebraically closed field of characteristic zero, so that any seaweed is conjugate to a standard one. Note that while two standard parabolic subalgebras cannot be conjugate, two standard seaweeds can be.

In the case that q∩q is standard, let Equation, Equation, and denote the seaweed by Equation. Such a seaweed is parabolic if one of Ψ or Ψ′ or is the empty set, and called maximal parabolic if it is of the form Equation, respectively.

Type-A Seaweeds

Let Equation be the algebra of n×n matrices with trace zero and consider the triangular decomposition of Equation as above. Let = {β1,…, βn−1} be the set of simple roots of Equation with the standard ordering and let Equation denote a seaweed subalgebra of Equation where Ψ and Ψ′ are subsets of Π. Let compn denote the set of sequences of positive integers whose sum is n (i.e., compn is the set of compositions of n). It will be convenient to index seaweeds of Equation by pairs of elements of compn. Let Equation denote the power set of a set X. Let Equation be the usual bijection from compn to a set of cardinality n −1. That is, given Equation, defineEquation by

Equation

Now, following the notational conventions established in [3], define the type of the seaweed Equation to be the symbol

Equation

By construction, the sequence of numbers in a determines the heights of triangles below the main diagonal in Equation, which may have nonzero entries, and the sequence of numbers in b determines the heights of triangles above the main diagonal. For example, the seaweed Equation of type Equation has the following shape, where * indicates the possible nonzero entries from the ground field. See Figure 1, where we have chosen such a large example to fully illustrate the winding-down moves of Section 4, but also to provide a seaweed with an interesting homotopy type.

generalized-theory-applications-a-seaweed-type

Figure 1: A seaweed of type Equation.

Meanders, Index Formulas, and Homotopy Type

Type-A meanders

Following Dergechev and A. Kirillov [5], we associate a planar graph to each seaweed of type Equation as follows. Line up n vertices horizontally and label them Equation. Partition the set of vertices into two set partitions, called top and bottom. The top partition groups together the first a1 vertices, then the next a2 vertices, and so on, lastly grouping together the last am vertices. In a similar way, the bottom partition is determined by the sequence b1,…,bt. We call each set within a set partition a block. For each block in the top (likewise bottom) partition we build up the graph by adding edges in the same way. First, add an edge from the first vertex of a block to the last vertex of the same block drawn concave down (respectively concave up in the bottom part case). The edge addition is then repeated between the second vertex and the second to last and so on within each block of both partitions. More explicitly, given vertices Equation in a top block of size ai, there is an edge between them if and only if j + k = 2(a1 + a2 +…+ ai−1) + ai + 1. If Equation are in a bottom block of size bi, there is an edge between them if and only if j + k = 2(b1 + b2 +…+ bi−1) + bi + 1. The resulting undirected planar graph is called the meander associated to the given seaweed. We say that the meander has the same type as its associated seaweed See Figure 2.

generalized-theory-applications-a-meander-type

Figure 2: A meander of type Equation.

Type-A Index Formulas

Evidently, every meander consists of a disjoint union of cycles, paths, and points (degenerate paths). The main result of [5] is that the index of the meander can be computed by counting the number and type of each of these components.

Theorem 3.1 (Theorem 5.1, [5]): If p is a seaweed subalgebra of Equation, then

Equation

where C is the number of cycles and P is the number of paths in the associated meander.

This elegant result, and the Type-C analogue ([Theorem 4.5) are difficult to apply in practice. However, in certain cases, the following index formulas allow us to ascertain the index directly from the block sizes of the flags that define the seaweed. The following formulas were developed in the first two articles in this series. We hasten to add that the formula in Theorem 3.2, which follows as a corollary to Theorem 3.3, was known early on to Elashvilli [14].

Theorem 3.2 (Theorem 7, [3]): A seaweed of type Equation has index gcd(a; b)−1

Theorem 3.3 (Theorem 8, [3]): A seaweed of type Equation, or type Equation, has index gcd(a + b; b + c) − 1.

The following result establishes that the formulas in Theorems 3.2 and 3.3 the only nontrivial linear ones that are available in the parabolic case.

Theorem 3.4 (Theorem 5.3, [11]): If m ≥ 4 and p is a seaweed of typeEquation, then there do not exist homogeneous polynomials f1, Equation of arbitrary degree, such that the index of Equation is given by gcd(f1(a1,…, am), f2(a1,…, am)).

Homotopy type

Definition 3.5: We say that a planar graph has homotopy type H(a1, a2,…,am) if its homotopy type is equivalent to the meander of type Equation. That is, a union of m non-concentric subgraphs, where each subgraph has homotopy type Equation concentric circles if ai is even, and Equation concentric circles with a point in the center if ai is odd.

Example: A planar graph with homotopy type H(5, 1, 2) is homotopically equivalent to the graph in the following Figure 3.

generalized-theory-applications-a-planar-homotopy

Figure 3: A planar graph with homotopy type H(5; 1; 2) .

When it makes sense, we define the homotopy type of a seaweed to be the homotopy type of its corresponding meander. More pointedly, we note that, unlike the index, the homotopy type of a Lie algebra g is not defined directly in terms of g's Lie Theory. It is therefore not a priori clear to what extent the homotopy type is an algebraic invariant. However, the following theorem follows from Theorem 5.3 in the recent paper by Moreau and Yakimova [11].

Theorem 3.6: Conjugate seaweed subalgebras of sl(n) have the same homotopy type.

Winding Down and the Signature

In this section, we recall two technical Lemmas See [8,9]. The first Lemma (Winding Down) can be used to discern the homotopy type of a meander. The second Lemma (Winding Up), in cooperation with the Winding-Down Lemma, will be used in the proof of the classification Theorems 5.1 and 5.2. In these theorems, we classify the homotopy types in the cases where there exist linear greatest common divisor index formulas for the associated seaweed.

The Winding-Down Lemma establishes that, through a deterministic sequence of graph theoretic moves, each meander can be contracted or “wound down” to reveal its homotopy type. Since the sequence of moves is uniquely determined by the meander, we refer to this sequence as the meander's signature. Essentially, the winding-down moves, and the attendant signature, are a graph-theoretic recasting of Panyushev's reduction algorithm which was used in [14] to develop inductive formulas for the index of seaweeds in Equation.

Lemma 4.1 (Winding Down): Given a meander M of type Equation, create a meander M′ by one of the following moves. For all moves except the Component Elimination, M and M′ have the same homotopy type.

1. Flip (F): If a1 < b1, then Equation

2. Component Elimination (C(c)): If a1 = b1 = c, then Equation of typeEquation.

3. Rotation Contraction (R): If b1 < a1 < 2b1, then Equation of typeEquation.

4. Block Elimination (B): If a1 = 2b1, then Equation of type Equation.

5. Pure Contraction (P): If a1 > 2b1, then Equation of typeEquation.

Example: We continue with our running example, and wind down the meander of Figure 2 in Figure 4 below.

generalized-theory-applications-winding-meander

Figure 4: Winding down the meander Equation, with signature RPC(4)FBC(3).

Note that each of the winding-down moves can be reversed to yield a winding-up move. The winding-up moves, which we record in the following Lemma, can be used to build up any meander, of any size and block configuration.

Lemma 4.2 (Winding Up): Every meander is the result of a sequence of the following moves applied to the empty meander. For all moves, except Component Creation, M and M′ have the same homotopy type.

Given a meander M of type Equation, create a meander M′ by one of the following moves:

1. Flip Equation of type Equation.

2. Component Creation Equation: If a1 = b1 = c, then Equation of typeEquation.

3. Rotation Expansion Equation: If b1 <a1< 2b1, then Equation of typeEquation.

4. Block Creation Equation of type Equation.

5. Pure Expansion Equation of typeEquation.

Classification Theorems

In this section, we present the promised homotopy type classification theorems. The proofs are inductive in nature and based on the Winding-Up lemma.

Theorem 5.1. If Equation is a seaweed subalgebra of Equation of type Equation or Equation, with gcd(a,b) = k, then its homotopy type is H(k).

Proof. Given any integer m≥2, we prove by induction that the theorem holds for all such seaweeds created using m winding-up moves. For the base of induction, the only seaweeds of this type that can be created by two moves are those created by Component Creation followed by Block Creation. Such a meander has type Equation. The homotopy type is H(a), and gcd(a,a) = a.

Now, let m > 2 and assume the theorem holds for all such seaweeds created using m−1 winding-up moves. Let p be a seaweed of type Equation or Equation created by m-1 winding-up moves. Also let k = gcd(a, b). Let Equation be the seaweed obtained from p by applying the appropriate windingdown move, so that Equation is created using m-1 winding-up moves. We break the proof into cases depending on which winding-down move is applied. For all cases except Component Elimination, Equation and Equation have the same homotopy type.

Case 1: A Flip is applied to Equation.

This case is trivial.

Case 2: A Component Elimination is applied to Equation.

This case is vacuously true − a Component Elimination cannot be applied to a seaweed of type Equation or Equation.

Case 3: A Rotation Contraction is applied to Equation.

In order to apply this move, Equation must have type Equation with n < 2a, thus Equation has type Equation. For any positive integers x and y we have gcd(x, y) = gcd(x, x + y). Therefore,

gcd(b, 2an) = gcd(b, b + 2an) = gcd(b, a) = k.

The claim then follows from the inductive hypothesis.

Case 4: A Block Elimination is applied to Equation.

In order to apply this move, Equation must have type Equation with n=2a, thus b = a and k = gcd(a; b) = a. The claim now follows from the fact that Equation has type Equation.

Case 5: A Pure Contraction is applied to Equation

In order to apply this move, Equation must have type Equation with n>2a, thus Equation has typeEquation. Now

gcd(a, n−2a) = gcd(a, n−2a + a) = gcd(a, b) = k.

So, the claim follows from the inductive hypothesis.

Theorem 5.2. Let Equation seaweed subalgebra of Equation of typeEquation, Equation. Let k = gcd(a + b, b + c), and let ra mod k, and sb mod k, where 0 ≤r < k and 0≤s < k.

1. If r = 0 or s = 0, then the homotopy type is H(k).

2. If r and s are nonzero, then the homotopy type is H(r, s).

Proof. Note that k divides (a + b)−(a + c) = ac, so rac mod k. In the case that a seaweed has type Equation, we have 2n = (b + c) + (a + d) and so, k = gcd(n, b + c) = gcd(n, a + d). Also, k divides (a + b)−(a + d) = bd, so sbd mod k.

Given any integer m≥2, we prove, by induction, that the theorem holds for all such seaweeds created using m winding-up moves. For the base of induction, the only seaweeds of this type that can be created by two moves are those created by two Component Creation moves. Such a meander has type Equation and the homotopy type is H(a, b). Since k = gcd(a + b, b + a) = a + b, we have r = a and s = b, as desired.

Now, let m > 2 and assume the theorem holds for all such seaweeds created using m−1 winding-up moves. Let Equation be a seaweed of type Equation , Equation created by m winding-up moves. Also, let k = gcd(a + b, b + c) and Equation be the seaweed obtained from p by applying the appropriate winding-down move, so that Equation is created using m−1 winding-up moves. We break the proof into cases depending on which windingdown move is applied. For all cases except Component Elimination, Equation and Equation have the same homotopy type.

Case 1: A Flip is applied to Equation.

This case is trivial.

Case 2: A Component Elimination is applied to Equation.

To apply this move, Equation must have type Equation. As noted above, the homotopy type is H(r, s), as desired.

Case 3: A Rotation Contraction is applied to Equation.

If Equation has type Equation, then this move cannot be applied.

If Equation has type Equation, then we must have n < 2a in order to apply this move, and Equation has type Equation. Note that

gcd(c+b, b+2an) = gcd(b+c, ac) = gcd(b+c, b+c+ac) = k.

So, by induction, Equation has homotopy type H(k) or H(r, s) depending on the values of rc mod k and sb mod k, respectively.

If Equation has type Equation, then we must have c < a < 2c in order to apply this move, and Equation has typeEquation.

Now, gcd(c+b, c+d) = gcd(c+b, n) = k, so, by induction, Equation has homotopy type H(k) or H(r; s) depending on the values of rc mod k and sd mod k, respectively.

Case 4: A Block Elimination is applied to Equation.

If Equation has typeEquation, then this move cannot be applied.

If Equation has typeEquation, then we must have n = 2a in order to apply this move. Therefore, b + c = a = n/2. Since Equation has typeEquation, we know by Theorem 5.1 that the homotopy type is H(j) where j = gcd(c,b), but since j divides b+c = a, we also see that j divides a+b, so jk. Now, since k = gcd(a+b, b+c) = gcd(n/2+b, n/2), we have that k divides b, and so k also divides a and c. This implies kj. We conclude that k = j. Note that r = s = 0 and the homotopy type of Equation is H(k), as desired.

If Equation has type ., then we must have a = 2c in order to apply this move, and Equation has typeEquation. Again, by Theorem 5.1, the homotopy type is H(j) where j = gcd(c, b). Clearly, j divides b + c, and j also divides a + b = 2c + b, so jk. Now, k divides (a + b)−(b + c) = ac = c, so k also divides b and a. This implies that kj. We conclude that k = j. Once again r = s = 0 and the homotopy type of Equation is H(k), as desired.

Case 5: A Pure Contraction is applied to Equation

If Equation has type Equation, then this move cannot be applied.

If Equation has type Equation, then we must have n > 2a in order to apply this move, and Equation has typeEquation. Now gcd(b + c, a + b) = gcd(b + c, n) = k, so, by induction, Equation has homotopy type H(k) or H(r, s) depending on the values of rc mod k and sb mod k, respectively.

If Equation has type ., then we must have a > 2c in order to apply this move, and Equation has typeEquation. Note that

gcd(a−2c+c, c+b) = gcd(ac, b+c) = gcd(ac+b+c, b+c) = k.

So, by induction, Equation has homotopy type H(k) or H(r, s) depending on the values of rc mod k and sb mod k, respectively.

The following example demonstrates that the homotopy type can sometimes distinguish between algebras when grosser invariants are unable to detect differences.

Example: The seaweeds of type Equation and Equation both have dimension (27), rank (7), and index (1), but have homotopy types H(1, 1) and H(2), respectively.

Summary and Looking Ahead

We define a meander to be a planar graph associated to a seaweed algebra which is designed in such a way that the index of the seaweed may be computed by counting the number and type of connected components of the graph. Such combinatorial formulas have been established in the Type-A case by Dergachev and A. Kirillov [4] and in the Type-C case by Coll et al. [11] (and independently by Panyushev and Yakimova [5]). These elegant formulas are difficult to apply in practice, but in certain cases can be replaced by closed form linear greatest common divisor formulas based on the flags that define the seaweed in its standard representation. For maximal parabolics in the Type-A case, this was done early on by Elashvili, and later, as a corollary to their combinatorial result, by Dergachev and A. Kirillov, and still later by Coll et al., as a consequence of a more generalized formula. The Type-C case was addressed by Coll et al. in [10]. Investigations by Karnauhova and Liebsher [13] show that these formulas are the full complement of linear greatest common divisor index formulas available in the Type-A and Type-C cases.

The simplicial homotopy type of a meander is evidently welldefined. Recent work of Moreau and Yakimova [12] establishes that the homotopy type of a Type-A seaweed is also well-defined, up to conjugation. In a forthcoming note, and after the fashion of this article, we will consider the Type-C case. In particular, we will establish that the homotopy type of a Type-C seaweed is a conjugation invariant. Furthermore, we will classify the homotopy types in all cases where linear greatest common divisor formulas for the index exist.

Finally, it follows from Joseph ([13], Theorem 8.4) that the index of a Type-B seaweed is the same as the index of a Type-C seaweed having the same flags. Consequently, the construction of Type-C meanders and related index formulas for Type-C seaweeds [10, 5], as well as results on the homotopy type of Type-C meanders, carry over Mutatas Mutandis to the Type-B case.

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