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Meander Graphs and Frobenius Seaweed Lie Algebras II

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Meander Graphs and Frobenius Seaweed Lie Algebras II Georgia Southern University Digital Commons@Georgia Southern Mathematical Sciences Faculty Publications Mathematical Sciences, Department of 7-29-2015 Meander Graphs and Frobenius Seaweed Lie Algebras II Vincent Coll Lehigh University, [email protected] Matthew Hyatt Pace University Colton Magnant Georgia Southern University, [email protected] Hua Wang Georgia Southern University, [email protected] Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/math-sci-facpubs Part of the Education Commons, and the Mathematics Commons Recommended Citation Coll, Vincent, Matthew Hyatt, Colton Magnant, Hua Wang. 2015. "Meander Graphs and Frobenius Seaweed Lie Algebras II." Journal of Generalized Lie Theory and Applications, 9 (1): 1-6. doi: 10.4172/ 1736-4337.1000227 https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/361 This article is brought to you for free and open access by the Mathematical Sciences, Department of at Digital Commons@Georgia Southern. It has been accepted for inclusion in Mathematical Sciences Faculty Publications by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact [email protected]. Coll et al., J Generalized Lie Theory Appl 2015, 9:1 Journal of Generalized Lie http://dx.doi.org/10.4172/1736-4337.1000227 Theory and Applications Research Article Open Access Meander Graphs and Frobenius Seaweed Lie Algebras II Vincent Coll1*, Matthew Hyatt2, Colton Magnant3 and Hua Wang3 1Department of Mathematics, Lehigh University, Bethlehem, PA, USA 2Mathematics Department, Pace University, Pleasantville, NY, USA 3Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, USA Abstract We provide a recursive classification of meander graphs, showing that each meander is identified by a unique sequence of fundamental graph theoretic moves. This sequence is called the meander’s signature and can be used to construct arbitrarily large sets of meanders, Frobenius or otherwise, of any size and configuration. In certain special cases, the signature is used to produce an explicit formula for the index of seaweed Lie subalgebra of sl(n) in terms of elementary functions. Keywords: Biparabolic; Frobenius; Lie algebra; Meander; Seaweed Seaweed algebras algebra Definition 2. Let k be an arbitrary field of characteristic0 and n k {}b Introduction a positive integer. Fix two ordered partitions {}aii=1 and jj=1 of the number n. Let {}e n be the standard basis in kn. A subalgebra of sl(n) Dergachev and A. Kirilov [1] introduced meanders as planar graph ii=1 that preserves the vector spaces {V = span (e , . , e +…+a )} and representations of bi-parabolic, or seaweed, subalgebras of sl(n) and i 1 a1 i also provided a combinatorial method of computing the index of such {Wj = span (eb1+…+bj+1, ..., en)} is called a seaweed algebra of type aa| ||… a seaweeds from the number and type of the connected components 12 due to their suggestive shape when exhibited in matrix of their associated meander graphs. Of particular interest are those bb12|||… bm seaweed algebras whose associated meander graph consists of a single form (Given in left hand side of Figure 1). path. Such algebras have index zero. More generally, algebras with Remark. A basis free definition is available but not necessary for index zero are called Frobenius and have been extensively studied in the present discussion. Also, the notion of seaweed algebra has been the context of invariant theory [2-5] and more recently from the point extended to reductive algebras by Panychev [7]. of view of quantum group theory [3]. Meanders Extending the work of Elashvilli, Coll et al. [2,6] showed that a abc|| seaweed of type was Frobenius precisely when gcd (a + b, b + Definition 3. To each seaweed algebra, a planar graph, called a n c)=1. However, the methods used there do not extend to seaweeds with meander, can be constructed as follows: Line up n vertices and label them with v , v , . , v and partition this set into two ordered partitions, more than four blocks. Also, the question of what the index actually is 1 2 n for the non-Frobenius four block case was left unaddressed. We take called top and bottom. For each part in the top (likewise bottom) we up these questions in this follow-up note, by first providing a recursive build up the graph by adding edges in the same way. This involves adding classification of meander graphs, showing that each meander is the edge from the first vertex of a part to the last vertex of the same part identified by a unique sequence of fundamental graph theoretic moves. drawn concave down (respectively concave up in the bottom part case). The edge addition is then repeated between the second vertex and the This sequence is called the meander’s signature. The signature second to last and so on within each part of both partitions. The parts of provides a fast algorithm for the computation of the index of Lie these partitions are called blocks. With top blocks of sizes a1, a2, . , aℓ algebra associated with the meander, and can therefore be used to test and bottom blocks of sizes b , b , . , b , we say such a meander has type any relatively prime conditions based on the type of the meander that 1 2 m aa12| ||… a might define Frobenius seaweed Lie algebra. In particular, we find that (Given in right hand side of Figure 1). bb12|||… bm an easy induction on the moves of the signature gives that the index abc|| Recursive Classification and Winding Down of a seaweed of type is gcd (a + b, b + c)−1. The signature also n yields, with relative ease, new infinite families of Frobenius seaweed Lie In this section, we show that any meander can be contracted or algebras of any size and type. “Wound Down” to the empty meander through a sequence of graph- Definitions In this section, we detail the notions of the index of Lie algebra, and *Corresponding author: Vincent Coll, Department of Mathematics, Lehigh seaweed algebras and the meanders associated with them. University, Bethlehem, PA, USA, Tel: 610-758-3732; E-mail: [email protected] March 14, 2015; July 26, 2015; July 29, 2015 Index Received Accepted Published Citation: Coll V, Hyatt M, Magnant C, Wang H (2015) Meander Graphs and Definition 1. Let g be Lie algebra over a field of characteristic zero. Frobenius Seaweed Lie Algebras II. J Generalized Lie Theory Appl 9: 227. * doi:10.4172/1736-4337.1000227 For any functional F g one can associate the Kirillov form BF (x, y) = F [x, y] which is skew-symmetric and bilinear. The index of g is defined Copyright: © 2015 Coll V, et al. This is an open-access article distributed under ∈ * the terms of the Creative Commons Attribution License, which permits unrestricted to be the minimum dimension of the kernel of BF as F ranges over g . The Lie algebra g is Frobenius if its index is zero. use, distribution, and reproduction in any medium, provided the original author and source are credited. J Generalized Lie Theory Appl Volume 9 • Issue 1 • 1000227 ISSN: 1736-4337 GLTA, an open access journal Citation: Coll V, Hyatt M, Magnant C, Wang H (2015) Meander Graphs and Frobenius Seaweed Lie Algebras II. J Generalized Lie Theory Appl 9: 227. doi:10.4172/1736-4337.1000227 Page 2 of 5 v1 v2 v3 v4 v5 v6 v7 | | 2|2|3 (b) A meander of type 2 2 3 (a) A seaweed of type 5|2 5|2 Figure 1: A seaweed and corresponding meander. theoretic moves; each of which is uniquely determined by the structure Remark. Although not our concern here, one of the major of the meander at the time of move application. We find that there are questions for both open and closed meanders is their enumeration. five such moves, only one of which affects the component structure Using a different family of index preserving operators on meanders of the graph and is therefore the only move capable of modifying the graphs, Dufflo and Yu have recently developed an approach which can, index of the graph, here defined to be twice the number of cycles plus in certain cases, effect this enumeration via polynomials [8]. the number of paths minus one. Dergachev and Kirillov showed that the index of the graph is precisely the index of the associated seaweed The signature subalgebra. Since the sequence of moves which contracts a meander We call the cases in Lemma 4 moves since they move one meander to the empty meander uniquely identifies the graph, we call this to another. Notice that in each of these moves except the Flip, the sequence the meander’s signature. Although developed independently, number of vertices in the graph is reduced. Also note that in each move we find that the signature is essentially a graph theoretic recasting except for the Component Elimination move, the component structure of Panyushev’s reduction algorithm, which [7] was used to develop of the meander is maintained. Given a meander, there exists a unique inductive formulas for the index of seaweeds in gl(n). Here these sequence of moves (elements of the set {F, C(c), B, R, P}) which reduce inductive formulas are expressed in terms of elementary functions, the given meander down to the empty meander. Such a list is called which are laid plain by the explicit nature of the signature. the signature of the meander. Since the only move that changes the Lemma 4 (Winding Down). If M is a general meander of type component structure is C(c), we get the following corollary, which provides a recursive classification of Frobenius meanders. aa| ||| a… a M = 123 n , bb|||| b… b Theorem 5 (Recursive classification).
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