Combinatorial index formulas for Lie algebras of seaweed type

Alex Cameron; Jr.; Matthew Hyatt; Vincent E. Coll

REVIEW 2 major objections 4 minor 1 cited by

Combinatorial index formulas for Lie algebras of seaweed type

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For a type-D seaweed algebra, the index equals 2C + P-tilde, where C counts cycles and P-tilde counts tail-meeting paths in the meander.

desk verdict Valuable type-D meander program with a repairable proof gap and a false Frobenius criterion; deserves refereeing but needs a major revision. read the letter →

arxiv 1908.03105 v1 pith:GA6JWFT3 submitted 2019-08-08 math.RA math.CO

Alex Cameron , Vincent E. Coll , Jr. , Matthew Hyatt This is my paper · ORCID

classification math.RAmath.CO MSC 17B08

keywords FrobeniusLiealgebraspecialorthogonalseaweedindexmeandertailgreatestcommondivisorformulatypeD

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper completes a program that computes the index of seaweed Lie algebras by counting pieces of an associated planar graph, now for the type-D family $so(2n)$. It proves that for a seaweed defined by two partial compositions $a$ and $b$, the index is $2C + \tilde P$, where $C$ is the number of cycles in the meander and $\tilde P$ is the number of path components that meet the distinguished tail in zero or two vertices. From this single formula the paper extracts greatest-common-divisor index formulas for low-complexity seaweeds, characterizes Frobenius type-D seaweeds, and shows exactly where closed-form gcd formulas stop being possible. A reader should care because the same meander machinery that worked in types A and C now covers the special orthogonal case, making the index an elementary counting problem rather than a computation in the dual of a Lie algebra.

What carries the argument

The load-bearing object is the meander, a planar graph whose top edges are laid out according to one composition and bottom edges according to the other, so that each vertex is incident with at most one edge of each kind and the graph decomposes into cycles and paths. For type D the paper adds a distinguished set of vertices, the tail $T_n^D(a|b)$, defined by the difference between the two composition lengths and taking one of three configurations depending on parity and on whether the first composition sums to $n$. The index formulas count cycles and tail-meeting paths of this graph, and the closed-form results additionally use the winding-down moves and homotopy types that reduce a meander to a sequence of component eliminations, together with the observation that in one-parameter cases the relevant permutation is generated by a single difference $\Delta$.

What would settle it

Take the paper's own example $p^D_{14}((5|9)/(9), III)$, whose meander has index zero. With the representative 5 for $\Delta/2 \bmod 7$, the fractional part in Theorem 5.38 is $5^5/14 \approx 0.214$, matching the Frobenius verdict; the congruent representative 12 gives $12^5/14 \approx 0.714$, which would declare the same algebra non-Frobenius. Any convention that fixes representatives must choose the first value, and a direct index calculation on any example where the two representatives straddle 1/2 settles whether the stated condition is correct.

Watch

Extended reading notes

Core claim

The paper's central discovery is a graph-theoretic formula for the index of a type-D seaweed subalgebra of $so(2n)$, stated as Theorem 5.10: if $M_n^D(a|b)$ is the meander built from the two partial compositions $a$ and $b$, and $T$ is the associated tail, then $\operatorname{ind} p_n^D(a|b) = 2C + \tilde P$, where $C$ is the number of cycles in the meander and $\tilde P$ is the number of path components containing either zero or two vertices of $T$. Equivalently, the index counts cycles of the top-bottom permutation containing zero or two tail elements. From this base formula the paper derives explicit greatest-common-divisor and congruence tests for Frobenius seaweeds, including the one-part closed forms and the three-part classifications, and proves that four-part seaweeds with a proper second parabolic cannot have their index given by any polynomial gcd formula. It also reduces non-seaweed-shaped type-D seaweeds to seaweed-shaped ones, with the index changing by zero or two, and identifies Frobenius examples among them with type-A meanders of homotopy type $H(2)$.

Load-bearing premise

The Frobenius classification for the four-tail case in Theorem 5.38 depends on treating $\Delta/2$, which is only defined modulo $n/2$, as a concrete integer when forming the fractional part in the theorem's condition; different representatives of the same residue class can change the verdict.

Editorial extensions

If this is right

A type-D seaweed is Frobenius exactly when its meander is a forest rooted in the tail, so index-zero algebras can be recognized by looking at the picture.
Tail configuration II can never be Frobenius, so the search for Frobenius algebras reduces to configurations I and III.
For one-part and three-part seaweeds, index and Frobenius status are decided by gcds and congruence or fractional-part conditions on the part sizes.
Four-part seaweeds with a proper second parabolic admit no polynomial gcd index formula, so the list of closed forms is complete at four parts.
Seaweeds without seaweed shape have index equal to that of a seaweed-shaped cousin, or that value minus two, and the Frobenius ones correspond exactly to type-A meanders of homotopy type $H(2)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The fractional-part tests in the paper can be read as a comparison of two modular inverses; this suggests they are checkable by fast modular exponentiation once a representative of $\Delta/2$ is fixed, giving a polynomial-time Frobenius test for one-parameter type-D seaweeds.
The standard reduction from type C to type B suggests that the same tail formula should yield gcd and congruence classifications for the odd orthogonal family $so(2n+1)$, which the paper does not tabulate explicitly.
The switch relating non-seaweed-shaped and seaweed-shaped algebras points to an algorithm that needs only the Dynkin subsets, not the matrix realization: replace the exceptional root, compute the seaweed-shaped index, and adjust by 0 or 2 according to whether two specified vertices lie on a path or a cycle.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Here is my read. The paper is a real step forward for the meander-index program in type D. It gives a tail-based graph formula (Thm 5.10), works out the three tail configurations, and produces a battery of closed-form gcd formulas including Frobenius criteria. The no-polynomial-gcd theorem, if made rigorous, would be a satisfying type-D analogue of Karnauhova-Liebscher. The section on type-D seaweeds that do not have seaweed shape (Thm 5.1, Thm 5.42) is genuinely new and useful.

The good news: Theorem 5.10 is proved by an induction that is mostly coherent, and the surrounding graph machinery is well organized. The authors are honest about relying on Panyushev-Yakimova and Dvorsky for the inductive index reductions; self-citation is not a problem here.

The soft spots are real. First, the proof of Thm 5.10 says that by Cor 5.9 it suffices to treat the case sum a_i = n. But Cor 5.9 is an index identity, not a statement about the graph statistic f = 2C + P~. Unless you add a lemma showing f(M^D_{n+k}(a|b)) = k + (or k-1+) f(M^C_n(a|b)), the formula is only verified in the sum a_i = n case. I do not think the gap is fatal, but it is a genuine missing step.

Second, Thm 5.38 as stated is false. The theorem compares the fractional part of (Delta/2)^(phi(n/2)-1)/n to 1/2, but the proof compares residues modulo n/2. Those two conditions are not equivalent. The reader's example n=22, a=5, b=17, c=17 satisfies the stated gcd and fractional-part conditions while the seaweed has index 2. This is a concrete counterexample, not a stylistic slip. The criterion can probably be repaired by replacing n with n/2 in the denominator, but as written it cannot stand.

Third, Thm 5.40(ii) is argued by informal reduction to type A. The intuition is plausible, but the proof is a sketch. If the theorem is important to the paper's scope, it needs a real transfer argument.

Net: the paper deserves a serious referee. The main framework is sound and the errors are local, but a referee should demand a fixed Thm 5.38, a proof of the missing comparison lemma in Thm 5.10, and a rigorous version of Thm 5.40(ii) before the classification results are quoted. I would not cite the Frobenius classification as it stands.

Referee Report

2 major / 4 minor

Summary. The paper develops meander-theoretic index formulas for seaweed subalgebras of so(2n), in parallel with earlier work in types A, B, and C. The authors introduce type-D meanders and a type-D tail with three configurations, prove a combinatorial formula for the index (Theorem 5.10), and use it to derive explicit gcd-type closed formulas and a classification of Frobenius type-D seaweeds, including a negative result in the spirit of Karnauhova-Liebscher. The paper also treats type-D seaweeds without seaweed shape and shows their indices differ from those of associated seaweed-shaped algebras by a constant.

Significance. If correct, the paper would supply a complete tail-based calculus for type-D seaweeds and would advance the program, initiated by Dergachev-Kirillov and Coll-Hyatt-Magnant, of computing indices of seaweed algebras from meander combinatorics in all classical types. The explicit gcd formulas and the type-D analogue of the no-polynomial-gcd obstruction are useful and natural contributions. However, the current version contains a false classification theorem and an unproved reduction in the proof of the central index formula, so the significance is conditional on substantial repair.

major comments (2)

[Theorem 5.38 and its proof, Section 5.4.3] The stated condition (ii) is not the condition proved. The proof compares residues modulo n/2: after the display beginning 'Let σ1 ...', the congruences are modulo n/2, and the Euler argument yields k1 and k2 modulo n/2. But condition (ii) of the theorem evaluates the fractional part of (Δ/2)^(φ(n/2)-1)/n, which is a statement modulo n. These are not equivalent. For a concrete counterexample, take n=22, a=5, b=17, c=17. Then gcd(a+b,b+c)=gcd(22,34)=2, and Δ=a-c≡10 (mod 22), so Δ/2=5; moreover (5^9)/22 has fractional part 9/22≈0.409<0.5, so the theorem's hypotheses hold. Yet the residue modulo 11 is 9, which is larger than 11/4, and the meander has index 2 rather than 0. Thus Theorem 5.38 is false as stated. The denominator in (ii), and in the scholium following the proof, should be n/2, and the proof must also handle the excluded residue n/4 explicitly.
[Theorem 5.10, proof, Section 5.4.1] The reduction to the case ∑ai=n is asserted but not proved. Corollary 5.9 is a statement about the index ind, not about the statistic f=2C+P~ used in Theorem 5.10. For a type-D meander with ∑ai<n (tail configurations I with t even, II, and III), no argument is given that f(M^D_n(a|b)) equals the type-C statistic on the reduced data plus the appropriate constant k or k−1. The induction that follows starts only after the reduction, in the case ∑ai=n. This leaves a genuine family of cases unverified in the proof of the central formula. The gap may be repairable, for instance by an explicit edge-contraction lemma in the spirit of Lemma 6.1, but the current text does not supply it.

minor comments (4)

[Proof of Theorem 5.34] The proof refers to 'equations (3) and (4)' when the displayed congruences are labeled (6) and (7).
[Scholium after Theorem 5.38] The scholium repeats the modulus error, writing φ(n) instead of φ(n/2) and dividing by n; it should be corrected together with Theorem 5.38.
[Corollary 5.9] There is a typo in the definition of t: the last summand is written as 'br' rather than 'bi', and the summation index should be made consistent.
[Theorem 5.26] The word 'Futhermore' should be 'Furthermore'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the type-D meander index formula is anchored in external inductive index formulas and prior graph-theoretic lemmas, with no input being renamed as a prediction.

full rationale

I walked the main derivation chain, focusing on Theorem 5.10 and the later gcd and Frobenius classifications. The proof of Theorem 5.10 reduces, via Corollary 5.9, to the case sum a_i = n and sum b_i <= n. Corollary 5.9 is not a restatement of the graph statistic 2C + P~; it is derived from the cited inductive formulas of Panyushev and Dvorsky (Theorems 5.6, 5.7, and 5.8), which are external results about the index, not about the meander statistic being proved. The inductive steps then compare f(G) with the index using those same external formulas and the graph-contraction moves of Lemma 6.1, cited from the authors' prior work [2]. That lemma is a parameter-free statement about meander moves; its assumptions do not include the target type-D index formula, so citing it is independent mathematical support rather than circularity. Similarly, Theorem 5.26 imports a homotopy-type characterization from [2], but that characterization concerns type-A meanders and gcd conditions, not the type-D index formula being established, and it is used after the central meander formula is already available. I find no place where a fitted parameter is later called a prediction, no quantity defined in terms of the target result, and no uniqueness claim used to force a choice that is itself the paper's conclusion. The known weak point is different: Theorem 5.10's proof does not explicitly justify the f-statistic analogue of Corollary 5.9 when sum a_i < n, and Theorem 5.38's modular condition is not well-defined as stated because Delta/2 is a residue class while the displayed real-number fractional part depends on a representative. These are correctness or completeness gaps, not circular reductions. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters or new physical entities. The 'tail' is a defined subset of meander vertices. The central formula rests on established prior theory and the type-A homotopy-type transfer.

assumptions (4)

domain assumption Any seaweed subalgebra is conjugate to a standard one, so index computations may be restricted to standard seaweeds p(Ψ|Ψ').
Invoked in Section 2 to justify working with standard seaweeds only; standard in the literature (Panyushev [14]).
standard math The type-C index formula ind = 2C + P~ with the type-C tail (Theorem 4.5) and the inductive index formulas of Panyushev and Dvorsky (Theorems 5.6-5.8) are correct.
The type-D formula in Theorem 5.10 is proved by induction on these prior results; they are cited, not reproved.
domain assumption The signature and homotopy-type machinery of Coll et al. (Lemma 6.1, Theorem 5.26) transfers from type-A meanders to type-D meanders.
Theorems 5.24, 5.27, 5.34, and 5.38 use type-A homotopy types of pA_n(a|b,c|k) to infer component structure of type-D meanders.
standard math Euler's totient theorem applies to Delta (or Delta/2) as an integer representative, so the multiplicative inverse is given by Delta^(phi(n)-1).
Used in the proofs of Theorems 5.34 and 5.38; in Theorem 5.38 the representative of Delta/2 is not well defined modulo n, which is the source of the stated condition's error.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Combinatorial index formulas for Lie algebras of seaweed type." pith.science (2026). https://pith.science/paper/GA6JWFT3

@misc{pith2026190803105,
  author       = {Pith},
  title        = {Pith review of: Combinatorial index formulas for Lie algebras of seaweed type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GA6JWFT3}},
  note         = {Machine review of arXiv:1908.03105}
}

read the original abstract

Analogous to the types A, B, and C cases, we address the computation of the index of seaweed subalgebras in the type-D case. Formulas for the algebra's index can be computed by counting the connected components of its associated meander. We focus on a set of distinguished vertices of the meander, called the tail of the meander, and using the tail, we provide comprehensive combinatorial formulas for the index of a seaweed in all the classical types. Using these formulas, we provide all general closed-form index formulas where the index is given by a polynomial greatest common divisor formula in the sizes of the parts that define the seaweed.

Figures

Figures reproduced from arXiv: 1908.03105 by the authors.

**Figure 2.** The shape of elements from p C 3 ((3) | (1)) Remark 4.2. Similar to the type-A case (see Remark 3.1), type-C seaweeds have seaweed shape: Let P Da be the subalgebra of block-diagonal matrices whose blocks have sizes a1 × a1, . . . , am × am, 2(n − ai) × 2(n − Pai), am × am, . . . , a1 × a1 and similarly for Db. A type-C seaweed has seaweed shape if it is the subalgebra of gl(n) spanned by the intersection of Da with… view at source ↗

**Figure 3.** The meander MC 12((2, 1, 2, 6) | (3, 2, 1, 2)) The following theorem is the type-C analogue of the combinatorial formula for the index of type-A seaweeds given in Theorem 3.2. Theorem 4.5 (Coll et al. [4], Theorem 4.5). Consider the seaweed p C n (a | b), and let T = Tn(a | b). The index of p C n (a | b) is equal to 2C + Pe where C is the number of cycles in MC n (a | b) and Pe is the number of connected components … view at source ↗

Figures from the paper (25 more)

**Figure 4.** Figure 4: The meander MC 14 7 | 7 11 with components highlighted The following corollary gives a necessary condition for a symplectic seaweed to have minimal index. Corollary 4.9 (Coll et al. [4], Corollary 4.7). If ind p C n (a | b) = 0, then Pai = n, and Pbi = n−r < n, and the…

**Figure 5.** Figure 5: The seaweed p D 8 ({α1, α2, α4, α5, α6, α7, α8} | {α1, α2, α3, α4, α5, α6, α7}) 5.2 Type-D seaweed-shaped seaweeds Curiously, type-D seaweeds do not necessarily have seaweed shape in their natural representation. Consequently, not all type-D seaweeds have the block tr…

**Figure 6.** Figure 6: The parabolic of so(10) with Ψ = {α1, α2, α3, α5} (left) does not have seaweed shape, while the parabolic of so(10) with Ψ = {α1, α2, α3, α4} (right) does have seaweed shape. All other type-D parabolics have seaweed shape. However, making this type of “switch” does not…

**Figure 7.** Figure 7: p D 5 ({α2, α3, α5} | {α1, α3, α4}) Remark 5.2. There is a nice visual representation for when a type-D seaweed does not have seaweed shape using a split Dynkin diagram: any of α1, ..., αn−2 can be included in either parabolic, indicated by the gray vertices, but the e…

**Figure 8.** Figure 8: A type-D seaweed without seaweed shape While a seaweed without seaweed shape does not have block triangular form from which compositions can be obtained, type-D seaweeds with seaweed shape share this property with seaweeds of all other classical types. As in types B an…

**Figure 9.** Figure 9: The meander for the seaweed p D 8 3 | 5 4 ,I 1 2 3 4 5 6 7 8 9 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

**Figure 10.** Figure 10: The meander for the seaweed p D 9 7 3 | 3 ,II 1 2 3 4 5 6 7 8 9 [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

**Figure 11.** Figure 11: The meander for the seaweed p D 9 4 | 3 | 2 2 | 2 | 2 ,III 5.4 Type-D formulas In this section, we establish a combinatorial formula for the index of a type-D seaweed analogous to Theorem 3.2 and Theorem 4.5. We use this to classify Frobenius type-D seaweeds and ex…

**Figure 12.** Figure 12: The meander for the seaweed p D 8 5 | 3 5 ,III has b = n − c. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 13.** Figure 13: The meander for the seaweed p D 10 7 | 3 5 ,III has b < n − c. 1 2 3 4 5 6 7 8 9 10 [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

**Figure 14.** Figure 14: The meander for the seaweed p D 10 4 | 6 7 ,III has b > n − c. 5.4.3 Seaweeds p D n a | b c , III The analysis of these seaweeds breaks into three cases, illustrated by the examples in Figures 12, 13, and 14, respectively. Case 1: b = n − c If b = n − c, then p …

**Figure 15.** Figure 15: The seaweed p D 9 4 | 3 | 2 6 ,III is Frobenius. This example, together with Theorem 5.18, gives some insight into Frobenius seaweeds of the form p D n a|b|k c ,III for specific k with k < n − c. Theorem 5.22. The seaweed p D n a|b|k c ,III with k = 2 or 3 is…

**Figure 16.** Figure 16: The seaweed p D 9 3 | 6 6 ,III has type-A homotopy type H(3). 1 2 3 4 5 6 7 8 9 10 [PITH_FULL_IMAGE:figures/full_fig_p017_16.png]

**Figure 17.** Figure 17: The seaweed p D 10 4 | 6 7 ,III has type-A homotopy type H(1). Letting d = k in Theorem 5.2 of [2] gives the following useful corollary. Theorem 5.26. The seaweed p = p A n a | b c | k has homotopy type H(k) if and only if gcd(a + b, b + c) = k. Futhermore, a, b an…

**Figure 18.** Figure 18: The seaweed p D 10 6 | 4 7 ,III has type-A homotopy type H(1). Comparing this example to the seaweed in [PITH_FULL_IMAGE:figures/full_fig_p018_18.png]

**Figure 19.** Figure 19: ) with ∆1 = 2, which appears two times, ∆2 = 7, which appears three times, and ∆3 = 5, which appears four times. The top-bottom map defines, in the obvious way, a permutation on the set S = {1, . . . , 9} to yield the permutation cycle σS = (4 9 2 7 5 3 8 1 6). Now, d…

**Figure 20.** Figure 20: MA 8 3 | 5 8 and its chart of ∆’s Remark 5.33. For Frobenius seaweeds p D n a | b c ,III with a + b = n and c < n, we have 19 [PITH_FULL_IMAGE:figures/full_fig_p019_20.png]

**Figure 21.** Figure 21: The wound down meander MA n a | b c | 5 As a corollary of Theorem 5.37, if the seaweed p D n a | b c ,III is Frobenius, it must have type-A homotopy type H(1, 1). Such seaweeds necessarily have gcd(a + b, b + c) = 2; however, this does not provide a sufficient char…

**Figure 22.** Figure 22: The seaweed p D 14 5 | 9 9 ,III has type-A homotopy type H(1, 1) and index zero. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_22.png]

**Figure 23.** Figure 23: The seaweed p D 22 9 | 13 17 ,III has type-A homotopy type H(1, 1) and index two. To differentiate between these, we make the following observations about the type-A meanders from the previous examples: 1. Each meander consists of two paths, a blue path and an oran…

**Figure 24.** Figure 24: The seaweeds p D 15 5 | 4 | 6 12 and p A 18 3 | 5 | 4 | 6 18 5.5 Type-D seaweeds without seaweed shape Finally, we analyze type-D seaweeds without seaweed shape. Recall that the classification from Theorem 5.1 and [PITH_FULL_IMAGE:figures/full_fig_p024_24.png]

**Figure 25.** Figure 25: The index of p D 5 ({α2, α3, α5} | {α1, α3, α4}) (left) and p D 5 ({α2, α3, α4} | {α1, α3, α4}) (right) are both one. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_25.png]

**Figure 26.** Figure 26: The index of p D 4 ({α1, α4} | {α1, α2, α3}) (left) is zero, and the index of p D 4 ({α1, α3} | {α1, α2, α3}) (right) is two. As a corollary of Theorem 5.42, we can classify Frobenius type-D seaweeds without seaweed shape. Theorem 5.44. There is a bijection between Fr…

**Figure 27.** Figure 27: The meander MA 10 3 | 7 2 | 5 | 3 has signature RBF P F P C(1)C(3). The component elimination moves in Lemma 6.1 give the homotopy type of the meander. A meander has homotopy type H(a1, a2, . . . , am) if its signature contains C(ai) exactly once for all integers i ∈ …

**Figure 28.** Figure 28: The meander for p A 10 3 | 7 2 | 5 | 3 is homotopically equivalent to the meander MA 4 1|3 1|3 . Note that each of the moves in Lemma 6.1 can be reversed to yield a “winding-up” move. These moves, which we record in the following lemma, can be used to build any meande…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The index of Lie poset algebras
math.CO 2019-08 conditional novelty 7.0 of 10

For type-A Lie poset algebras from posets of height at most two, the paper gives closed-form index formulas, classifies the Frobenius cases, and proves they are absolutely rigid.

Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages · cited by 1 Pith paper

[1]

Belavin and V

A. Belavin and V. Drinfeld. Solutions of the Classical Ya ng-Baxter Equation for Simple Lie Algebras. Funktsional. Anal. i Prilozhen, 16:1–29, 1982

work page 1982
[2]

V. Coll, A. Dougherty, M. Hyatt, and N. Mayers. Meander Gr aphs and Frobenius Seaweed Lie Algebras III. Journal of Generalized Lie Theory and Applications, 11(2), 2017

work page 2017
[3]

V. Coll, A. Giaquinto, C. Magnant, et al. Meanders and Fro benius Seaweed Lie Algebras. Journal of Generalized Lie Theory and Applications, 5, 2011

work page 2011
[4]

V. Coll, M. Hyatt, and C. Magnant. Symplectic Meanders. Communications in Algebra, 45(11):4717–4729, 2017

work page 2017
[5]

V. Coll, M. Hyatt, C. Magnant, and H. Wang. Meander Graphs and Frobenius Seaweed Lie Algebras II. Journal of Generalized Lie Theory and Applications, 9(1), 2015

work page 2015
[6]

Dergachev and A

V. Dergachev and A. Kirillov. Index of Lie Algebras of Sea weed Type. J. Lie Theory, 10(2):331–343, 2000

work page 2000
[7]

Diximier

J. Diximier. Algebres Enveloppantes. Gauthier-Villars, 1974

work page 1974
[8]

A. Dvorsky. Index of Parabolic and Seaweed Subalgebras o f so(2n). Linear Algebra and its Applications, 374:127–142, 2003

work page 2003

Show all 17 references

[9]

Elashvili

A. Elashvili. On the Index of Parabolic Subalgebras of Se misimple Lie Algebras. unpublished preprint, 1990

1990
[10]

Gerstenhaber and A

M. Gerstenhaber and A. Giaquinto. Boundary Solutions o f the Classical Yang-Baxter Equation. Letters in Mathematical Physics, 40(4):337–353, 1997

1997
[11]

Gerstenhaber and A

M. Gerstenhaber and A. Giaquinto. Graphs, Frobenius Fu nctionals, and the Classical Yang-Baxter Equation. arXiv:0808.2423, 2008

2008 arXiv
[12]

A. Joseph. On Semi-Invariants and Index for Biparaboli c (Seaweed) Algebras, I. J. Algebra, 305(1):487–515, 2006

2006
[13]

Karnauhova and S

A. Karnauhova and S. Liebscher. Connected Components o f Meanders: I. Bi-Rainbow Meanders. arXiv:1504.03099, 2015

2015 arXiv
[14]

Panyushev

D. Panyushev. Inductive Formulas for the Index of Seawe ed Lie Algebras. Moscow Mathematical Journal, 1(2):221–241, 2001

2001
[15]

Panyushev and O

D. Panyushev and O. Yakimova. On Seaweed Subalgebras an d Meander Graphs in Type C. Paciﬁc Journal of Mathematics, 285(2):485–499, 2016

2016
[16]

Panyushev and O

D. Panyushev and O. Yakimova. On Seaweed Subalgebras an d Meander Graphs in Type D. Journal of Pure and Applied Algebra, 222(02):3414–3431, 2017

2017
[17]

Tauvel and R

P. Tauvel and R. W. Yu. Sur l’indice de Certaines Alebras de Lie. Annales de l’Institute Fourier, 54:1793–1810, 2004. 27

2004

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Belavin and V

A. Belavin and V. Drinfeld. Solutions of the Classical Ya ng-Baxter Equation for Simple Lie Algebras. Funktsional. Anal. i Prilozhen, 16:1–29, 1982

work page 1982

[2] [2]

V. Coll, A. Dougherty, M. Hyatt, and N. Mayers. Meander Gr aphs and Frobenius Seaweed Lie Algebras III. Journal of Generalized Lie Theory and Applications, 11(2), 2017

work page 2017

[3] [3]

V. Coll, A. Giaquinto, C. Magnant, et al. Meanders and Fro benius Seaweed Lie Algebras. Journal of Generalized Lie Theory and Applications, 5, 2011

work page 2011

[4] [4]

V. Coll, M. Hyatt, and C. Magnant. Symplectic Meanders. Communications in Algebra, 45(11):4717–4729, 2017

work page 2017

[5] [5]

V. Coll, M. Hyatt, C. Magnant, and H. Wang. Meander Graphs and Frobenius Seaweed Lie Algebras II. Journal of Generalized Lie Theory and Applications, 9(1), 2015

work page 2015

[6] [6]

Dergachev and A

V. Dergachev and A. Kirillov. Index of Lie Algebras of Sea weed Type. J. Lie Theory, 10(2):331–343, 2000

work page 2000

[7] [7]

Diximier

J. Diximier. Algebres Enveloppantes. Gauthier-Villars, 1974

work page 1974

[8] [8]

A. Dvorsky. Index of Parabolic and Seaweed Subalgebras o f so(2n). Linear Algebra and its Applications, 374:127–142, 2003

work page 2003