Meander Graphs and Frobenius Seaweed Lie Algebras III

Coll, Vincent E.; Dougherty, A; Hyatt, Matthew; Mayers, Nicholas

doi:10.4172/1736-4337.1000266

Cited by 13 publications

(19 citation statements)

References 15 publications

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“…The Cremmer-Gervais r-matrices have explicit formulas which we describe in Section 2.2. For example when i = 1 and n = 3, we have r CG (1, 3) = 2e 12 ∧ e 32 + e 12 ∧ e 21 + e 13 ∧ e 31 + e 23 ∧ e 32 + 1 3 (e 11 − e 22 ) ∧ (e 22 − e 33 ) ∈ sl 3 ∧ sl 3 .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Subprime solutions of the classical Yang–Baxter equation

Johnson

2019

Journal of Algebra

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We introduce a new family of classical r-matrices for the Lie algebra sln that lies in the Zariski boundary of the Belavin-Drinfeld space M of quasi-triangular solutions to the classical Yang-Baxter equation. In this setting M is a finite disjoint union of components; exactly φ(n) of these components are SLn-orbits of single points. These points are the generalized Cremmer-Gervais r-matrices r i,n which are naturally indexed by pairs of positive coprime integers, i and n, with i < n. A conjecture of Gerstenhaber and Giaquinto states that the boundaries of the Cremmer-Gervais components contain r-matrices having maximal parabolic subalgebras p i,n ⊆ sln as carriers. We prove this conjecture in the cases when n ≡ ±1 (mod i). The subprime linear functionals f ∈ p * i,n and the corresponding principal elements H ∈ p i,n play important roles in our proof. Since the subprime functionals are Frobenius precisely in the cases when n ≡ ±1 (mod i), this partly explains our need to require these conditions on i and n. We conclude with a proof of the GG boundary conjecture in an unrelated case, namely when (i, n) = (5, 12), where the subprime functional is no longer a Frobenius functional.2010 Mathematics Subject Classification. 16T25; 17B62.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Some interesting families of Frobenius subalgebras of sl n include certain types of parabolic and biparabolic Lie algebras (see e.g. [3,5,13]). In this paper we focus on the maximal parabolic subalgebras p(i, n) ⊆ sl n , where the i-th negative root vector is deleted.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Subprime solutions of the classical Yang–Baxter equation

Johnson

2019

Journal of Algebra

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“…, β a1+a2+···+am−1 }. Now, following the notational conventions established in [3] define the type of the seaweed p A n (ϕ A (a) | ϕ A (b)) to be the symbol…”

Section: Seaweedsmentioning

confidence: 99%

“…We let M S denote the set of indices (i, j) such that there is a directed edge from i to j in M . (See the right side of Figure 2 where the directed edges of M are {(2, 1), (6,3), (5,4), (2,3), (4, 6)} and F MS = e * 2,1 + e * 6,3 + e * 5,4 + e * 2,3 + e * 4,6 .) The Dergachev-Kirillov functional is regular in the sense that F MS realizes the smallest possible dimension for the kernel of F ([y, −]) where F ranges over all linear functionals.…”

Section: Frobenius Functionalsmentioning

confidence: 99%

The unbroken spectrum of type-A Frobenius seaweeds

2017

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If g is a Frobenius Lie algebra, then for certain F ∈ g * the natural map g −→ g * given by x −→ F [x, −] is an isomorphism. The inverse image of F under this isomorphism is called a principal element. We show that if g is a Frobenius seaweed subalgebra of An−1 = sl(n) then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicites have a symmetric distribution. Our proof methods are constructive and combinatorial in nature.Mathematics Subject Classification 2010 : 17B20, 05E15

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“…Frobenius Lie algebras form a large class and they appear naturally in different areas. For example many parabolic and biparabolic (seaweed) subalgebras of semisimple Lie algebras are Frobenius[CGM,CMW,DY,E1,E2,E3,CV,JS,PY2,O3], including most Borel subalgebras of simple Lie algebras[EO, p. 146]. A Frobenius biparabolic Lie algebra g satisfies interesting properties.…”

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confidence: 99%

The polynomiality of the Poisson center and semi-center of a Lie algebra and Dixmier's fourth problem

Ooms

2017

Journal of Algebra

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Abstract.Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We provide necessary and also some sufficient conditions in order for its Poisson center and semi-center to be polynomial algebras over k. This occurs for instance if g is quadratic of index 2 with [g, g] = g and also if g is nilpotent of index at most 2. The converse holds for filiform Lie algebras of type L n , Q n , R n and W n . We show how Dixmier's fourth problem for an algebraic Lie algebra g can be reduced to that of its canonical truncation g Λ . Moreover, Dixmier's statement holds for all Lie algebras of dimension at most eight. The nonsolvable ones among them possess a polynomial Poisson center and semi-center.

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Meander Graphs and Frobenius Seaweed Lie Algebras III

Cited by 13 publications

References 15 publications

Subprime solutions of the classical Yang–Baxter equation

Subprime solutions of the classical Yang–Baxter equation

The unbroken spectrum of type-A Frobenius seaweeds

The polynomiality of the Poisson center and semi-center of a Lie algebra and Dixmier's fourth problem

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