The unbroken spectrum of Frobenius seaweeds II: type-B and type-C

Alex Cameron^∗, Vincent E. Coll, Jr.^∗∗, Matthew Hyatt^†, and Colton Magnant^††
July 23, 2019

^∗Department of Mathematics, Lehigh University, Bethlehem, PA, USA: [email protected] ^∗∗Department of Mathematics, Lehigh University, Bethlehem, PA, USA: [email protected] ^†FactSet Research Systems, New York, NY, USA: [email protected]

††

Department of Mathematics, Clayton State University, Morrow, GA, USA: [email protected]

Abstract

Analogous to the Type-A_n−1 = sl(n) case, we show that if g is a Frobenius seaweed subalgebra of B_n = so(2n + 1) or C_n = sp(2n), then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicities have a symmetric distribution.

Mathematics Subject Classification 2010: 17B20, 05E15

Key Words and Phrases: Frobenius Lie algebra, seaweed, biparabolic, principal element, Dynkin diagram, spectrum, regular functional, Weyl group

1 Introduction

Notation: All Lie algebras will be finite dimensional over C, and the Lie multiplication will be denoted by [-,-].

The index of a Lie algebra is an important algebraic invariant and, for seaweed algebras, is bounded by the algebra’s rank: ind g ≤ rk g, (see [11]). More formally, the index of a Lie algebra g is given by

ind g = min dim(ker(B_F )),

F ∈g^∗

where F is a linear form on g, and B_F is the associated skew-symmetric bilinear Kirillov form, defined by B_F (x, y) = F([x, y]) for x, y ∈ g. On a given g, index-realizing functionals are called regular and exist in profusion, being dense in both the Zariski and Euclidean topologies of g^∗.
Of particular interest are Lie algebras which have index zero. Such algebras are called Frobenius and have been studied extensively from the point of view of invariant theory [21] and are of special interest in deformation and quantum group theory stemming from their connection with the classical Yang-Baxter equation (see [13] and [14]). A regular functional F on a Frobenius Lie algebra g is called a Frobenius functional; equivalently, B_F (−, −) is non-degenerate. Suppose B_F (−, −) is non-degenerate and let [F] be the matrix of B_F (−, −) relative to some basis {x₁, . . . , x_n} of g. In [1], Belavin and Drinfeld showed that

X

−1

ij

[F] x_i ∧ x_j

i,j

is the infinitesimal of a Universal Deformation Formula (UDF) based on g. A UDF based on g can be used to deform the universal enveloping algebra of g and also the function space of any Lie group which contains g in its Lie algebra of derivations. Despite the existence proof of Belavin and Drinfel’d, only two UDF’s are known: the exponential and quasi-exponential. These are based, respectively, on the abelian [3] and non-abelian [4] Lie algebras of dimension two (see also [16]).
A Frobenius functional can be algorithmically produced as a by-product of the Kostant Cascade
(see [17] and [20]). If F is a Frobenius functional on g, then the natural map g → g^∗ defined by x → F([x, −]) is an isomorphism. The image of F under the inverse of this map is called a principal

b

element of g and will be denoted F. It is the unique element of g such that

• b
• b

F ◦ adF = F([F, −]) = F.

As a consequence of Proposition 3.1 in [22], Ooms established that the spectrum of the adjoint of a principal element of a Frobenius Lie algebra is independent of the principal element chosen to

b

compute it (see also [14], Theorem 3). Generally, the eigenvalues of adF can take on virtually any value (see [10] for examples). But, in their formal study of principal elements [15], Gerstenhaber and Giaquinto showed that if g is a Frobenius seaweed subalgebra of A_n−1 = sl(n), then the spectrum of the adjoint of a principal element of g consists entirely of integers.¹ Subsequently, the last three of the current authors showed that this spectrum must actually be an unbroken sequence of integers centered at one half [6].² Moreover, the dimensions of the associated eigenspaces are shown to have a symmetric distribution.
The goal of this paper is to establish the following theorem, which asserts that the abovedescribed spectral phenomena for type A is also exhibited in the classical type-B and type-C cases.

Theorem 1.1. If g is a Frobenius seaweed subalgebra of type B or type C (i.e., g is a subalgebra

b

of so(2n + 1) or sp(2n), respectively) and F is a principal element of g, then the spectrum of

b

adF consists of an unbroken set of integers centered at one-half. Moreover, the dimensions of the associated eigenspaces form a symmetric distribution.

Remark 1.2. In the first article of this series [6], the type-A unbroken symmetric spectrum result is established using a combinatorial argument based on the graph-theoretic meander construction of Dergachev and Kirillov [11]. Here, the combinatorial arguments heavily leverage the results of [6], but the inductions are predicated on the more sophisticated “orbit meander” construction of Joseph [19].

2 Seaweeds

Let g be a simple Lie algebra equipped with a triangular decomposition

g = u₊ ⊕ h ⊕ u₋,

(1)

¹Joseph, seemingly unaware of the type-A result of [15] used different methods to strongly extend this integrity result to all seaweed subalgebras of semisimple Lie algebras [19].
²The paper of Coll et al. appears as a folllow-up to the Lett. in Math. Physics article by Gerstenhaber and
Giaquinto [15], where they claim that the eigenvalues of the adjoint representation of a semisimple principal element of a Frobenius seaweed subalgebra of sl(n) consists of an unbroken sequence of integers. However, M. Dufflo, in a private communication to those authors, indicated that their proof contained an error.

2where h is a Cartan subalgebra of g. Let ∆ be its root system where ∆₊ are the positive roots on u₊ and ∆₋ are the negative roots roots on u₋, and let Π denote the set of simple roots of g. Given β ∈ ∆, let g_β denote its corresponding root space, and let x_β denote the element of weight α in a Chevalley basis of g. Given a subset π₁ ⊆ Π, let p_π denote the parabolic subalgebra of g generated

1

by all g_β such that −β ∈ Π or β ∈ π₁. Such a parabolic subalgebra is called standard with respect to the Borel subalgebra u₋ ⊕ h, and it is known that every parabolic subalgebra is conjugate to exactly one standard parabolic subalgebra.
Formation of a seaweed subalgebra of g requires two weakly opposite parabolic subalgebras, i.e., two parabolic subalgebras p and p^′ such that p + p^′ = g. In this case, p ∩ p^′ is called a seaweed, or in the nomenclature of Joseph [19], a biparabolic subalgebra of g. Given a subset π₂ ⊆ Π, let p⁻

π₂

denote the parabolic subalgebra of g generated by all g_β such that β ∈ Π or −β ∈ π₂. Given two subsets π₁, π₂ ⊆ Π, we have p_π + p_π = g. We now define the seaweed

• 1
• 2

p(π₁ | π₂) = p_π ∩ p⁻_π ,

1
2

which is said to be standard with respect to the triangular decomposition in (1). Any seaweed is conjugate to a standard one, so it suffices to work with standard seaweeds only. Note that an arbitrary seaweed may be conjugate to more than one standard seaweed (see [23]).
We will often assume that π₁ ∪π₂ = Π, for if not then p(π₁ | π₂) can be expressed as a direct sum of seaweeds. Additionally, we use superscripts and subscripts to specify the type and rank of the containing simple Lie algebra g. For example p^C_n (π₁ | π₂) is a seaweed subalgebra of C_n = sp(2n), the symplectic Lie algebra of rank n.
It will be convenient to visualize the simple roots of a seaweed by constructing a graph, which we call a split Dynkin diagram. Suppose g has rank n, and let Π = {α_n, . . . , α₁}, where α₁ is the exceptional root for types B and C. Draw two horizontal lines of n vertices, say v_n⁺, . . . , v₁⁺ on top and v_n⁻, . . . , v₁⁻ on the bottom. Color v_i⁺ black if α_i ∈ π₁, color v_i⁻ black if α_i ∈ π₂, and color all other vertices white. Furthermore, if α_i, α_j ∈ π₁ are not orthogonal, connect v_i⁺ and v_j⁺ with an edge in the standard way used in Dynkin diagrams. Do the same for bottom vertices according to the roots in π₂. A maximally connected component of a split Dynkin diagram is defined in the obvious way, and such a component is of type B or C if it contains the exceptional root α₁; otherwise the component is of type A. See Figure 1.

Figure 1: The split Dynkin diagram of p^C₈ ({α₈, α₇, α₆, α₃, α₂, α₁ | α₈, α₇, α₅, α₄, α₃, α₂}) Given a seaweed p(π₁ | π₂) of a simple Lie algbera g, let W denote the Weyl group of its root system ∆, generated by the reflections s_α such that α ∈ Π. For j = 1, 2 define W_π to be the

j

subgroup of W generated by s_α such that α ∈ π_j. Let w_j denote the unique longest (in the usual Coxeter sense) element of W_π , and define an action

j

(

−w_jα, for all α ∈ π_j;

i_jα = α,

for all α ∈ Π \ π_j.
Note that i_j is an involution. For components of types B and C, the longest element w_j = −id.
To visualize the action of i_j, we append dashed edges to the split Dynkin diagram of a seaweed.
Specifically, we draw a dashed edge from v_i⁺ to v_j⁺ if i₁α_i = α_j, and we draw a dashed edge from v_i⁻

3to v_j⁻ if i₂α_i = α_j. For simplicity, we will omit drawing the looped edge in the case that i_jα_i = α_i. We call the resulting graph the orbit meander of the associated seaweed. See Figure 2.

Figure 2: The orbit meander of p^C₈ ({α₈, α₇, α₆, α₃, α₂, α₁ | α₈, α₇, α₅, α₄, α₃, α₂})
It turns out that the index of p(π₁ | π₂) is governed by the orbits of the cyclic group < i₁i₂ > acting on Π (see Section 7.16 in [18]). Since we are interested in only Frobenius seaweeds, we do not require the full power of that theorem, needing only the following corollary.

Theorem 2.1 (Joseph [18], Section 7.16). Given subsets π₁, π₂ ⊆ Π such that π₁ ∪ π₂ = Π, let π_∪ = Π \ (π₁ ∩ π₂). The seaweed p(π₁ | π₂) is Frobenius if and only if every < i₁i₂ > orbit contains exactly one element from π_∪.

Example 2.2. (Frobenius seaweed) The seaweed from Figure 2 is Frobenius as the < i₁i₂ > orbits

of p^C₈ ({α₈, α₇, α₆, α₃, α₂, α₁ | α₈, α₇, α₅, α₄, α₃, α₂}) are {α₇, α₆, α₈}, {α₅, α₂}, {α₄, α₃}, {α₁}, and

each orbit contains exactly one element from π_∪ = {α₆, α₅, α₄, α₁}. Example 2.3. For an example of a seaweed that is not Frobenius, consider the orbit meander of p^A₇ ({α₇, α₆, α₅, α₄, α₃, α₂ | α₇, α₆, α₄, α₃, α₂, α₁}) shown in Figure 3 below. The < i₁i₂ > orbits {α₇, α₃} and {α₆, α₂} contain no elements from π_∪ = {α₅, α₁}, and the orbit {α₅, α₄, α₁} contains two elements from π_∪.

Figure 3: The orbit meander of p^A₇ ({α₇, α₆, α₅, α₄, α₃, α₂ | α₇, α₆, α₄, α₃, α₂, α₁})

3 Principal Elements

b

Given a Frobenius seaweed with Frobenius functional F and corresponding principal element F,

b

the eigenvalues of adF are independent of which Frobenius functional is chosen [22]. We call these eigenvalues the spectrum of the seaweed. In this section, we describe an algorithm for computing them.
Let p(π₁ | π₂) be Frobenius and F_{π ,π} be an associated Frobenius functional with principal

• 1
• 2

• b
• b

element F_{π ,π} . (We will simply write F and F when the seaweed is understood.) Let σ be a maximally connected component of either π₁ or π₂, and for convenience let σ = {α_k, α_k−1, . . . , α₁} where α₁ is the exceptional root if σ is of type B or C.

• 1
• 2

4

• b
• b

Each eigenvalue of adF can be expressed as a linear combination of elements α_i(F) where α_i is a simple root. We call such numbers simple eigenvalues. In many cases, the simple eigenvalues are determined.

b

Lemma 3.1 (Joseph [18], Section 5). In Table 1 below, the given value is α_i(F) if σ is a maximally

b

connected component of π₁, and it is −α_i(F) if σ is a maximally connected component of π₂. In either case it is assumed that α_i ∈ σ.

• b
• b

• Type
• α_i(F)
• α_i(F)

A_k : k ≥ 1
1, if i_jα_i = α_i
B_2k−1 : k ≥ 2 (−1)ⁱ⁻¹, if 1 ≤ i ≤ 2k − 1

• B_2k : k ≥ 2
• (−1)ⁱ, if 2 ≤ i ≤ 2k
• 0, if i = 1

• C_k : k ≥ 2
• 0, if 2 ≤ i ≤ k
• 1, if i = 1

b

Table 1: Values of α_i(F)
For the cases not covered by Table 1, that is, for maximally connected components of type A_k with k ≥ 2 and i_jα_i = α_i, the following lemma (which includes a corrected typo from [18]) can be applied.

Lemma 3.2 (Joseph [18], Section 5). For each equation below, σ is assumed to be a maximally connected component of π₁. If σ is a maximally connected component of π₂, then replace α_i → −α_i and i₁ → i₂.
Let σ be of type A, then

(

1, if σ is of type A_k and (α_i, i₁α_i) < 0;

b

α_i(F) + i₁α_i(F) =

b

0, if σ is of type A_k and (α_i, i₁α_i) = 0.

Notice σ contains α_i with (α_i, i₁α_i) < 0 only when σ is of type A_2k with k ≥ 1.
Example 3.3. Using Table 1 and applying Lemma 3.2, we compute the simple eigenvalues of the seaweed p^C₈ ({α₈, α₇, α₆, α₃, α₂, α₁ | α₈, α₇, α₅, α₄, α₃, α₂}). See Figure 4 where the simple eigenvalues are noted above or below the appropriate vertex in the orbit meander for this seaweed.