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Vincent E. Coll, Jr.

Publications and source records attributed to Vincent E. Coll, Jr..

14 recordsLinked to original sources
arXiv2026

Rigidity and Cohomology of Seaweed Lie Algebras

Seaweed (biparabolic) subalgebras form a large and structurally rich class of subalgebras of simple Lie algebras. We determine their adjoint cohomology. If $\mathfrak{s}$ is an indecomposable seaweed subalgebra of a complex simple Lie algebra, then \[ H^\ast(\mathfrak{s},\mathfrak{s})=0, \] and hence $\mathfrak{s}$ is absolutely rigid. If $\mathfrak{s}$ is decomposable, then the Coll--Gerstenhaber decomposition for Lie semidirect products gives, for each $n\ge 0$, a canonical description of $H^n(\mathfrak{s},\mathfrak{s})$ in terms of exterior powers of $\mathcal{Z}(\mathfrak{s})^\ast$ and the zero-weight cohomology of $\mathfrak{s}/\mathcal{Z}(\mathfrak{s})$. In particular, the center is the unique source of nontrivial adjoint cohomology. These results identify indecomposability as the precise condition for cohomological rigidity and give a uniform description of adjoint cohomology for seaweed Lie algebras.

math.RA↗
arXiv2023

Contact seaweeds II: type C

This paper is a continuation of earlier work on the construction of contact forms on seaweed algebras. In the prequel to this paper, we show that every index-one seaweed subalgebra of $A_{n-1}=\mathfrak{sl}(n)$ is contact by identifying contact forms that arise from Dougherty's framework. We extend this result to include index-one seaweed subalgebras of $C_{n}=\mathfrak{sp}(2n)$. Our methods are graph-theoretic and combinatorial.

math.RA↗
arXiv2023

Classification of contact seaweeds

A celebrated result of Gromov ensures the existence of a contact structure on any connected, non-compact, odd dimensional Lie group. In general, such structures are not invariant under left translation. The problem of finding which Lie groups admit a left-invariant contact structure resolves to the question of determining when a Lie algebra $\mathfrak{g}$ is contact; that is, admits a one-form $φ\in\mathfrak{g}^*$ such that $φ\wedge(dφ)^k\neq 0.$ In full generality, this remains an open question; however we settle it for the important category of the evocatively named seaweed algebras by showing that an index-one seaweed is contact precisely when it is quasi-reductive. Seaweeds were introduced by Dergachev and Kirillov who initiated the development of their index theory -- since completed by Joseph, Panyushev, Yakimova, and Coll, among others. Recall that a contact Lie algebra has index one -- but not characteristically so. Leveraging recent work of Panyushev, Baur, Moreau, Duflo, Khalgui, Torasso, Yakimova, and Ammari, who collectively classified quasi-reductive seaweeds, our equivalence yields a full classification of contact seaweeds. We remark that since type-A and type-C seaweeds are de facto quasi-reductive (by a result of Panyushev), in these types index one alone suffices to ensure the existence of a contact form.

math.RA↗
arXiv2022

The breadth of Lie poset algebras

The breadth of a Lie algebra $L$ is defined to be the maximal dimension of the image of $ad_x=[x,-]:L\to L$, for $x\in L$. Here, we initiate an investigation into the breadth of three families of Lie algebras defined by posets and provide combinatorial breadth formulas for members of each family.

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arXiv2021

The unbroken spectra of Frobenius seaweeds

We show that if $\mathfrak{g}$ is a Frobenius seaweed, then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicities have a symmetric distribution. Our methods are combinatorial.

math.CO↗
arXiv2020

Regular functionals on seaweed Lie algebras

The index of a Lie algebra $\mathfrak{g}$ is defined by ind $\mathfrak{g}=$ $\min_{f\in \mathfrak{g}^*}\dim(\ker (B_f))$, where $f$ is an element of the linear dual $\mathfrak{g}^*$ and $B_f(x,y)=f([x,y])$ is the associated skew-symmetric Kirillov form. We develop a broad general framework for the explicit construction of regular (index realizing) functionals for seaweed subalgebras of $\mathfrak{gl}(n)$ and the classical Lie algebras: $A_n=\mathfrak{sl}(n+1),$ $B_n=\mathfrak{so}(2n+1)$, and $C_n=\mathfrak{sp}(2n)$. Until now, this problem has remained open in $\mathfrak{gl}(n)$ -- and in all the classical types.

math.RA↗
arXiv2020

The index and spectrum of Lie poset algebras of type B, C, and D

In this paper, we define posets of types B, C, and D. These posets encode the matrix forms of certain Lie algebras which lie between the algebras of upper-triangular and diagonal matrices. Interestingly, such type-B, C, and D Lie poset algebras can be related to Reiner's notion of a parset. Our primary concern is the index and spectral theories of type-B, C, and D Lie poset algebras. For an important restricted class, we develop combinatorial index formulas and, in particular, characterize posets corresponding to Frobenius Lie algebras. In this latter case we show that the spectrum is binary; that is, consists of an equal number of 0's and 1's.

math.CO↗
arXiv2020

The classification of flat Riemannian metrics on the plane

We classify all smooth flat Riemannian metrics on the two-dimensional plane. In the complete case, it is well-known that these metrics are isometric to the Euclidean metric. In the incomplete case, there is an abundance of naturally-arising, non-isometric metrics that are relevant and useful. Remarkably, the study and classification of all flat Riemannian metrics on the plane -- as a subject -- is new to the literature. Much of our research focuses on conformal metrics of the form $e^{2φ}g_0$, where $φ: \mathbb{R}^2 \rightarrow \mathbb{R)$ is a harmonic function and $g_0$ is the standard Euclidean metric on $\mathbb{R}^2$. We find that all such metrics, which we call "harmonic", arise from Riemann surfaces.

math.DG↗
arXiv2019

The index of Lie poset algebras

We provide general closed-form formulas for the index of type-A Lie poset algebras corresponding to posets of restricted height. Furthermore, we provide a combinatorial recipe for constructing all posets corresponding to type-A Frobenius Lie poset algebras of heights zero, one, and two. A finite Morse theory argument establishes that the simplicial realization of such posets is contractible. It then follows, from a recent theorem of Coll and Gerstenhaber, that the second Lie cohomology group of the corresponding Lie poset algebra with coefficients in itself is zero. Consequently, the Lie poset algebra is absolutely rigid and cannot be deformed. We also provide matrix representations for Lie poset algebras in the other classical types. By so doing, we are able to give examples of deformable Lie algebras which are both solvable and Frobenius. This resolves a question of Gerstenhaber and Giaquinto about the existence of such algebras.

math.CO↗
arXiv2019

Combinatorial index formulas for Lie algebras of seaweed type

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.

math.RA↗
arXiv2019

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

Analogous to the Type-$A_{n-1}=\mathfrak{sl}(n)$ case, we show that if $\mathfrak{g}$ is a Frobenius seaweed subalgebra of $B_{n}=\mathfrak{so}(2n+1)$ or $C_{n}=\mathfrak{sp}(2n)$, then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicities have a symmetric distribution.

math.RT↗
arXiv2019

The Flat Plane and a Constructive Proof of Minding's Theorem

Minding's most celebrated result is his namesake theorem of 1839 which established that all surfaces having the same constant curvature must be locally isometric. Today, Minding's theorem is a staple in differential geometry textbooks. But, to the best of our knowledge, all published proofs of it, inclusive of Minding's original argument are existential in nature. In this note, we give a constructive proof of Minding's theorem in the flat case. The proof requires only some basic facts about harmonic functions and complex analytic functions.

math.DG↗
arXiv2016

Symplectic meanders

Analogous to the sl(n) case, we address the computation of the index of seaweed subalgebras of sp(2n) by introducing graphical representations called symplectic meanders. Formulas for the algebra's index may be computed by counting the connected components of its associated meander. In certain cases, formulas for the index can be given in terms of elementary functions.

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