arXiv subjects

Ronald Umble

Publications and source records attributed to Ronald Umble.

17 recordsLinked to original sources

arXiv2024

Gerstenhaber-Schack Bialgebras

A *Gerstenhaber-Schack (G-S) bialgebra* consists of a graded Hopf algebra $H$ together with multilinear operations $\{\omega^1_3,\omega^2_2,\omega^3_1\}\subset \{Hom^{-1}(H^{\otimes m},H^{\otimes n}): m+n=4\},$ whose sum is the degree $-1$ component of a $2$-cocycle in the G-S complex of $H$. A *G-S extension* of a graded Hopf algebra $H$ is a G-S bialgebra containing $H$. G-S extensions of $H$ are classified up to isomorphism by the degree $-1$ component of the G-S cohomology group $H_{GS}^{2}(H;H)$. We exhibit a space $X$ and a non-trivial topologically induced G-S bialgebra structure on $H^{\ast}\left( \Omega X;\mathbb{Z}_{2}\right) .$

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arXiv2022

Comparing Diagonals on the Associahedra

We prove that the formula for the diagonal approximation $\Delta_{K}$ on J. Stasheff's $n$-dimensional associahedron $K_{n+2}$ derived by the current authors in 2004 agrees with the "magical formula" for the diagonal approximation $\Delta_{K}^{\prime}$ derived by Markl and Shnider in 2006, by Loday in 2011, and by Masuda, Thomas, Tonks, and Vallette in 2021.

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arXiv2021

Computing the Dimension of a Bipartition Matrix

The dimension of a bipartition matrix (BPM) is the sum of the dimensions of its indecomposable factors. The dimension of an indecomposable BPM is the sum of its row, column, and entry dimensions. To compute these dimensions, we apply four routines of independent interest: (1) Factor a bipartition as a product of indecomposables; (2) recover a bipartition from its indecomposable factorization; (3) factor a BPM as a product of indecomposables; and (4) compute the "transpose-rotation" (the column dimension of a BPM is the row dimension of its transpose-rotation).

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arXiv2021

Framed Matrices and $A_{\infty}$-Bialgebras

We complete the construction of the biassociahedra $KK$, construct the free matrad $\mathcal{H}_{\infty}$, realize $\mathcal{H}_{\infty}$ as the cellular chains of $KK,$ and define an $A_{\infty}$-bialgebra as an algebra over $\mathcal{H}_{\infty}.$ We construct the bimultiplihedra $JJ,$ construct the relative free matrad $r\mathcal{H}_{\infty}$ as a $\mathcal{H}_{\infty}$-bimodule, realize $r\mathcal{H}_{\infty}$ as the cellular chains of $JJ$, and define a morphism of $A_{\infty}$-bialgebras as a bimodule over $\mathcal{H}_{\infty}$. We prove that the homology of every $A_{\infty}$-bialgebra over a commutative ring with unity admits an induced $A_{\infty}$-bialgebra structure. We extend the Bott-Samelson isomorphism to an isomorphism of $A_{\infty}$-bialgebras and determine the $A_{\infty} $-bialgebra structure of $H_{\ast}\left( \Omega\Sigma X;\mathbb{Q}\right) $. For each $n\geq2$, we construct a space $X_{n}$ and identify an induced nontrivial $A_{\infty}$-bialgebra operation $\omega_{2}^{n}: H^{\ast}\left(\Omega X_{n};\mathbb{Z}_{2}\right) ^{\otimes2}\rightarrow H^{\ast}\left(\Omega X_{n};\mathbb{Z}_{2}\right) ^{\otimes n}$.

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arXiv2019

Periodic Orbits on Obtuse Edge Tessellating Polygons

A periodic orbit on a frictionless billiard table is a piecewise linear path of a billiard ball that begins and ends at the same point with the same angle of incidence. The period of a primitive periodic orbit is the number of times the ball strikes a side of the table as it traverses its trajectory exactly once. In this paper we find and classify the periodic orbits on a billiard table in the shape of a 120-isosceles triangle, a 60-rhombus, a 60-90-120-kite, and a 30-right triangle. In each case, we use the edge tessellation (also known as tiling) of the plane generated by the figure to unfold a periodic orbit into a straight line segment and to derive a formula for its period in terms of the initial angle and initial position.

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arXiv2018

An $A_{\infty}$-coalgebra Structure on a Closed Compact Surface

Let $P$ be an $n$-gon with $n\geq3.$ There is a formal combinatorial $A_\infty$-coalgebra structure on cellular chains $C_*(P)$ with non-vanishing higher order structure when $n\geq5$. If $X_g$ is a closed compact surface of genus $g\geq2$ and $P_g$ is a polygonal decomposition, the quotient map $q:P_g\to X_g$ projects the formal $A_\infty$-coalgebra structure on $C_*(P_g)$ to a quotient structure on $C_*(X_g)$, which persists to homology $H_{\ast}\left( X_g;\mathbb{Z}_{2}\right) $, whose operations are determined by the quotient map $q$, and whose higher order structure is non-trivial if and only if $X_g$ is orientable or unorientable with $g\geq3$. But whether or not the $A_{\infty}$-coalgebra structure on homology observed here is topologically invariant is an open question.

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arXiv2013

Cups Products in Z2-Cohomology of 3D Polyhedral Complexes

Let $I=(\mathbb{Z}^3,26,6,B)$ be a 3D digital image, let $Q(I)$ be the associated cubical complex and let $\partial Q(I)$ be the subcomplex of $Q(I)$ whose maximal cells are the quadrangles of $Q(I)$ shared by a voxel of $B$ in the foreground -- the object under study -- and by a voxel of $\mathbb{Z}^3\smallsetminus B$ in the background -- the ambient space. We show how to simplify the combinatorial structure of $\partial Q(I)$ and obtain a 3D polyhedral complex $P(I)$ homeomorphic to $\partial Q(I)$ but with fewer cells. We introduce an algorithm that computes cup products on $H^*(P(I);\mathbb{Z}_2)$ directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in $\mathbb{R}^3$.

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arXiv2012

Morphisms of A-infinity Bialgebras and Applications

We define the notion of a relative matrad and realize the free relative matrad as a free H_\infty-bimodule structure on cellular chains of bimultiplihedra JJ={JJ_{n,m} = JJ_{m,n}}. We define a morphism G:A => B of A_\infty-bialgebras as a bimodule over H_\infty and prove that the homology of every A_\infty-bialgebra over a commutative ring with unity admits an induced A_\infty-bialgebra structure. We extend the Bott-Samelson isomorphism to an isomorphism of A_\infty-bialgebras and identify the A_\infty-bialgebra structure of H_*(ΩΣX; Q). For each n>1, we construct a space X_n and identify an induced nontrivial A_\infty-bialgebra operation ω_2^n : H^*(ΩX_n; Z_2)^2 -> H^*(ΩX_n; Z_2)^n.

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arXiv2012

A topologically induced 2-in/2-out operation on loop cohomology

We apply the Transfer Algorithm introduced in arXiv:1106.5090 to transfer an A_\infty-algebra structure that cannot be computed using the classical Basic Perturbation Lemma. We construct a space X whose (base pointed) loop cohomology H = H^*(ΩX; Z_2) comes equipped with a nontrivial operation ω: H x H --> H x H.

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arXiv2012

Tensor Products of $A_\infty$-algebras with Homotopy Inner Products

We show that the tensor product of two cyclic $A_\infty$-algebras is, in general, not a cyclic $A_\infty$-algebra, but an $A_\infty$-algebra with homotopy inner product. More precisely, we construct an explicit combinatorial diagonal on the pairahedra, which are contractible polytopes controlling the combinatorial structure of an $A_\infty$-algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic $A_\infty$-algebra can be thought of as an $A_\infty$-algebra with homotopy inner products whose higher inner products are trivial. However, the higher inner products on the tensor product of cyclic $A_\infty$-algebras are not necessarily trivial.

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arXiv2010

Matrads, Biassociahedra and A_{\infty}-Bialgebras

We introduce the notion of a matrad M = {M_{n,m}} whose submodules M_{*,1} and M_{1,*} are non-Sigma operads. We define the free matrad H_{\infty} generated by a singleton in each bidegree (m,n) and realize H_{\infty} as the cellular chains on biassociahedra KK_{n,m} = KK_{m,n}, of which KK_{n,1} = KK_{1,n} is the associahedron K_{n}. We construct the universal enveloping functor from matrads to PROPs and define an A_{\infty}-bialgebra as an algebra over H_{\infty}.

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arXiv2010

A-infinity Bialgebras of Type (m,n)

An A-infinity bialgebra of type (m,n) is a Hopf algebra H equipped with a "compatible" operation ω: H^{\otimes m} \to H^{\otimes n} of positive degree. We determine the structure relations for A-infinity bialgebras of type (m,n) and construct a purely algebraic example for each m \geq 2 and m+n \geq 4.

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arXiv2005

Structure Relations in Special A_\infty-bialgebras

We compute the structure relations in special A_\infty-bialgebras whose operations are limited to those defining the underlying A_\infty-(co)algebra substructure. Such bialgebras appear as the homology of certain loop spaces. Whereas structure relations in general A_\infty-bialgebras depend upon the combinatorics of permutahedra, only Stasheff's associahedra are required here.

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arXiv2005

The Biderivative and A_\infty-bialgebras

An A_\infty-bialgebra is a DGM H equipped with structurally compatible operations {ω^{j,i} : H^{\otimes i} --> H^{\otimes j}} such that (H,ω^{1,i}) is an A_\infty-algebra and (H,ω^{j,1}) is an A_\infty-coalgebra. Structural compatibility is controlled by the biderivative operator Bd, defined in terms of two kinds of cup products on certain cochain algebras of pemutahedra over the universal PROP U = End(TH).

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arXiv2004

Diagonals on the Permutahedra, Multiplihedra and Associahedra

We construct an explicit diagonal Δ_P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks' projection P --> K and its factorization through J. We introduce the notion of a permutahedral set Z and lift Δ_P to a diagonal on Z. We show that the double cobar construction Ω^2(C_*(X)) is a permutahedral set; consequently Δ_P lifts to a diagonal on Ω^2(C_*(X)). Finally, we apply the diagonal on K to define the tensor product of A_\infty-(co)algebras in maximal generality.

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arXiv2004

A Diagonal on the Associahedra

Let C_*(K) denote the cellular chains on the Stasheff associahedra. We construct an explicit combinatorial diagonal Δ: C_*(K) --> C_*(K) \otimes C_*(K); consequently, we obtain an explicit diagonal on the A_\infty-operad. We apply the diagonal Δto define the tensor product of A_\infty-(co)algebras in maximal generality.

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arXiv2002

The Deformation Complex For DG Hopf Algebras

Let H be a differential graded Hopf algebra over a field k. This paper gives an explicit construction of a triple cochain complex that defines the Hochschild-Cartier cohomology of H. A certain truncation of this complex is the appropriate setting for deforming H as an H(q)-structure. The direct limit of all such truncations is the appropriate setting for deforming H as a strongly homotopy associative structure. Sign complications are systematically controlled. The connection between rational perturbation theory and the deformation theory of certain free commutative differential graded algebras is clarified.

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