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- SAMSON SANEBLIDZE, RONALD UMBLE
- 2002

We construct an explicit diagonal on the permutahedra {Pn}. Related diagonals on the multiplihedra {Jn} and the associahedra {Kn} are induced by Tonks' projection θ : Pn → K n+1 [19] and its factorization through Jn. We use the diagonal on {Kn} to define the tensor product of A∞-(co)algebras. We introduce the notion of a permutahedral set Z, observe that… (More)

- RONALD UMBLE
- 2000

To Jim Stasheff on the occasion of his 65th birthday

- RONALD N. UMBLE
- 1996

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… (More)

- RONALD UMBLE
- 2004

An A∞-bialgebra is a DGM H equipped with structurally compatible operations ω j,i : H ⊗i → H ⊗j such that H, ω 1,i is an A∞-algebra and H, ω j,1 is an A∞-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… (More)

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: MSC: Primary: 18D50 (operads); 55P20 (Eilenberg–Mac Lane spaces); 55P43 (spectra with… (More)

- Rocío González-Díaz, Javier Lamar, Ronald Umble
- IWCIA
- 2011

Let I be a 3D digital image, and let Q(I) be the associated cubical complex. In this paper we show how to simplify the combina-torial structure of Q(I) and obtain a homeomorphic cellular complex P (I) with fewer cells. We introduce formulas for a diagonal approximation on a general polygon and use it to compute cup products on the cohomology H * (P (I)).… (More)

- RONALD UMBLE
- 2005

We introduce the notion of a matron M = {Mn,m} whose sub-modules M * ,1 and M 1, * are non-Σ operads. We construct a functor from PROP to matrons and its inverse, the universal enveloping functor. We define the free matron H∞, generated by a singleton in each bidegree (m, n) = (1, 1), and define an A∞-bialgebra as an algebra over H∞. We realize H∞ as the… (More)

- Ronald N. Umble
- 2009

- RONALD UMBLE
- 2005

We compute the structure relations in special A∞-bialgebras whose operations are limited to those defining the underlying A∞-(co)algebra sub-structure. Such bialgebras appear as the homology of certain loop spaces. Whereas structure relations in general A∞-bialgebras depend upon the com-binatorics of permutahedra, only Stasheff's associahedra are required… (More)

An A∞-bialgebra of type (m, n) is a Hopf algebra H equipped with a " compatible " operation ω n m : H ⊗m → H ⊗n of positive degree. We determine the structure relations for A∞-bialgebras of type (m, n) and construct a purely algebraic example for each m ≥ 2 and m + n ≥ 4. Let R be a (graded or ungraded) commutative ring with unity, and let m and n be… (More)