In this paper the Hochschild-cochain-complex of an A∞-algebra A with values in an A∞-bimodule M over A and maps between them is defined. Then, an∞-inner-product on A is defined to be an A∞-bimodule-map between A and its dual A∗. There is a graph-complex associated to A∞-algebras with ∞-inner-product.

the decomposition $gamma=bh$ of a group $gamma$ into a subset $b$ and a subgroup $h$ of $gamma$ induces, under general conditions, a group-like structure for $b$, known as a gyrogroup. the famous concrete realization of a gyrogroup, which motivated the emergence of gyrogroups into the mainstream, is the space of all relativistically admissible velocities along with a binary operation given by t...

Flat sub-Riemannian structures are local approximations — nilpotentizations — of sub-Riemannian structures at regular points. Lie algebras of symmetries of flat maximal growth distributions and sub-Riemannian structures of rank two are computed in dimensions 3, 4, and 5. 1. Sub-Riemannian structures A sub-Riemannian geometry is a triple (M,∆, 〈·, ·〉), where M is a smooth manifold, ∆ ⊂ TM is a s...

Let PN (R) be the space of all real polynomials in N variables with the usual inner product 〈 , 〉 on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form representing this inner product on the space of coefficient vectors of all polynomials in PN (R) of degree ≤ M . We exhibit two applications of this formula. First, given a...