Let α be a polynomial Poisson bivector on a finite-dimensional vector space V over C. Then Kontsevich [K97] gives a formula for a quantization f ⋆ g of the algebra S(V )∗. We give a construction of an algebra with the PBW property defined from α by generators and relations. Namely, we define an algebra as the quotient of the free tensor algebra T (V ∗) by relations xi ⊗ xj − xj ⊗ xi = Rij(~) wh...

We analyze the structure of the Malvenuto-Reutenauer Hopf algebraSSym of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. In addition, we describe the structure constants of the...

When R is a commutative ring, the minimal free resolution of a map Ra → Rb and symmetric and skew-symmetric maps Ra → Ra, under suitable generality conditions, are well known and have been developed by a series of authors. See [2, A2.6] for an overview. In this note we do an analog for the exterior algebra E = ⊕ ∧i V on a finite dimensional vector space V and general graded maps Ea → E(1)b, and...

We construct a differential and a Lie bracket on the space {Hom(A, A)},k,l≥0 for any associative algebra A. The restriction of this bracket to the space {Hom(A, A)},k≥0 is exactly the Gerstenhaber bracket. We discuss some formality conjecture related with this construction. We also discuss some applications to deformation theory. 0. There exists a well-known way (due to Jim Stasheff) to define ...

We show that the category of algebraically cofibrant objects in a combinatorial and simplicial model category A has a model structure that is left-induced from that on A. In particular it follows that any presentable model category is Quillen equivalent (via a single Quillen equivalence) to one in which all objects are cofibrant. Let A be a cofibrantly generated model category. Garner’s version...

CoCasl [11], a recently developed coalgebraic extension of the algebraic specification language Casl [2], allows for modelling systems in terms of inductive datatypes as well as of co-inductive process types. Here, we demonstrate how to specify process algebras, namely CCS [10] and CSP [8,17], within such an algebraic-coalgebraic framework. It turns out that CoCasl can deal with the fundamental...

c) restricts to an operation of type [(P )! AbstractIFace[(P ); ]]. Of course, it has to be proven that the model satis es the assertions and creationconditions of the abstract class, as expressed by the following lemma. Abstract_refine : LEMMA AbstractAssert?(abstract(c)) AND AbstractCreate?(abstract(c))(new) As an example, we can prove that DoubleCounter with parameter n re nes a counter modu...