arXiv subjects
Publications and source records attributed to Steve Shnider.
We show that the method of stochastic reduction of linear superpositions can be applied to the process of disentanglement for the spin-0 state of two spin-1/2 particles. We describe the geometry of this process in the framework of the complex projective space
Our aim is to construct a functorial tensor product of $A_\infty$-algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These construction were in fact already indicated by R. Umble and S. Saneblidze in [9]; we will try to give a more satisfactory presentation. We also prove that there does not exist an associative tensor product of $A_\infty$-algebras.
Coherence phenomena appear in two different situations. In the context of category theory the term `coherence constraints' refers to a set of diagrams whose commutativity implies the commutativity of a larger class of diagrams. In the context of algebra coherence constrains are a minimal set of generators for the second syzygy, that is, a set of equations which generate the full set of identities among the defining relations of an algebraic theory. A typical example of the first type is Mac Lane's coherence theorem for monoidal categories, an example of the second type is the result of Drinfel'd saying that the pentagon identity for the `associator' of a quasi-Hopf algebra implies the validity of a set of identities with higher instances of this associator. We show that both types of coherence are governed by a homological invariant of the operad for the underlying algebraic structure. We call this invariant the (space of) coherence constraints. In many cases these constraints can be explicitly described, thus giving rise to various coherence results, both classical and new.
The main results of our paper deal with the lifting problem for multilinear differential operators between complexes of horizontal de Rham forms on the infinite jet bundle. We answer the question when does an n-multilinear differential operator from the space of (N,0)-forms (where N is the dimension of the base) to the space of (N-s,0)-forms allow an n-multilinear extension of degree (-s,0) defined on the whole horizontal de Rham complex. To study this problem we define a differential graded operad DEnd of multilinear differential endomorphisms, which we prove to be acyclic in positive degrees (negative mapping degrees) and describe the cohomology group in degree zero in terms of the characteristic. As a corollary, we solve the lifting problem. An important application to mathematical physics is the proof of existence of a strongly homotopy Lie algebra structure extending a Lie bracket on the space of functionals.
We study one and two parameter quantizations of the function algebra on a semisimple orbit in the coadjoint representation of a simple Lie group subject to the condition that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group. We prove that the corresponding Poisson bracket must be the sum of the so-called R-matrix bracket and an invariant bracket. We classify such brackets for all semisimple orbits and show that they form a family of dimension equal to the rank equal to the second cohomology group of the orbit and then we quantize these brackets. A two parameter (or double) quantization corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on the orbit. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases.
In our paper [Markl, Shnider: Drinfel'd Algebra Deformations and the Associahedra, IMRN 1994, no. 4, 169-176] we announced a construction of a cohomology controlling deformations of quasi-coassociative (or Drinfel'd) bialgebras. The full version of the paper will appear as [Markl, Shnider: Drinfel'd Algebra Deformations, Homotopy Comodules and the Associahedra] in Trans. Amer. Math. Soc. The construction in the paper was based on very explicit arguments using deep combinatorial properties of the associahedra. The present paper gives an alternative, general nonsense approach to the construction. So, we just prove the existence of such a cohomology without explicitly constructing it. This should be compared with the two approaches to the cohomology of associative algebras: we either describe explicitly the Hochschild complex and say "Behold! this is the cohomology" or we prove the existence of a projective resolution and define the cohomology as the derived functor.
The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel'd algebra. The task is accomplished in three steps. The first step is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involve a new, highly non-trivial, construction. The first construction, essentially combinatorial, defines a differential graded Lie algebra structure on the simplicial chain complex of the associahedra. The second construction, of a more algebraic nature, is the definition of map of differential graded Lie algebras from the complex defined above to the algebra of derivations on the bar resolution. Using the existence of this map and the acyclicity of the associahedra we can define a so-called homotopy comodule structure on the bar resolution of a general Drinfeld algebra. This in turn allows us to define the desired cohomology theory in terms of a complex which consists, roughly speaking, of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution. The complex is bigraded but not a bicomplex as in the Gerstenhaber-Schack theory for bialgebra deformations. The new components of the coboundary operator are defined via the constructions mentioned above. As an application we show that the Drinfel'd deformation of the universal enveloping algebra of a simple Lie algebra is not a jump deformation. The results of the paper were announced in the paper "Drinfel'd algebra deformations and the associahedra" (IMRN, Duke Math. Journal, 4(1994), 169-176, appeared also as preprint hep-th/9312196).
We construct a cohomology theory controlling the deformations of a general Drinfel'd algebra. The picture presented here has two sides -- the combinatorial one related with the fact of the existence of a graded Lie algebra structure on the simplicial cochain complex of the associahedra, and the algebraic one related with the algebra of derivations on the bar construction.