642 research outputs found
Symmetric Functions in Noncommuting Variables
Consider the algebra Q> of formal power series in countably
many noncommuting variables over the rationals. The subalgebra Pi(x_1,x_2,...)
of symmetric functions in noncommuting variables consists of all elements
invariant under permutation of the variables and of bounded degree. We develop
a theory of such functions analogous to the ordinary theory of symmetric
functions. In particular, we define analogs of the monomial, power sum,
elementary, complete homogeneous, and Schur symmetric functions as will as
investigating their properties.Comment: 16 pages, Latex, see related papers at
http://www.math.msu.edu/~sagan, to appear in Transactions of the American
Mathematical Societ
Magnetic fields in noncommutative quantum mechanics
We discuss various descriptions of a quantum particle on noncommutative space
in a (possibly non-constant) magnetic field. We have tried to present the basic
facts in a unified and synthetic manner, and to clarify the relationship
between various approaches and results that are scattered in the literature.Comment: Dedicated to the memory of Julius Wess. Work presented by F. Gieres
at the conference `Non-commutative Geometry and Physics' (Orsay, April 2007
The primitives and antipode in the Hopf algebra of symmetric functions in noncommuting variables
We identify a collection of primitive elements generating the Hopf algebra
NCSym of symmetric functions in noncommuting variables and give a combinatorial
formula for the antipode.Comment: 8 pages; footnote added; references added; further remarks adde
Encryption methods using formal power series rings
Recently there has been a great deal of work on noncommutative algebraic cryptography. This involves the use of noncommutative algebraic objects as the platforms for encryption systems. Most of this work, such as the Anshel-Anshel-Goldfeld scheme, the Ko-Lee scheme and the Baumslag-Fine-Xu Modular group scheme use nonabelian groups as the basic algebraic object. Some of these encryption methods have been successful and some have been broken. It has been suggested that at this point further pure group theoretic research, with an eye towards cryptographic applications, is necessary.In the present study we attempt to extend the class of noncommutative algebraic objects to be used in cryptography. In particular we explore several different methods to use a formal power series ring R << x1; :::; xn >> in noncommuting variables x1; :::; xn as a base to develop cryptosystems. Although R can be any ring we have in mind formal power series rings over the rationals Q. We use in particular a result of Magnus that a finitely generated free group F has a faithful representation in a quotient of the formal power series ring in noncommuting variables
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