Home Online Manual
Top
Back: Groebner bases for two-sided ideals in free associative algebras
Forward: Example of use of Letterplace
FastBack: Graded commutative algebras (SCA)
FastForward: Non-commutative libraries
Up: LETTERPLACE
Top: Singular Manual
Contents: Table of Contents
Index: Index
About: About this document

7.6.3 Letterplace correspondence

Our work as well as the name letteplace has been inspired by the work of Rota.

Already Feynman and Rota encoded the monomials (words) of the free algebra $x_{i_1} x_{i_2} \dots x_{i_m} \in K\langle x_1,\ldots,x_n \rangle$ via the double-indexed letterplace (that is encoding the letter (= variable) and its place in the word) monomials $x(i_1 \vert 1) x(i_2 \vert 2) \dots x(i_m \vert m) \in K[X\times N]$, where $X=\{x_1,\ldots,x_n\}$ and $N$ is the monoid of natural numbers, starting with 0 which cannot be used as a place.

Note, that the letterplace algebra $K[X \times N]$ is an infinitely generated commutative polynomial $K$-algebra. Since $K<$ $x_1$,..., $x_n$ $>$ is not Noetherian, it is common to perform the computations with modules up to a given degree. In that case the truncated letterplace algebra is finitely generated commutative ring.

In [LL09] a natural shifting on letterplace polynomials was introduced and used. Indeed, there is 1-to-1 correspondence between graded two-sided ideals of a free algebra and so-called letterplace ideals in the letterplace algebra, see [LL09] for details. All the computations take place in the letterplace algebra.

A letterplace monomial of length $m$ is a monomial of a letterplace algebra, such that its $m$ places are exactly 1,2,..., $m$. In particular, such monomials are multilinear with respect to places. A letterplace polynomial is an element of the $K$-vector space, spanned by letterplace monomials. A letterplace ideal is generated by letterplace polynomials subject to two kind of operations:

the $K$-algebra operations of the letterplace algebra and simultaneous shifting of places by any natural number $n$.