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COMMUTATION RELATIONS FOR ARBITRARY QUANTUM K. R. GOODEARL
 

Summary: COMMUTATION RELATIONS FOR ARBITRARY QUANTUM
MINORS
K. R. GOODEARL
Abstract. Complete sets of commutation relations for arbitrary pairs of quan-
tum minors are computed, with explicit coefficients in closed form.
Introduction
The title of this paper begins with what may seem a misnomer the term com-
mutation relation, in current usage, does not refer to a commutativity condition,
xy = yx, but has evolved to encompass various "skew commutativity" conditions
that have proved to be useful replacements for commutativity. Older types of com-
mutation relations include conditions of the form xy - yx = z, used in defining
Weyl algebras and enveloping algebras. In quantized versions of classical algebras,
relations such as xy = qyx (known as q-commutation) appear, along with mix-
tures of both types. Thus, it has become common to refer to any equation of the
form xy = yx + z, where is a nonzero scalar, as a commutation relation for
x and y. One important use of such a relation, especially in enveloping algebras,
is that if the algebra supports a filtration such that deg(z) < deg(x) + deg(y),
then the images of x and y in the associated graded algebra, call them x and y,
commute up to a scalar: xy = yx. Similarly, the cosets of x and y modulo the
ideal generated by z commute up to . Such coset relations are key ingredients in

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics