Categorical Description of Some Differentiation Algorithms

Abstract

It is presented a complete categorical description of one of the main differentiation algorithms for representations of posets with involution — Differentiation II — constructed originally by the second author at the end of the 80's on the base of the matrix approach. A similar categorical description of some simpler additional differentiations is given as well.

On the apparatus of differentiation DI–DV for posets

“…In the present section we recall main results and notation regarding classification of ordinary posets and with involution [6,23,24,28,29,32].…”

“…As an example in Fig. 2, we show a poset with involution (P, ) where a < b * , a * < b, a * < b * and = {(a, a * ), (b, b * )}) [6].…”

“…Let k be an arbitrary field, U 0 be some finite-dimensional vector space over k and K ∈ be some class then U K 0 is the direct sum of |K |-copies of U 0 . In this case, the copy of U 0 which corresponds to a point x ∈ K is denoted by [6]. For each class K ∈ and each point x ∈ K , we consider the natural injection and projection:…”

“…Categorical properties of the main algorithms of differentiation have been studied by Gabriel, Zavadskij, and the first author et al Gabriel founded this line of investigation giving in 1973 the categorical properties of the algorithm of differentiation with respect to a maximal point establishing that there is a bijection between sets of indecomposable representations of corresponding quotient categories, Zavadskij proved categorical properties of the algorithm of differentiation with respect to a suitable pair of points in 1991 describing also the Auslander-Reiten quiver of posets of finite growth representation type, he also gave categorical properties of his generalization of DI to posets with relations in 2005, jointly with the first author, Zavadskij gave the categorical properties of the algorithm of differentiation DII in 2006 and the first author et al described categorical properties of some versions of the algorithms of differentiation DVII-DIX for equipped posets in 2013. We recall that the apparatus of differentiation DI, DVII, DVIII and DIX was used by Zavadskij in order to classify equipped posets of finite growth representation type [6][7][8]11,28,32].…”

The School of Kiev in Colombia; the legacy of Alexander Zavadskij

“…Gabriel gave the categorical interpretation of the algorithm of differentiation with respect to a maximal point of Nazarova and Roiter in [16]. On the other hand, in [7,9,10,34] Cañadas and Zavadskij investigated some categorical properties of some algorithms of differentiation for posets with involution, equipped posets and Figure 11. A slide of the talk given by Zavadskij in Bielefeld-1998 (ICRA) where the original idea of equipped posets is explained and Zavadskij's finite growth representation type criterion.…”