Factorizations of integer matrices as products of idempotents and nilpotents

AbstractIt is proved that for n ⩾ 3, every n ×n matrix with integer entries and determinant zero is the product of 36n + 217 idempotent matrices with integer entries

Group inversion in certain finite-dimensional algebras generated by two idempotents

Invertibility in Banach algebras generated by two idempotents can be checked with the help of a theorem by Roch, Silbermann, Gohberg, and Krupnik. This theorem cannot be used to study generalized invertibility. The present paper is devoted to group invertibility in two types of finite-dimensional algebras which are generated by two idempotents, algebras generated by two tightly coupled idempotents on the one hand and algebras of dimension at most four on the other. As a side product, the paper gives the classification of all at most four-dimensional algebras which are generated by two idempotents. (c) 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved

Classifying large N limits of multiscalar theories by algebra

We develop a new approach to RG flows and show that one-loop flows in multiscalar theories can be described by commutative but non-associative algebras. As an example related to $D$-brane field theories and tensor models, we study the algebra of a theory with $M$ $SU(N)$ adjoint scalars and its large $N$ limits. The algebraic concepts of idempotents and Peirce numbers/Kowalevski exponents are used to characterise the RG flows. We classify and describe all large $N$ limits of algebras of multiadjoint scalar models: the standard `t Hooft matrix theory limit, a `multi-matrix' limit, each with one free parameter, and an intermediate case with extra symmetry and no free parameter of the algebra, but an emergent free parameter from a line of one-loop fixed points. The algebra identifies these limits without diagrammatic or combinatorial analysis.Comment: 23 pages, 5 figures, Added qualitative discussion of: two loops for couplings with vanishing one-loop beta function, early uses of the algebra, origin of the non-associativity, and algebra for the simple O(N) mode

Idempotents, nilpotents, rank and order in finite transformation semigroups

Let E, E₁ denote, respectively, the set of singular idempotents in T[sub]n (the semigroup of all full transformations on a finite set X[sub]n = {1,..., n}) and the set of idempotents of defect 1. For a singular element in Tn, let k(),k₁ () be defined by the properties ∈ Eᵏ⁽ᵃ⁾, ∉ Eᵏ⁽ᵃ⁾⁻¹, ∈ E₁ᵏ¹⁽ᵃ⁾, ∉ E₁ᵏ¹⁽ᵃ⁾⁻¹. In this Thesis, we obtain results analogous to those of Iwahori (1977), Howie (1980), Saito (1989) and Howie, Lusk and McFadden (1990) concerning the values of k() and k₁() for the partial transformation semigroup P[sub]n. The analogue of Howie and McFadden's (1990) result on the rank of the semigroup K(n,r) = { ∈ T [sub]n: |im | ≤ r,2 ≤ r ≤ n-1} is also obtained. The nilpotent-generated subsemigroup of P[sub]n was characterised by Sullivan in 1987. In this work, we have obtained its depth and rank. Nilpotents in IO[sub]n and PO[sub]n (the semigroup of all partial one-one order-preserving maps, and all partial order-preserving maps) are studied. A characterisation of their nilpotent-generated subsemigroups is obtained. So also are their depth and rank. We have also characterised their nilpotent-generated subsemigroup for the infinite set X = {1,2,...}. The rank of the semigroup L(n,r) = {a ∈ S : |im | ≤r, 1 ≤ r ≤ n - 2} is investigated for S = O[sub]n,PO[sub]n,SPO[sub]n and I[sub]n (where O[sub]n is the semigroup of all order-preserving full transformations, SPO[sub]n the semigroup of all strictly partial order- preserving maps, and In the semigroup of one-one partial transformation)