Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf-Galois extensions. These connections and some of their consequences are explored in this paper. We produce several new families of solutions related in many different ways with rings, near-rings and groups. We also study the solutions of the YBE that skew braces naturally produce. We prove, for example, that the order of the canonical solution associated with a finite skew brace is even: it is two times the exponent of the additive group modulo its center.

We study series of left ideals of skew left braces that are analogs of upper central series of groups. These concepts allow us to define left and right nilpotent skew left braces. Several results related to these concepts are proved and applications to infinite left braces are given. Indecomposable solutions of the Yang–Baxter equation are explored using the structure of skew left braces.

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We describe several methods of constructing R-matrices that are dependent upon many parameters, for example unitary R-matrices and R-matrices whose entries are functions. As an application, we construct examples of R-matrices with prescribed singular values. We characterise some classes of indecomposable set-theoretic solutions of the quantum Yang-Baxter equation (QYBE) and construct R-matrices related to such solutions. In particular, we establish a correspondence between one-generator braces and indecomposable, non-degenerate involutive set-theoretic solutions of the QYBE, showing that such solutions are abundant. We show that R-matrices related to involutive, non-degenerate solutions of the QYBE have special form. We also investigate some linear algebra questions related to R-matrices.constructed using set-theoretic solutions of the quantum Yang-Baxter equation, and have only one nonzero element in each column. This type of matrices appear frequently in the literature, for example in [17], where Baxterisation of some R-matrices of this type was obtained, and in [29], where they appear as universal gates for the quantum computation related to the circuit model (see Theorem 1,[29]). In [38,24,21] they appear in connection with Braid groups and topological quantum computation (see also [7,35,36,31]). They also appear as combinatorial R-matrices in the theory of crystal bases, geometric crystals and box-bell systems. The R-matrices related to involutive, non-degenerate settheoretic solutions of the QYBE give cocycles into abelian groups [20], therefore they can be given as an input in the construction of universal R-matrices and twists for Hopf algebras, for example as in Theorem 4.2, [18].Most of the R-matrices constructed in our paper are unitary. A unitary R-matrix leads to a unitary representation of the Braid group, and the resulting unitary matrices associated to braids can be used to process quantum information [38,15,21]. In connection with the topological quantum computation, it was conjectured in [24,42] that a single unitary R-matrix can generate only finite representations of Braid groups, and in [24] it was confirmed in several important classes of R-matrices. In [40], Rowell made the following comment: "From the quantum information point of view, a unitary R-matrix can be used to directly simulate topological quantum computers on the quantum circuit model. More recently people have begun to study what extra gates one needs to supplement braiding with in order to achieve universality (see e.g. [16]). If a single R-matrix can only generate a (nearly) finite group, can an additional small gate lead to a universal gate set?". This question provides the inspiration for our construction of R-matrices with many parameters. All of our examples are locally monomial BVS (we recall the definition in Section 2).Recall that locally monomial braided vector spaces (BVS) were introduced by Galindo and Rowell in [24], and localisation was introduced by Rowell and Wang in Definition 2.3 in [42]. A related notion of br...