This study of graded rings includes the first systematic account of the graded Grothendieck group, a powerful and crucial invariant in algebra which has recently been adopted to classify the Leavitt path algebras. The book begins with a concise introduction to the theory of graded rings and then focuses in more detail on Grothendieck groups, Morita theory, Picard groups and K-theory. The author extends known results in the ungraded case to the graded setting and gathers together important results which are currently scattered throughout the literature. The book is suitable for advanced undergraduate and graduate students, as well as researchers in ring theory.
A Leavitt path algebra associates to a directed graph a Z-graded algebra and in its simplest form it recovers the Leavitt algebra L(1, k). In this note, we first study this Z-grading and characterize the (Z-graded) structure of Leavitt path algebras, associated to finite acyclic graphs, Cn-comet, multi-headed graphs and a mixture of these graphs (i.e., polycephaly graphs). The last two types are examples of graphs whose Leavitt path algebras are strongly graded. We give a criterion when a Leavitt path algebra is strongly graded and in particular characterize unital Leavitt path algebras which are strongly graded completely, along the way obtaining classes of algebras which are group rings or crossed-products. In an attempt to generalize the grading, we introduce weighted Leavitt path algebras associated to directed weighted graphs which have natural Zgrading and in their simplest form recover the Leavitt algebras L(n, k). We then show that the basic properties of Leavitt path algebras can be naturally carried over to weighted Leavitt path algebras.
Abstract. Employing Bak's dimension theory, we investigate the non-stable quadratic K-group K 1,2n
Using the E-algebraic branching systems, various graded irreducible representations of a Leavitt path K-algebra L of a directed graph E are constructed. The concept of a Laurent vertex is introduced and it is shown that the minimal graded left ideals of L are generated by the Laurent vertices or the line points leading to a detailed description of the graded socle of L. Following this, a complete characterization is obtained of the Leavitt path algebras over which every graded irreducible representation is finitely presented. A useful result is that the irreducble representation V [p] induced by infinite paths tail-equivalent to an infinite path p (we call this a Chen simple module) is graded if and only if p is an irrational path. We also show that every one-sided ideal of L is graded if and only if the graph E contains no cycles. Supplementing the theorem of one of the co-authors that every Leavitt path algebra L is graded von Neumann regular, we show that L is graded self-injective if and only if L is a graded semi-simple algebra, made up of matrix rings of arbitrary size over the field K or the graded field K[x n , x −n ] where n ∈ N.
We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff groupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital left modules over the Steinberg algebra of the skew-product groupoid arising from the grading. To do this, we show that the Steinberg algebra of the skew product is graded isomorphic to a natural generalisation of the the Cohen-Montgomery smash product of the Steinberg algebra of the underlying groupoid with the grading group. In the second part of the paper, we study the minimal (that is, irreducible) representations in the category of graded modules of a Steinberg algebra, and establish a connection between the annihilator ideals of these minimal representations, and effectiveness of the groupoid.Specialising our results, we produce a representation of the monoid of graded finitely generated projective modules over a Leavitt path algebra. We deduce that the lattice of order-ideals in the K 0 -group of the Leavitt path algebra is isomorphic to the lattice of graded ideals of the algebra. We also investigate the graded monoid for Kumjian-Pask algebras of row-finite k-graphs with no sources. We prove that these algebras are graded von Neumann regular rings, and record some structural consequences of this.
We revisit localisation and patching method in the setting of Chevalley groups. Introducing certain subgroups of relative elementary Chevalley groups, we develop relative versions of the conjugation calculus and the commutator calculus in Chevalley groups G(Φ, R), rk(Φ) ≥ 2, which are both more general, and substantially easier than the ones available in the literature. For classical groups such relative commutator calculus has been recently developed by the authors in [34,33]. As an application we prove the mixed commutator formula,for two ideals a, b R. This answers a problem posed in a paper by Alexei Stepanov and the second author. O Life, you put thousand traps in my wayDare to try, is what you clearly say Omar Khayam IntroductionOne of the most powerful ideas in the study of groups of points of reductive groups over rings is localisation. It allows to reduce many important problems over arbitrary commutative rings, to similar problems for semi-local rings. Localisation comes in a number of versions. The two most familiar ones are localisation and patching, proposed by Daniel Quillen [55] and Andrei Suslin [65], and localisationcompletion, proposed by Anthony Bak [8].Originally, the above papers addressed the case of the general linear group GL(n, R). Soon thereafter, Suslin himself, Vyacheslav Kopeiko, Marat Tulenbaev, Giovanni Taddei, Leonid Vaserstein, Li Fuan, Eiichi Abe, You Hong, and others proposed workingThe work of the second author was supported by RFFI projects
This note revisits localisation and patching method in the setting of generalised unitary groups. Introducing certain subgroups of relative elementary unitary groups, we develop relative versions of the conjugation calculus and the commutator calculus in unitary groups, which are both more general, and substantially easier than the ones available in the literature. For the general linear group such relative commutator calculus has been recently developed by the first and the third authors. As an application we prove the mixed commutator formula, [EU(2n, I, Γ), GU(2n, J, ∆)] = [EU(2n, I, Γ), EU(2n, J, ∆)], for two form ideals (I, Γ) and (J, ∆) of a form ring (A, Λ). This answers two problems posed in a paper by Alexei Stepanov and the second author. 1 • Absolute standard unitary commutator formulae, Bak-Vavilov [9], Theorem 1.1 and Vaserstein-Hong You [50].• Relative unitary commutator formula at the stable level, under some additional stability assumptions, Habdank [18,19].• Relative commutator formula for the general linear group GL(n, R), 53,26]. This case is obtained, as one sets in our Theorem, A = R⊕R 0 .Observe, that in the above generality (relative, without stability conditions) our results are new already for the following familiar cases.• The case of symplectic groups Sp(2l, R), when the involution is trivial, and Λ = R.• The case of split orthogonal groups SO(2l, R), when the involution is trivial and Λ = 0.• The case of classical unitary groups SU(2l, R), when Λ = Λ max . See [20] §5.2B for further discussion on the generalised unitary groups.Actually, in § §8,9 we give another proof of Theorem 1, imitating that of [53]. Namely, we show, that Theorem 1 can be deduced from the absolute standard commutator formula by careful calculation of levels of the above commutator groups, and some group-theoretic arguments.Nevertheless, we believe that our localisation proof, based on the relative conjugation calculus and commutator calculus, we develop in § §5,6 of the present paper, and especially the calculations themselves, are of independent value, and will be used in many further applications.The paper is organised as follows. In § §2-4 we recall basic notation, and some background facts, used in the sequel. The next two sections constitute the technical core of the paper. Namely, in §5, and in §6 we develop relative unitary conjugation calculus, and relative unitary
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