Good Gradings Research Articles

Very good gradings on matrix rings are epsilon-strong

We investigate properties of group gradings on matrix rings M n ( R ) , where R is an associative unital ring and n is a positive integer. More precisely, we introduce very good gradings and show that any very good grading on M n ( R ) is necessarily epsilon-strong. We also identify a condition that is sufficient to guarantee that M n ( R ) is an epsilon-crossed product, i.e. isomorphic to a crossed product associated with a unital twisted partial action. In the case where R has IBN, we provide a characterization of when M n ( R ) is an epsilon-crossed product. Our results are illustrated by several examples.

The implementation of a digital group intervention for individuals with subthreshold borderline personality disorder

Context: Systems Training for Emotional Predictability and Problem Solving for Emotional Intensity (STEPPS-EI), a 13-week skills-based group intervention for individuals with subthreshold borderline personality disorder (BPD) has been deemed feasible and clinically effective in primary care [1] [2]. To modernize the service, STEPPS-EI lesson content has recently built onto an eHealth platform (Minddistrict). Due to Covid-19 restrictions, group sessions were additionally delivered remotely via Zoom. This project evaluates the implementation of this digitally blended version of STEPPS-EI within two Sussex Partnership NHS Foundation Trust (SPFT) primary care services.Methods: Service users and pracitioners who participated in the first two groups from March to July 2021 were invited to take part in a feasibility evaluation investigating recruitment, retention, and attendance rates, in addition to self-reported symptoms (BSL-23, QuEST), quality of life (ReQoL), system usability, and qualitative and quantitative measures designed to shed light on the experience and opinions of service users and practitioners during the intervention. Service users participating in following groups (from July to December 2021) were invited to share their symptom and quality of life outcome data only.Results: 14 service users and 5 practitioners agreed to take part in the primary evaluation. Results suggested that 86% of these service users attended at least 75% of the group sessions, and that service users completed on average 70% of the online material. Usability ratings revealed good gradings for Zoom from all participants, yet lower gradings for Minddistrict. Further analyses revealed a generally positive attitude towards digital STEPPS-EI from all parties and practical suggestions on how to improve the intervention. 11 service users from following groups agreed to share their data. Bayesian analyses were conducted for the data of service users who provided ratings at both timepoints. Evidence was found for a decrease in BSL-23 scores and an increase in ReQoL ratings from baseline to post-intervention. Incomplete self-report data-sets limits conclusions.Conclusions: It was found that the implementation of STEPPS-EI delivered in a blended digital format was feasible. The online delivery might increase service users' engagement with the material and group sessions. Yet more training and support on the use of Minddistrict may be required to increase usability.Implications: It may be possible to effectively implement digital interventions for individuals with subthreshold BPD. However, more research on the effect of these on symptom outcomes should follow.

Induced good gradings of structural matrix rings

Our approach to structural matrix rings defines them over preordered directed graphs. A grading of a structural matrix ring is called a good grading if its standard unit matrices are homogeneous. For a group G, a G-grading set is a set of arrows with the property that any assignment of these arrows to elements of G uniquely determines an induced good grading. One of our main results is that a G-grading set exists for any transitive directed graph if G is a group of prime order. This extends a result of Kelarev. However, an example of Molli Jones shows there are directed graphs which do not have G-grading sets for any cyclic group G of even order greater than 2. Finally, we count the number of nonequivalent elementary gradings by a finite group of a full matrix ring over an arbitrary field.

Classifying good gradings on structural matrix algebras

ABSTRACTLet k be a field and let be a partial order on the set . If G is a group, we classify the good G-gradings on the associated structural matrix algebra as the orbits of the action of the automorphism group of on a certain set of functions. We explicitly count the number of isomorphism types of good gradings in some cases where G is finite and the graph associated with has a cyclic associated undirected graph.

A cohomological point of view on gradings on algebras with multiplicative basis

In this paper we generalize the concepts of good and elementary gradings for an associative algebra A with a fixed multiplicative basis B. When the group G considered in the grading is abelian, we equip the set of good G-gradings of A with a structure of abelian group, which is denoted by G(B,G). Moreover, when A admits elementary G-gradings, we show the set E(B,G) of all elementary G-gradings of A is a subgroup of G(B,G). In this case, we introduce a cohomology for the pair (A,B) and we show that G(B,G)/E(B,G) is isomorphic to the first cohomology group of (A,B) with coefficients in G.

Corrigendum: Good gradings of generalized incidence rings

There is a gap in the proof of [1, Theorm 3.4(2)]. Example 1 shows the unstated condition R⊆S1 is necessary.Example 1. Suppose G is a group and R is a G-graded ring such that 1∈R1 and Rg≠0 for some...

Group Gradings on Polynomial Algebras

Let k be a field. We consider gradings on a polynomial algebra k[X1,…, Xn] by an arbitrary abelian group G, such that the indeterminates are homogeneous elements of nontrivial degree. We classify the isomorphism types of such gradings, and we count them in the case where G is finite. We present some examples of good gradings and find a minimal set of generators of the subalgebra of elements of trivial degree.

Good Gradings on Upper Block Triangular Matrix Algebras

We investigate group gradings of upper block triangular matrix algebras over a field such that all the matrix units lying there are homogeneous elements. We describe these gradings as endomorphism algebras of graded flags and classify them as orbits of a certain biaction of a Young subgroup and the group G on the set G n , where G is the grading group and n is the size of the matrix algebra. In particular, the results apply to algebras of upper triangular matrices.

Good Gradings of Generalized Incidence Rings

This inquiry is based on both the construction of generalized incidence rings due to Gene Abrams and the construction of good group gradings of incidence algebras due to Molli Jones. We provide conditions for a generalized incidence ring to be graded isomorphic to a subring of an incidence ring over a preorder. We also extend Jones's construction to good group gradings for incidence algebras over preorders with crosscuts of length one or two.

Good gradings of basic Lie superalgebras

We classify good ℤ-gradings of basic Lie superalgebras over an algebraically closed field \(\mathbb{F}\) of characteristic zero. Good ℤ-gradings are used in quantum Hamiltonian reduction for affine Lie superalgebras, where they play a role in the construction of super W-algebras. We also describe the centralizer of a nilpotent even element and of an \(\mathfrak{s}\mathfrak{l}_2\)-triple in \(\mathfrak{g}\mathfrak{l}\left( {\left. m \right|n} \right)\) and \(\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {\left. m \right|2n} \right)\).

Group Gradings in Incidence Algebras

If X is a bounded countable locally finite partially ordered set, R an integral domain, and G a group having the property that no non-identity element has order a unit of R. Then it is shown that any G-grading of the incidence algebra I(X, R) is equivalent to a good grading. Further, an example is given showing that not all group gradings of incidence algebras are equivalent to good gradings.

Elementary and Good Group Gradings of Incidence Algebras over Partially-Ordered Sets with Cross-Cuts

Let G be a group, X a locally-finite partially-ordered set, and F a field. We provide an algorithmic method for finding all good G-gradings of the incidence algebra I(X,F) when X has a cross-cut of length one or two. In these cases, we show that the good gradings are determined by a “freeness” property. It is shown that the number of good gradings of the incidence algebra I(X,F), when X has a cross-cut of length one or two, depends only on the size of G and not on its structure, but this is no longer true when the shortest cross-cut of X has length greater than two. If X has a cross-cut of length one, then every good grading of I(X,F) is an elementary grading, but, when the shortest cross-cut of X has length greater than one, there may exist good gradings of I(X,F) that are not elementary gradings. Finally, we establish bounds on the number of good gradings of I(X,F) for any finite partially-ordered set X.

Group Gradings on Full Matrix Rings

We study G-gradings of the matrix ring Mn(k), k a field, and give a complete description of the gradings where all the elements ei,j are homogeneous, called good gradings. Among these, we determine the ones that are strong gradings or crossed products. If G is a finite cyclic group and k contains a primitive |G|th root of 1, we show how all G-gradings of Mn(k) can be produced. In particular we give a precise description of all C2-gradings of M2(k) and show that for algebraically closed k, any such grading is isomorphic to one of the two good gradings.