# Rings and Algebras

## New submissions

[ total of 12 entries: 1-12 ]
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### New submissions for Tue, 20 Mar 18

[1]
Title: Maximal subalgebras of the exceptional Lie algebras in low characteristic
Authors: Thomas Purslow
Subjects: Rings and Algebras (math.RA); Representation Theory (math.RT)

In this thesis we consider the maximal subalgebras of the exceptional Lie algebras in algebraically closed fields of positive characteristic. This begins with a quick recap of the article by Herpel and Stewart which considered the Cartan type maximal subalgebras in the exceptional Lie algebras for good characteristic, and then the article by Premet considering non-semisimple maximal subalgebras in good characteristic.
For $p=5$ we give an example of what appears to be a new maximal subalgebra in the exceptional Lie algebra of type $E_8$. We show that this maximal subalgebra is isomorphic to the $p$-closure of the non-restricted Witt algebra $W(1;2)$.
After this, we focus completely on characteristics $p=2$ and $p=3$ giving examples of new non-semisimple maximal subalgebras in the exceptional Lie algebras. We consider the Weisfeiler filtration associated to these maximal subalgebras and leave many open questions. There are one or two examples of simple maximal subalgebras in $F_4$ for $p=3$ and $E_8$ for $p=2$.

[2]
Title: Rings additively generated by idempotents and nilpotents
Subjects: Rings and Algebras (math.RA)

A ring R is a strongly 2-nil-clean if every element in R is the sum of two idempotents and a nilpotent that commute. A ring R is feebly clean if every element in R is the sum of two orthogonal idempotents and a unit. In this paper, strongly 2-nil-clean rings are studied with an emphasis on their relations with feebly clean rings. This work shows new interesting connections between strongly 2-nil-clean rings and weakly exchange rings

[3]
Title: Local derivations on Solvable Lie algebras
Subjects: Rings and Algebras (math.RA)

We show that in the class of solvable Lie algebras there exist algebras which admit local derivations which are not ordinary derivation and also algebras for which every local derivation is a derivation. We found necessary and sufficient conditions under which any local derivation of solvable Lie algebras with abelian nilradical and one-dimensional complementary space is a derivation. Moreover, we prove that every local derivation on a finite-dimensional solvable Lie algebra with model nilradical and maximal dimension of complementary space is a derivation.

[4]
Title: On n-Hom-Leibniz algebras and cohomology
Subjects: Rings and Algebras (math.RA); Mathematical Physics (math-ph); Quantum Algebra (math.QA)

The purpose of this paper is to provide a cohomology of $n$-Hom-Leibniz algebras. Moreover, we study some higher operations on cohomology spaces and deformations.

[5]
Title: Graded Identities and Isomorphisms on Algebras of Upper Block-Triangular Matrices
Subjects: Rings and Algebras (math.RA)

Let $G$ be an abelian group and $\mathbb{K}$ an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the $G$-gradings on upper block-triangular matrix algebras provided that $G$ is finite. We prove that their result holds for any abelian group $G$: any grading is isomorphic to the tensor product $A\otimes B$ of an elementary grading $A$ on an upper block-triangular matrix algebra and a division grading $B$ on a matrix algebra. We then consider the question of whether graded identities $A\otimes B$, where $B$ is an algebra with a division grading, determine $A\otimes B$ up to graded isomorphism. In our main result, Theorem 3, we reduce this question to the case of elementary gradings on upper block-triangular matrix algebras which was previously studied by O. M. Di Vincenzo and E. Spinelli.

### Cross-lists for Tue, 20 Mar 18

[6]  arXiv:1803.06463 (cross-list from math.QA) [pdf, ps, other]
Title: Multiplication formulas and semisimplicity for q-Schur superalgebras
Comments: 22 pages. Nagoya J. Math. (to appear)
Subjects: Quantum Algebra (math.QA); Group Theory (math.GR); Rings and Algebras (math.RA); Representation Theory (math.RT)

We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for q-Schur superalgebras. This gives a combinatorialisation of the relative norm approach developed by the first two authors. We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for q-Schur superalgebras. We also construct infinitesimal and little q-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.

[7]  arXiv:1803.06590 (cross-list from math.RT) [pdf, ps, other]
Title: Cell Decompositions for Rank Two Quiver Grassmannians
Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG); Rings and Algebras (math.RA)

We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these "truncated preprojectives" are also established. We also provide two natural combinatorial labelings for these cells. On the one hand, they are labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cells are in bijection with compatible pairs in a maximal Dyck path as predicted by the theory of cluster algebras. The natural bijection between these two labelings gives a geometric explanation for the appearance of Dyck path combinatorics in the theory of quiver Grassmannians.

[8]  arXiv:1803.06754 (cross-list from math.RT) [pdf, ps, other]
Title: Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
Subjects: Representation Theory (math.RT); Quantum Algebra (math.QA); Rings and Algebras (math.RA)

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal subcategories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part on the corresponding quantum cluster algebra structures. Moreover, we prove that our isomorphisms are specialized at $t = 1$ to the isomorphisms of (classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by other methods. As a consequence, we prove a conjecture formulated by the first author in 2002 : the multiplicities of simple modules in standard modules in the categories above for type $B_n^{(1)}$ are given by the specialization of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive.

### Replacements for Tue, 20 Mar 18

[9]  arXiv:1609.04201 (replaced) [pdf, ps, other]
Title: Quotients of orders in algebras obtained from skew polynomials with applications to coding theory
Authors: Susanne Pumpluen
Comments: The title changed from "Quotients of orders in algebras obtained from skew polynomials and possible applications" to "Quotients of orders in algebras obtained from skew polynomials with applications to coding theory". This version contains some minor corrections of the previous one, mostly typos
Subjects: Rings and Algebras (math.RA)
[10]  arXiv:1802.07219 (replaced) [pdf, ps, other]
Title: Leibniz algebras as non-associative algebras
Authors: Jorg Feldvoss
Subjects: Rings and Algebras (math.RA)
[11]  arXiv:1712.06356 (replaced) [pdf, other]
Title: On convergence of infinite matrix products with alternating factors from two sets of matrices
Authors: Victor Kozyakin
Comments: 7 pages, 13 bibliography references, expanded Introduction and Section 4 "Remarks and Open Questions", accepted for publication in Discrete Dynamics in Nature and Society
Subjects: Optimization and Control (math.OC); Rings and Algebras (math.RA)
[12]  arXiv:1802.07049 (replaced) [pdf, ps, other]
Title: The Bieri-Neumann-Strebel invariants via Newton polytopes
Authors: Dawid Kielak
Comments: 39 pages, no figures. v2. improved exposition
Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Rings and Algebras (math.RA)
[ total of 12 entries: 1-12 ]
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