Nonfinitely Based Varieties of Right Alternative Metabelian Algebras

Since 1976, it is known from the paper by V. P. Belkin that the variety RA₂ of right alternative metabelian (solvable of index 2) algebras over an arbitrary field is not Spechtian (contains nonfinitely based subvarieties). In 2005, S. V. Pchelintsev proved that the variety generated by the Grassmann RA₂-algebra of finite rank r over a field ?, for char(?) ≠ 2, is Spechtian iff r = 1. We construct a nonfinitely based variety 𝔐 generated by the Grassmann 𝒱-algebra of rank 2 of certain finitely based subvariety 𝒱 ? RA₂ over a field ?, for char(?) ≠ 2, 3, such that 𝔐 can also be generated by the Grassmann envelope of a five-dimensional superalgebra with one-dimensional even part.     

A New Functor from E 6-mod to E 7-mod

We find a new representation of the simple Lie algebra of type E ₇ on the polynomial algebra in 27 variables. Using this representation and Shen's idea of mixed product, we construct a new functor from E ₆-Mod to E ₇-Mod. A condition for the functor to map a finite-dimensional irreducible E ₆-module to an infinite-dimensional irreducible E ₇-module is obtained. Our general framework also gives a direct polynomial extension from irreducible E ₆-modules to irreducible E ₇-modules, which can be used to derive Gel'fand–Zetlin bases for E ₇ from those for E ₆ that can be obtained from those for D ₅ according to our earlier work.     

Monoid Valuations and Value Ordered Supervaluations

We complement two papers on supertropical valuation theory ([11 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 . Preprint at arXiv:1003.1101 .[Crossref], [Web of Science ®] , [Google Scholar]], [12 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2013 ). Dominance and transmissions in supertropical valuation theory . Commun. Algebra 41 ( 7 ): 2736 – 2782 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]) by providing natural examples of m-valuations (= monoid valuations), and afterwards of supervaluations and transmissions between them. These supervaluations have values in totally ordered supertropical semirings, and the transmissions discussed respect the orderings. We develop the basics of the theory of such semirings and transmissions.     

Tree Oriented Pullback

Given two epimorphisms of algebras A ? B and C ? B, we consider the pullback R. We introduce a particular class of algebras, the tree oriented pullback, where there is a close relationship between the category of indecomposable modules of these algebras. This leads us to prove that if A and C are hereditary algebras, then R is a tilted algebra.     

Root Systems In Finite Symplectic Vector Spaces

We study realizations of root systems in possibly degenerate symplectic vector spaces over finite fields, up to symplectic isomorphisms. The main result of this article is the classification of such realizations for the field 𝔽₂. Thereby, each root system requires a specific degree of degeneracy of the symplectic vector space. Our main motivation for this article is that for each such realization of a root system we can construct a Nichols algebra over a nonabelian group.     

Ideal Structure of Leavitt Path Algebras with Coefficients in a Unital Commutative Ring

For a (countable) graph E and a unital commutative ring R, we analyze the ideal structure of the Leavitt path algebra L _R (E) introduced by Mark Tomforde. We first modify the definition of basic ideals and then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of L _R (E), and we then determine prime graded basic ideals and left (or right) primitive graded ideals of L _R (E). In particular, when E satisfies Condition (K) and R is a field, they imply that the set of prime ideals and the set of primitive ideals of L _R (E) coincide.     

积和式及行列式的一个推广

基于对积和式性质的讨论,给出了积和式的"化长为方"计算方法;基于对积和式概念的研究,作为行列式的推广,给出了一般m×n矩阵的行式定义,讨论了行式的性质和计算方法,推广了克莱姆法则.     

效应代数的表示理论

The aim of this paper is to characterize representable and weak representable effect algebras and establish a representation theory of effect algebras. An effect algebra E is said to be representable if there exists a Hilbert space H and a monomorphism π from E into the Hilbert space effect algebra ε(H) and it is said to be weakly representable if there exists an injective morphism from E into some ε(H). It is proved that an effect algebra E with the nonempty state space S(E) is representable if and only if x, y ∈ E, f(x)+f(y) ≤ 1 implies x⊕y is defined; it is weakly representable if and only if the state space S(E) separates the points of E. Some operational properties of representable effect algebras are established, and some applications of the obtained results are listed.