Mathematics

We discuss several constructions of swap polynomials, that is 2--tensor valued matrix polynomials which are multiples of the swap or switch operator.

Cancellative dimer algebras on a torus are noncommutative crepant resolutions, and in particular have many nice algebraic and homological properties. All of these properties disappear, however, for dimer algebras on higher genus surfaces. We consider a new class of quiver algebras on surfaces, called 'geodesic ghor algebras', that reduce to cancellative dimer algebras on a torus, yet continue to possess nice properties on higher genus surfaces. We show that noetherian localizations of these algebras are endomorphism rings of modules over their centers, and establish a rich interplay between their central geometry and the topology of the surface in which they are embedded.

Motivated by trying to find a new proof of Artin's theorem on positive polynomials, we state and prove a Positivstellensatz for preordered semirings in the form of a local-global principle. It relates the given algebraic order on a suitably well-behaved semiring to the geometrical order defined in terms of a probing by homomorphisms to "test algebras". We introduce and study the latter as structures intended to capture the behaviour of a semiring element in the infinitesimal neighbourhoods of a real point of the real spectrum. As first applications of our local-global principle, we prove two abstract non-Archimedean Positivstellensätze. The first one is a non-Archimedean generalization of the classical Positivstellensatz of Krivine-Kadison-Dubois, while the second one is deeper. A companion paper will use our second Positivstellensatz to derive an asymptotic classification of random walks on locally compact abelian groups. As an important intermediate result, we develop an abstract Positivstellensatz for preordered semifields which states that a semifield preorder is always the intersection of its total extensions. We also introduce quasiordered rings and develop some of their theory. While these are related to Marshall's T -modules, we argue that quasiordered rings offer an improved definition which puts them among the basic objects of study for real algebra.

Let R be a ring. It is proved that an R -module M is Baer (resp. dual Baer) if and only if every exact sequence 0→X→M→Y→0 with Y∈ Cog ( M R ) (resp. X∈ Gen ( M R ) ) splits. This shows that being (dual) Baer is a Morita invariant property. As more applications, the Baer condition for the R -module M + = Hom Z (M,Q/Z) is investigated and shown that R is a von Neumann regular ring, if R + is a Baer R -module. Baer modules with (weak) chain conditions are studied and determined when a Baer (resp. dual baer) module is a direct sum of mutually orthogonal prime (resp. co-prime) modules. Finitely generated dual Baer modules over commutative rings are studeid

A new generalized inverse for a square matrix H∈ C n×n , called CCE-inverse, is established by the core-EP decomposition and Moore-Penrose inverse H † . We propose some characterizations of the CCE-inverse. Furthermore, two canonical forms of the CCE-inverse are presented. At last, we introduce the definitions of CCE-matrices and k -CCE matrices, and prove that CCE-matrices are the same as i -EP matrices studied by Wang and Liu in [The weak group matrix, Aequationes Mathematicae, 93(6): 1261-1273, 2019].

For an n -dimensional Leibniz/Lie algebra h over a field k we introduce a new invariant A(h) , called the \emph{universal algebra} of h , as a quotient of the polynomial algebra k[ X ij |i,j=1,⋯,n] through an ideal generated by n 3 polynomials. We prove that A(h) admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on h . The new object A(h) is the key tool in answering two open problems in Lie algebra theory. First, we prove that the automorphism group Aut Lbz (h) of h is isomorphic to the group U(G(A(h ) o )) of all invertible group-like elements of the finite dual A(h ) o . Secondly, for an abelian group G , we show that there exists a bijection between the set of all G -gradings on h and the set of all bialgebra homomorphisms A(h)→k[G] . Based on this, all G -gradings on h are explicitly classified and parameterized. A(h) is also used to prove that there exists a universal commutative Hopf algebra associated to any finite dimensional Leibniz algebra h .

Recently Takahiro Yabe gave an almost complete classification of primitive symmetric 2 -generated axial algebras of Monster type. In this note, we construct a new infinite-dimensional primitive 2 -generated symmetric axial algebra of Monster type (2, 1 2 ) over a field of characteristic 5 , and use this algebra to complete the last case left open.

Let F be a finite field of charF=p and size |F|=q . Let E be the unitary infinity dimensional Grassmann algebra. In this short note, we describe the Z 2 × Z 2 -graded identities of E k ∗ ⊗E , where E k ∗ is the Grassmann algebra with a specific Z 2 -grading. In the end, we discuss about the Z 2 × Z 2 -graded GK-dimension of E k ∗ ⊗E in m variables.

We prove algebraic versions of recent results, proved by Brown, Fuller, Pitts, and Reznikoff, regarding regular and gauge-invariant ideals of graph C*-algebras. Precisely, for Leavitt path algebras of row-finite graphs, we describe the vertex set of a regular graded ideal. We show that for row finite graphs that satisfy Condition~(L), a regular ideal of the associated Leavitt path algebra is also graded. As a consequence we obtain the the quotient of the algebra by such an ideal is again a Leavitt path algebra. Finally, we show that for row-finite graphs, Condition~(L) is preserved by quotients by regular graded ideals.

In this short note, we construct a right adjoint to the functor which associates to a ring R equipped with a group action its twisted group ring. This right adjoint admits an interpretation as semilinearization, in that it sends an R -module to the group of semilinear R -module automorphisms of the module. As an immediate corollary, we provide a novel proof of the classical observation that modules over a twisted group ring are modules over the base ring together with a semilinear action.