Mathematics

We describe derivations and automorphisms of infinite tensor products of matrix algebras. Using this description we show that for a countable--dimensional locally matrix algebra A over a field F the dimension of the Lie algebra of outer derivations of A and the order of the group of outer automorphisms of A are both equal to |F | ℵ 0 , where |F| is the cardinality of the field F.

The space of derivations of finite dimensional evolution algebras associated to graphs over a field with characteristic zero has been completely characterized in the literature. In this work we generalize that characterization by describing the derivations of this class of algebras for fields of any characteristic.

Let R[G] be the group ring of a group G over an associative ring R with unity such that all prime divisors of orders of elements of G are invertible in R. If R is finite and G is a Chernikov (torsion FC-) group, then each R-derivation of R[G] is inner. Similar results also are obtained for other classes of groups G and rings R.

In this paper we establish decomposition theorems for derivations of group rings. We provide a topological technique for studying derivations of a group ring A[G] in case G has finite conjugacy classes. As a result, we describe all derivations of algebra A[G] for the case when G is a finite group, or G is an FC-group. In addition, we describe an algorithm to explicitly calculate all derivations of a group ring A[G] in case G is finite. As examples, derivations of Z 4 [ S 3 ] and F 2 m D 2n are considered.

The existence of a full strong exceptional sequence in the derived category of a smooth quadric hypersurface was proved by Kapranov. In this paper, we present a skew generalization of this result. Namely, we show that if S is a standard graded (±1) -skew polynomial algebra in n variables with n≥3 and f= x 2 1 +⋯+ x 2 n ∈S , then the derived category D b (qgrS/(f)) of the noncommutative scheme qgrS/(f) has a full strong exceptional sequence. The length of this sequence is given by n−2+ 2 r where r is the nullity of a certain matrix over F 2 . As an application, by studying the endomorphism algebra of this sequence, we obtain the classification of D b (qgrS/(f)) for n=3,4 .

The classical theorem of Milnor on pullback rings states that the category of projective modules over a pullback ring is equivalent to a certain category of gluing triples consisting of projective modules. We prove an analogous result on the level of derived categories, where the equivalence has to be replaced by an epivalence.

Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field F q are studied. The number of n×n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n . Subsequently, the enumeration of nonsingular n×n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n,q)=1 . In some cases, the number of singular n×n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinant of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted

Evolution algebras are non-associative algebras that describe non-Mendelian hereditary processes and have connections with many other areas. In this paper we obtain necessary and sufficient conditions for a given algebra A to be an evolution algebra. We prove that the problem is equivalent to the so-called SDC problem , that is, the simultaneous diagonalisation via congruence of a given set of matrices. More precisely we show that an n -dimensional algebra A is an evolution algebra if, and only if, a certain set of n symmetric n?n matrices { M 1 ,?? M n } describing the product of A are SDC . We apply this characterisation to show that while certain classical genetic algebras (representing Mendelian and auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations of these are evolution algebras. This is intriguing as evolution algebras model asexual reproduction unlike the classical ones.

This is the second paper devoted to construction of finitely presented infinite nil semigroup with identity x 9 =0 . This construction answers to the problem of Lev Shevrin and Mark Sapir. In the first part we constructed the sequence of complexes with some set of properties. Namely, all these complexes are uniform elliptic: any two points A and B with distance d can be connected with a system of shortest paths forming a disk of width λ⋅D for some global constant λ>0 . In the second part of the proof, a finite system of colors with determinism is introduced: for each minimum square that the complex consists of, the color of the three angles determines the color of the fourth corner. The present paper is devoted to the second part of the proof.

Extending work of Saneblidze-Umble and others, we use diagonals for the associahedron and multiplihedron to define tensor products of A-infinity algebras, modules, algebra homomorphisms, and module morphisms, as well as to define a bimodule analogue of twisted complexes (type DD structures, in the language of bordered Heegaard Floer homology) and their one- and two-sided tensor products. We then give analogous definitions for 1-parameter deformations of A-infinity algebras; this involves another collection of complexes. These constructions are relevant to bordered Heegaard Floer homology.