Related papers: Algebras and varieties
We define varieties of algebras for an arbitrary endofunctor on a cocomplete category using pairs of natural transformations. This approach is proved to be equivalent to the one of equational classes defined by equation arrows. Free…
We study varieties of complexes of projective modules with fixed ranks, and relate these varieties to the varieties of their homologies. We show that for an algebra of global dimension at most two, these two varieties are related by a pair…
We introduce the notion of relation type of an affine algebra and prove that it is well defined by using the Jacobi-Zariski exact sequence of Andr\'e-Quillen homology. In particular, the relation type is an invariant of an affine algebraic…
In this short note, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by L.…
We expand our previously founded basic theory of equiresidual algebraic geometry over an arbitrary commutative field, to a well-behaved theory of (equiresidual) algebraic varieties over a commutative field, thanks to the generalisation of…
We prove that a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine…
We classify (possibly non commutative) algebras of low rank over a domain R. We first review results for algebras of rank 2 and for finite-dimensional division algebras over the real numbers. These results motivate us to consider which…
Let A be a finitely generated associative algebra over an algebraically closed field. We characterize the finite dimensional modules over A whose orbit closures are regular varieties.
A new homological dimension is introduced to measure the quality of resolutions of `singular' finite dimensional algebras (of infinite global dimension) by `regular' ones (of finite global dimension). Upper bounds are established in terms…
The representation and the cohomology theory of associative 2-algebras are developed. We study the deformations and abelian extensions of associative 2-algebras in details.
We show that a class of algebras is closed under the taking of homomorphic images and direct products if and only if the class consists of all algebras that satisfy a set of (generally simultaneous) equations. For classes of regular…
In the present paper we study some algebraic properties of evolution algebras. Moreover, we reduce the study of evolution algebras of permutations to two special types of evolution algebras, idempotents and absolute nilpotent elements of…
We study algebraic subvarieties of strata of differentials in genus zero satisfying algebraic relations among periods. The main results are Ax-Schanuel and Andr\'e-Oort-type theorems in genus zero. As a consequence, one obtains several…
Universal algebraic geometry allows considering of geometric properties of every universal algebra. When two algebras have same algebraic geometry? We must consider the categories of algebraic closed sets of these algebras to answer this…
A metric algebra is a metric variant of the notion of $\Sigma$-algebra, first introduced in universal algebra to deal with algebras equipped with metric structures such as normed vector spaces. In this paper, we showed metric versions of…
In this paper, we give a purely cohomological interpretation of the extension problem for associative algebras; that is the problem of extending an associative algebra by another associative algebra. We then give a similar interpretation of…
Several results on presenting an affine algebraic group variety as a product of algebraic varieties are obtained.
We construct geometric examples of N-differential graded algebras such as the algebra of differential forms of depth $N$ on an affine manifold, and $N$-flat covariant derivatives.
We introduce the notion of amenability for affine algebras. We characterize amenability by Folner-sequences, paradoxicality and the existence of finitely invariant dimension-measures. Then we extend the results of Rowen on ranks, from…
We prove that a quasi-finite endomorphism of an algebraic variety over an algebraically closed field of characteristic zero, that is injective on the complement of a closed subvariety, is an automorphism. We also prove that an endomorphism…