Related papers: Unification theorems in algebraic geometry
This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed…
An $F$-algebra $A$ with unit $1$ is said to be a locally matrix algebra if an arbitrary finite collection of elements $a_1,$ $\ldots,$ $a_s $ from $ A$ lies in a subalgebra $B$ with $1$ of the algebra $A$, that is isomorphic to a matrix…
We construct a topological cellular operad such that the algebras over its cellular chains are the homotopy unital A-infinity algebras of Fukaya-Oh-Ohta-Ono.
We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e.…
It is shown that, given any finite dimensional, split basic algebra $\Lambda = K\Gamma/I$ (where $\Gamma$ is a quiver and $I$ an admissible ideal in the path algebra $K \Gamma$), there is a finite list of affine algebraic varieties, the…
We formulate gauge theories based on Leibniz(-Loday) algebras and uncover their underlying mathematical structure. Various special cases have been developed in the context of gauged supergravity and exceptional field theory. These are based…
Given a simple vertex algebra A and a reductive group G of automorphisms of A, the invariant subalgebra A^G is strongly finitely generated in most examples where its structure is known. This phenomenon is subtle, and is generally not true…
Let $R$ be an algebra over a ring $\Bbbk$, $T$ an $R$-algebra, $M$ a finitely generated projective $R$-module, and $N$ a $T$-module. Let $G$ be a linearly reductive group scheme over $\Bbbk$ equipped with a representation…
Let $A$ be an associative algebra over a field of characteristic $\neq 2$ that is generated by a finite collection of nilpotent elements. We prove that all Lie derived powers of $A$ are finitely generated Lie algebras.
If $\A$ is an algebra over a field $\F$ and $\s$ is a generating set of $\A$, the length of $\s$ indicates the maximal length needed to express an arbitrary element of $\A$ as a linear combination of words in the elements of $\s$. The…
In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Virasoro algebras. In particular,the derivation algebras, the automorphism groups and the second cohomology groups of these…
We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a generalization of metric affine geometry,…
In this paper we introduce a new kind of topological space, called 'structured space', which locally resembles various kinds of algebraic structures. This can be useful, for instance, to locally study a space that cannot be globally endowed…
The integrability condition called shape invariance is shown to have an underlying algebraic structure and the associated Lie algebras are identified. These shape-invariance algebras transform the parameters of the potentials such as…
We give a conceptual explanation of universal deformation formulas for unital associative algebras and prove some results on the structure of their moduli spaces. We then generalize universal deformation formulas to other types of algebras…
We construct a topology on a given algebraically closed field with a distinguished subfield which is also algebraically closed. This topology is finer than Zariski topology and it captures the sets definable in the pair of algebraically…
We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry…
This paper develops the geometry of locally bounded rational functions on non-singular real algebraic varieties. First various basic geometric and algebraic results regarding these functions are established in any dimension, culminating…
We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.
We investigate the defining ideal of the algebra over a field generated by the join-meet binomials coming from a finite distributive lattice. In the frame of algebras with straightening laws, the problem when the defining ideal is generated…