Constructive homomorphisms for classical groups

Let Omega be a quasisimple classical group in its natural representation over a finite vector space V, and let Delta be its normaliser in the general linear group. We construct the projection from Delta to Delta/Omega and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent Delta/Omega as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of Omega. A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spaces with forms. Our results are useful for the matrix group recognition project, can be used to solve element conjugacy problems, and can improve algorithms to construct maximal subgroups

Homomorphic encryption and some black box attacks

This paper is a compressed summary of some principal definitions and concepts in the approach to the black box algebra being developed by the authors. We suggest that black box algebra could be useful in cryptanalysis of homomorphic encryption schemes, and that homomorphic encryption is an area of research where cryptography and black box algebra may benefit from exchange of ideas

Formalizing Abstract Algebra in Constructive Set Theory

We present a machine-checked formalization of elementary abstract algebra in constructive set theory. Our formalization uses an approach where we start by specifying the group axioms as a collection of inference rules, defining a logic for groups. Then we can tell whether a given set with a binary operation is a group or not, and derive all properties of groups constructively from these inference rules as well as the axioms of the set theory. The formalization of all other concepts in abstract algebra is based on that of the group. We give an example of a formalization of a concrete group, the Klein 4-group

The Gelfand map and symmetric products

If A is an algebra of functions on X, there are many cases when X can be regarded as included in Hom(A,C) as the set of ring homomorphisms. In this paper the corresponding results for the symmetric products of X are introduced. It is shown that the symmetric product Sym^n(X) is included in Hom(A,C) as the set of those functions that satisfy equations generalising f(xy)=f(x)f(y). These equations are related to formulae introduced by Frobenius and, for the relevant A, they characterise linear maps on A that are the sum of ring homomorphisms. The main theorem is proved using an identity satisfied by partitions of finite sets.Comment: 14 pages, Late

Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective

We establish, generalizing Di Nola and Lettieri's categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-intepretability holding for particular classes of formulas: irreducible formulas, geometric sentences and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang's MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.Comment: 54 page

Minimal deformations of the commutative algebra and the linear group GL(n)

We consider the relations of generalized commutativity in the algebra of formal series $M_q (x^i )$, which conserve a tensor $I_q$-grading and depend on parameters $q(i,k)$ . We choose the $I_q$-preserving version of differential calculus on $M_q$ . A new construction of the symmetrized tensor product for $M_q$-type algebras and the corresponding definition of minimally deformed linear group $QGL(n)$ and Lie algebra $qgl(n)$ are proposed. We study the connection of $QGL(n)$ and $qgl(n)$ with the special matrix algebra \mbox{Mat} (n,Q) containing matrices with noncommutative elements. A definition of the deformed determinant in the algebra \mbox{Mat} (n,Q) is given. The exponential parametrization in the algebra \mbox{Mat} (n,Q) is considered on the basis of Campbell-Hausdorf formula.Comment: 14 page

A looping-delooping adjunction for topological spaces

Every principal G-bundle is classified up to equivalence by a homotopy class of maps into the classifying space of G. On the other hand, for every nice topological space Milnor constructed a strict model of loop space, that is a group. Moreover the morphisms of topological groups defined on the loop space of X generate all the bundles over X up to equivalence. In this paper, we show that the relationship between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles with a fixed structure group. Such a resul clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space construction, which are very important in topological K-theory, group cohomology and homotopy theory.Comment: v1: 24 pages; v2: 18 pages; Corrected typos; Revised structure in Introduction, and Sections 1 and 2; Results unchange