Gauge groups and bialgebroids

We study the Ehresmann–Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include: Galois objects of Taft algebras, a monopole bundle over a quantum sphere and a not faithfully flat Hopf–Galois extension of commutative algebras. For each of the latter two examples, there is in fact a suitable invertible antipode for the bialgebroid making it a Hopf algebroid.

POST-QUANTUM BLIND SIGNATURE PROTOCOL ON NON-COMMUTATIVE ALGEBRAS

A method for constructing a blind signature scheme based on a hidden discrete logarithm problem defined in finite non-commutative associative algebras is proposed. Blind signature protocols are constructed using four-dimensional and six-dimensional algebras defined over a ground finite field GF(p) and containing a global two-sided unit as an algebraic support. The basic properties of the used algebra, which determine the choice of protocol parameters, are described.

Hypercomplex Algebras and Calculi Derived from Generalized Kinematics

The paper provides a study of the commutative algebras generated by iteration of the cross products in $\mathbb{C}^3$. Focusing on particular real forms we also consider the analytical properties of the corresponding rings of functions and relate them to different physical problems. Familiar results from the theory of holomorphic and bi-holomorphic functions appear naturally in this context, but new types of hypercomplex calculi emerge as well. The parallel transport along smooth curves in $\mathbb{E}^3$ and the associated Maurer-Cartan form are also studied with examples from kinematics and electrodynamics. Finally, the dual extension is discussed in the context of screw calculus and Galilean mechanics; a similar construction is studied also in the multi-dimensional real and complex cases.

Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras

The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms Ωn(X,M). Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k=C.

The algebraic classification of nilpotent commutative algebras

<p style='text-indent:20px;'>This paper is devoted to the complete algebraic classification of complex <inline-formula><tex-math id="M1">\begin{document}$ 5 $\end{document}</tex-math></inline-formula>-dimensional nilpotent commutative algebras. Our method of classification is based on the standard method of classification of central extensions of smaller nilpotent commutative algebras and the recently obtained classification of complex <inline-formula><tex-math id="M2">\begin{document}$ 5 $\end{document}</tex-math></inline-formula>-dimensional nilpotent commutative <inline-formula><tex-math id="M3">\begin{document}$ \mathfrak{CD} $\end{document}</tex-math></inline-formula>-algebras.</p>

Miscellaneous reverse order laws and their equivalent facts for generalized inverses of a triple matrix product

<abstract><p>Reverse order laws for generalized inverses of products of matrices are a class of algebraic matrix equalities that are composed of matrices and their generalized inverses, which can be used to describe the links between products of matrix and their generalized inverses and have been widely used to deal with various computational and applied problems in matrix analysis and applications. ROLs have been proposed and studied since 1950s and have thrown up many interesting but challenging problems concerning the establishment and characterization of various algebraic equalities in the theory of generalized inverses of matrices and the setting of non-commutative algebras. The aim of this paper is to provide a family of carefully thought-out research problems regarding reverse order laws for generalized inverses of a triple matrix product $ ABC $ of appropriate sizes, including the preparation of lots of useful formulas and facts on generalized inverses of matrices, presentation of known groups of results concerning nested reverse order laws for generalized inverses of the product $ AB $, and the derivation of several groups of equivalent facts regarding various nested reverse order laws and matrix equalities. The main results of the paper and their proofs are established by means of the matrix rank method, the matrix range method, and the block matrix method, so that they are easy to understand within the scope of traditional matrix algebra and can be taken as prototypes of various complicated reverse order laws for generalized inverses of products of multiple matrices.</p></abstract>

Born geometry on ρ-commutative algebra

The aim of this paper is to establish a generalization of the Born geometry to [Formula: see text]-commutative algebras. We introduce the notion of Born [Formula: see text]-commutative algebras and study the existence and uniqueness of a torsion connection which preserves the Born structure. Also, an analogue of the fundamental theorem of Riemannian geometry will be proved for these algebras.