First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the corresponding Maurer-Cartan equation. In Section 6 we generalize the Maurer-Cartan equation to strongly homotopy Li...

The goal of this paper is to express the Bach tensor of a four dimensional conformal geometry of an arbitrary signature by the Cartan normal conformal (CNC) connection. We show that the Bach tensor can be identified with the Yang–Mills current of the connection. It follows from that result that a conformal geometry whose CNC connection is reducible in an appropriate way has a degenerate Bach te...

We decompose tensor products of the defining representation of a Cartan type Lie algebra W (n) in the case where the number of tensoring does not exceed the rank of the Lie algebra. As a result, we get a kind of Schur duality between W (n) and a finite dimensional non-semisimple algebra, which is the semi-group ring of the transformation semigroup Tm .

The Einstein-Cartan field equation of infinitely rotating rigid cylinder is obtained and solved using the techniques of Manko-Gutsunaev and Quevedo to describe gravitational field due to extended van-Stockum cylindrically perfect fluid distribution using improved energy-momentum tensor by Ray and Smalley.

First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the corresponding Maurer-Cartan equation. In Section 6 we generalize the Maurer-Cartan equation to strongly homotopy Li...

In 1977, M. Matsumoto and R. Miron [9] constructed an orthonormal frame for an n-dimensional Finsler space, called ‘Miron frame’. The present authors [1, 2, 3, 10, 11] discussed four-dimensional Finsler spaces equipped with such frame. M. Matsumoto [7, 8] proved that in a three-dimensional Berwald space, all the main scalars are h-covariant constants and the h-connection vector vanishes. He als...

In many engineering applications that use tensor analysis, such as tensor imaging, the underlying tensors have the characteristic of being positive definite. It might therefore be more appropriate to use techniques specially adapted to such tensors. We will describe the geometry and calculus on the Riemannian symmetric space of positive-definite tensors. First, we will explain why the geometry,...

We consider integrable category O representations of Borcherds–Kac–Moody algebras whose Cartan matrix is finite dimensional, and determine the necessary sufficient conditions for which tensor product irreducible from this isomorphic to another. This result generalizes a fundamental C. S. Rajan on unique factorization products dimensional simple Lie over complex numbers.

We show that the vertex algebra W1+∞ with central charge −1 is isomorphic to a tensor product of the simple W3 algebra with central charge −2 and a Heisenberg vertex algebra generated by a free bosonic field. We construct a family of irreducible modules of the W3 algebra with central charge −2 in terms of free fields and calculate the full character formulas of these modules with respect to the...

Let U be the quantised enveloping algebra associated to a Cartan matrix of finite type. Let W be the tensor product of a finite list of highest weight representations of U . Then EndU (W ) has a basis called the dual canonical basis and this gives an integral form for EndU (W ). We show that this integral form is cellular by using results due to Lusztig.