2 Using Maple in the study of the canonical formalism of general relativity 17 2. 1 2 CONTENTS Preview This is a review devoted to some results of Algebraic Programming (Computer Algebra-CA) used in treating several problems of general relativity, based mainly on already published articles (see the references). The first chapter presents procedures in REDUCE language using EXCALC package for al...

Hopf algebra structure on the differential algebra of the extended q-plane is defined. An algebra of forms which is obtained from the generators of the extended q-plane is introduced and its Hopf algebra structure is given. E-mail: [email protected] E-mail: [email protected]

We describe the basic Dolbeault cohomology algebra of canonical foliation on a class complex manifolds with torus symmetry group. This includes moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, maximal holomorphic action. also provide DGA model for ordinary algebra. The Hodge decomposition is proved by reducing to transversely Kähler (equivalently, polytopal) case using f...

The study of harmonic functions on a locally compact group G has recently been transferred to a “non-commutative” setting in two different directions: C.-H. Chu and A. T.-M. Lau replaced the algebra L∞(G) by the group von Neumann algebra VN(G) and the convolution action of a probability measure μ on L∞(G) by the canonical action of a positive definite function σ on VN(G); on the other hand, W. ...

The class of Borel sets in a locally compact space is the cr-ring generated by the compact sets [3, p. 219], while the class of weakly Borel sets is the (r-algebra generated by the closed sets [2]. The object of this note is to show that any measure defined on the class of Borel sets may be extended to the class of weakly Borel sets in a simple and canonical way (Theorem 1). There is a useful a...

Quantum canonical transformations are defined algebraically outside of a Hilbert space context. This generalizes the quantum canonical transformations of Weyl and Dirac to include non-unitary transformations. The importance of non-unitary transformations for constructing solutions of the Schrödinger equation is discussed. Three elementary canonical transformations are shown both to have quantum...

or its invariant space Inv(V1 ⊗ V2 ⊗ . . .⊗ Vn). The quantum group Uq(g) has representations and vector spaces of invariants which generalize these, and one can also study their bases, with or without the intention of specializing to q = 1. (For simplicity, we will usually consider Uq(g) as an algebra over C(q ), and we will only occassionally mention Z[q] as a ground ring.) Lusztig’s remarkabl...

We realize the current algebra of a Kac-Moody algebra as a quotient of a semi-direct product of the Kac-Moody Lie algebra and the free Lie algebra of the Kac-Moody algebra. We use this realization to study the representations of the current algebra. In particular we see that every ad-invariant ideal in the symmetric algebra of the Kac-Moody algebra gives rise in a canonical way to a representat...

We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many noncanonical sentences. Using probabi...

Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is constructed inside the quantized enveloping algebra of a KacMoody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution o...