# Journal of Nonlinear Mathematical Physics

Volume 13, Issue Supplement, August 2006

### Proceedings of the Baltic-Nordic Workshop Algebra, Geometry and Mathematical Physics

Tallinn, Estonia, October, 2005

**Research Article**

## 2. On a graded q-differential algebra

Viktor Abramov

Pages: 1 - 8

Given an associative unital ZN -graded algebra over the complex numbers we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-differential d of the graded q-differential algebra is a homogeneous endomorphism of degree 1 satisfying...

**Research Article**

## 3. Geometric approach to BRST-symmetry and ZN-generalization of superconnection

V. Abramov, O. Liivapuu

Pages: 9 - 20

We propose a geometric approach to the BRST-symmetries of the Lagrangian of a topological quantum field theory on a four dimensional manifold based on the formalism of superconnections. Making use of a graded q-differential algebra, where q is a primitive N-th root of unity, we also propose a notion...

**Research Article**

## 4. Non-coordinate case of graded differential algebra with ternary differential

Nadezda Bazunova

Pages: 21 - 26

In this article, we generalize a construction of graded q-differential algebra with ternary differential satisfying the property d3 = 0 and q-Leibniz rule on the non-coordinate case, that is on the case where the differentials of generators of underlying algebra do not coincide with generators of bimodule...

**Research Article**

## 5. On the matrix 3 × 3 exact solvable models of the type G2

C. Burdik, S. Posta, O. Navratil

Pages: 27 - 36

We study the exact solvable 3 × 3 matrix model of the type G2. We apply the construction similar to that one, which give the 2 × 2 matrix model. But in the studied case the construction does not give symmetric matrix potential. We conceive that the exact solvable 3 × 3 matrix potential model of the type...

**Research Article**

## 6. SO(2) and Hamilton-Dirac mechanics

Cestmir Burdik, Eugen Paal, Juri Virkepu

Pages: 37 - 43

Canonical formalism for plane rotations is developed. This group can be seen as a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and Hamiltonian are explicitly constructed and their physical interpretation are given. The Euler-Lagrange and Hamiltonian canonical equations coincide...

**Research Article**

## 7. The graphical calculus for ribbon categories: Algebras, modules, Nakayama automorphisms

Jurgen Fuchs

Pages: 44 - 54

The graphical description of morphisms in rigid monoidal categories, in particular in ribbon categories, is summarized. It is illustrated with various examples of algebraic structures in such categories, like algebras, (weak) bi-algebras, Frobenius algebras, and modules and bimodules. Nakayama automorphisms...

**Research Article**

## 8. Description of a Class of 2-Groups

Tatjana Gramushnjak, Peeter Puusemp

Pages: 55 - 65

Let n be an integer such that n 3 and Cm denote a cyclic group of order m . It is proved that there exist exactly 17 non-isomorphic groups of order 22n+1 which can be represented as a semidirect product (C2n × C2n ) C2. These groups are given by generators and defining relations.

**Research Article**

## 9. Rewriting in Operads and PROPs

Lars Hellström

Pages: 66 - 75

This paper is an informal collection of observations on how established rewriting techniques can be applied to or need to be adapted for use in non-associative algebras, operads, and PROPs.

**Research Article**

## 10. On Some Almost Quadratic Algebras Coming from Twisted Derivations

Daniel Larsson, Gunnar Sigurdsson, Sergei D. Silvestrov

Pages: 76 - 87

This paper explores the quasi-deformation scheme devised in [1, 3] as applied to the simple Lie algebra sl2(F) for specific choices of the involved parameters and underlying algebras. One of the main points of this method is that the quasi-deformed algebra comes endowed with a canonical twisted Jacobi...

**Research Article**

## 11. Note on operadic non-associative deformations

Eugen Paal

Pages: 87 - 92

Deformation equation of a non-associative deformation in operad is proposed. Its integrability condition (the Bianchi identity) is considered. Algebraic meaning of the latter is explained.

**Research Article**

## 12. Groups of Order Less Than 32 and Their Endomorphism Semigroups

Peeter Puusemp

Pages: 93 - 101

It is proved that among the finite groups of order less than 32 only the tetrahedral group and the binary tetrahedral group are not determined by their endomorphism semigroups in the class of all groups.

**Research Article**

## 13. Total Differentiation Under Jet Composition

Maido Rahula, Vitali Retsnoi

Pages: 102 - 109

Total differentiation operators as linear vector fields, their flows, invariants and symmetries form the geometry of jet space. In the jet space the dragging of tensor fields obeys the exponential law. The composition of smooth maps induces a composition of jets in corresponding jet spaces. The prolonged...

**Research Article**

## 14. Bosonic Realizations of the Colour Heisenberg Lie Algebra

Gunnar Sigurdsson, Sergei D. Silvestrov

Pages: 110 - 128

We describe realizations of the colour analogue of the Heisenberg Lie algebra by power series in non-commuting indeterminates satisfying Heisenberg's canonical commutation relations of quantum mechanics. The obtained formulas are used to construct new operator representations of the colour Heisenberg...

**Research Article**

## 15. Classical quasi-trigonometric r-matrices of Cremmer-Gervais type and their quantization

Julia Y.-Magnusson, Maxim Samsonov, Alexander Stolin

Pages: 129 - 136

We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang-Baxter equation. The method is based on so-called affinization of certain seaweed algebras and their quantum analogues.