Fuzzy σ algebras of physics

Following the idea of Zadeh, the concept of a statistical (or fuzzy) algebra is introduced. For two extreme cases of classical and quantum statistical algebras the representation theorems are proved. The basic feature distinguishing these two cases is the possibility of producing nontrivial superpositions of pure quantum states, which is absent in the classical case.A part of this work has been written during the author's stay at the Mathematics Department, University of Toronto (Canada). The financial support from the NSERC research grant No. A5206 is gratefully acknowledged.On leave of absence from the Institute of Theoretical Physics, University of Gdask, 80-952 Gdask, Poland.     

n-Extended Lorentzian Kac–Moody algebras

Letters in Mathematical Physics - We investigate a class of Kac–Moody algebras previously not considered. We refer to them as n-extended Lorentzian Kac–Moody algebras defined by their...     

Quasi-1 structure on q-Poincaré algebras

We use braided groups to introduce a theory of 1-structures on general inhomogeneous quantum groups, which we formulate as quasi-1 Hopf algebras. This allows the construction of the tensor product of unitary representations up to a quantum cocycle isomorphism, which is a novel feature of the inhomogeneous case. Examples include q-Poincaré quantum group enveloping algebras in R-matrix from appropriate to the previous q-Euclidean and q-Minkowski space-time algebras R₂₁x₁x₂ = x₂x₁R and R₂₁u₁Ru₂ = u₂R₂₁u₁R. We obtain unitarity of the fundamental differential representations. We further show that the Euclidean and Minkowski-Poincaré quantum groups are twisting equivalent by another quantum cocycle.     

Poisson–Nijenhuis structures on quiver path algebras

We introduce a notion of noncommutative Poisson–Nijenhuis structure on the path algebra of a quiver. In particular, we focus on the case when the Poisson bracket arises from a noncommutative symplectic form. The formalism is then applied to the study of the Calogero–Moser and Gibbons–Hermsen integrable systems. In the former case, we give a new interpretation of the bihamiltonian reduction performed in Bartocci et al. (Int Math Res Not 2010:279–296, 2010. arXiv:0902.0953).     

A class of Lorentzian Kac–Moody algebras

We consider a natural generalisation of the class of hyperbolic Kac–Moody algebras. We describe in detail the conditions under which these algebras are Lorentzian. We also construct their fundamental weights, and analyse whether they possess a real principal so(1,2) subalgebra. Our class of algebras include the Lorentzian Kac–Moody algebras that have recently been proposed as symmetries of M-theory and the closed bosonic string.     

Geometric approach to Kac–Moody and Virasoro algebras

In this paper we show the existence of a group acting infinitesimally transitively on the moduli space of pointed-curves and vector bundles (with formal trivialization data) and whose Lie algebra is an algebra of differential operators. The central extension of this Lie algebra induced by the determinant bundle on the Sato Grassmannian is precisely a semidirect product of a Kac–Moody algebra and the Virasoro algebra. As an application of this geometric approach, we give a local Mumford-type formula in terms of the cocycle associated with this central extension. Finally, using the original Mumford formula we show that this local formula is an infinitesimal version of a general relation in the Picard group of the moduli of vector bundles on a family of curves (without any formal trivialization).     

Classical and quantum algebras of non-local charges in σ models

We investigate the algebras of the non-local charges and their generating functionals (the monodromy matrices) in classical and quantum non-linear models. In the case of the classical chiral models it turns out that there exists no definition of the Poisson bracket of two monodromy matrices satisfying antisymmetry and the Jacobi identity. Thus, the classical non-local charges do not generate a Lie algebra. In the case of the quantum O(N) non-linear model, we explicitly determine the conserved quantum monodromy operator from a factorization principle together withP,T, and O(N) invariance. We give closed expressions for its matrix elements between asymptotic states in terms of the known two-particleS-matrix. The quantumR-matrix of the model is found. The quantum non-local charges obey a quadratic Lie algebra governed by a Yang-Baxter equation.Laboratoire associé au CNRS No. LA 280     

Homological unimodularity and Calabi–Yau condition for Poisson algebras

In this paper, we show that the twisted Poincaré duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives another definition of a unimodular Poisson algebra in terms of its Poisson Picard group. We also achieve twisted Poincaré duality for Hochschild (co)homology of Poisson bimodules using rigid dualizing complex. For a smooth Poisson affine variety with the trivial canonical bundle, we prove that its enveloping algebra is a Calabi–Yau algebra if the Poisson structure is unimodular.     

A note on the Lie algebras g(m∞)

The algebras g(m) are interpreted as realisations of the infinite rank affine Lie algebras g.     

The Miura transformation and Lie-Bäcklund algebras of exactly solvable equations

All the one-dimensional one-component local evolution equations connected via the Miura transformation are found. Exactly solvable equations and their Lie-Bäcklund algebras are shown to generate interesting transformations of infinite classes of evolution equations.     

Supersymmetric Chern–Simons vortex systems and extended supersymmetric quantum mechanics algebras

We study N=2$N=2$ supersymmetric Chern–Simons Higgs models in (2+1)$(2+1)$-dimensions and the existence of extended underlying supersymmetric quantum mechanics algebras. Our findings indicate that the fermionic zero modes quantum system in conjunction with the system of zero modes corresponding to bosonic fluctuations, are related to an N=4$N=4$ extended 1-dimensional supersymmetric algebra with central charge, a result closely connected to the N=2$N=2$ spacetime supersymmetry of the total system. We also add soft supersymmetric terms to the fermionic sector in order to examine how this affects the index of the corresponding Dirac operator, with the latter characterizing the degeneracy of the solitonic solutions. In addition, we analyze the impact of the underlying supersymmetric quantum algebras to the zero mode bosonic fluctuations. This is relevant to the quantum theory of self-dual vortices and particularly for the symmetries of the metric of the space of vortices solutions and also for the non-zero mode states of bosonic fluctuations.     

Möbius groups over general fields using clifford algebras associated with spheres

The space of 2-by-2 Hermitian matrices is isometric to Minkowski space. This is commonly used to exhibit the groupSL(2, ) as a twofold cover of the identity component of the Lorentz group. That these Hermitian matrices also represent equations of circles in the Euclidean plane leads to the groupPSL(2, ) as the Möbius group of the Euclidean plane. Clifford algebras naturally arise in the construction of covers of the orthogonal group by spin groups. By considering in addition the Clifford algebra of the space of equations of spheres, we are able to extend these ideas to the Möbius group of finite-dimensional vector spaces over general fields.