Quantum differential operators on the quantum plane

The universal enveloping algebra of a Lie algebra acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or quantum group) is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring. We present this situation for the quantum group of sl₂.     

The Socle of a Metric n-Lie Algebra

We define the socle of an n-Lie algebra as the sum of all the minimal ideals. An n-Lie algebra is called metric if it is endowed with an invariant nondegenerate symmetric bilinear form. We characterize the socle of a metric n-Lie algebra, which is closely related to the radical and the center of the metric n-Lie algebra. In particular, the socle of a metric n-Lie algebra is reductive, and a metric n-Lie algebra is solvable if and only if the socle coincides with its center. We also calculate the metric dimensions of simple and reductive n-Lie algebras and give a lower bound in the nonreductive case.     

On the Bishop boundary in topological algebras

We characterize the existence of the Bishop boundary in the context of an Urysohn (topological) algebra, by means ofG _δ points. Considering a subset of its spectrum and the restriction of its Gel'fand transform algebra on the previous subset, we realize its Bishop boundary, as the trace of the Bishop boundary of the initial algebra on the spectrum of the restriction algebra. We get a generalization of the latter by considering a continuous algebra morphism. This paper is based on and extends results of the author's Doctoral Thesis, written under the supervision of Professor A. Mallios (Univ. of Athens).     

Brauer algebras of simply laced type

The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A _{n − 1} on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph Q, called the Brauer algebra of type Q, and study its structure in the cases where Q is a Coxeter graph of simply laced spherical type (so its connected components are of type A _{n − 1}, D _n , E₆, E₇, E₈). We find its irreducible representations and its dimension, and show that the algebra is cellular. The algebra is generically semisimple and contains the group algebra of the Coxeter group of type Q as a subalgebra. It is a ring homomorphic image of the Birman-Murakami-Wenzl algebra of type Q; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.     

Degree of matrix relation algebras

We show that a relation algebra and its n-matrix relation algebra have the same degree for all positive integers n. An intermediate result relates the degree of a relation algebra to the degree of a relativization with respect to equivalence elements. Received July 18, 2001; accepted in final form April 24, 2002.     

Table algebras with central fusions

The character theory of table algebras is not as good as the character theory of finite groups. We introduce the notion of a table algebra with a central-fusion, in which the character theory has better properties. We study conditions under which a table algebra (A,B) has a central-fusion, and its central-fusion is exactly isomorphic to the wreath product of the central-fusion of a quotient table algebra of (A,B) and another table algebra. As a consequence, we obtain a complete characterization of table algebras with exactly one irreducible character whose degree and multiplicity are not equal. Applications to association schemes are also discussed.     

Inbreeding depression in a zygotic algebra

We study the algebraic behavior of a three dimensional zygotic algebra in the presence of parameters 0 < s < 1 and 0 < g < 1; s for selfing and g which reflects its associated inbreeding depression. We also study the dynamics of the system for which this algebra is a model. Our methods lean towards commutative algebra and algebraic geometry and find support on the computer program Macaulay2.Our results are best understood through the geometry of a rational function of the projective plane.     

Two explicit realizations of a Lie algebra

We introduce a Lie algebra whose some properties are discussed, including its proper ideals, derivations and so on. Then, we again give rise to its two explicit realizations by adopting subalgebra of the Lie algebra A₂ and a column-vector Lie algebra, respectively. Under the frame of zero curvature equations, we may use the realizations to generate the same Lax integrable hierarchies of evolution equations and their Hamiltonian structure.     

A $$\mathbb{Z}_{2}$$-orbifold model of the symplectic fermionic vertex operator superalgebra

We give an example of an irrational C ₂-cofinite vertex operator algebra whose central charge is −2d for any positive integer d. This vertex operator algebra is given as the even part of the vertex operator superalgebra generated by d pairs of symplectic fermions, and it is just the realization of the c = −2-triplet algebra given by Kausch in the case d = 1. We also classify irreducible modules for this vertex operator algebra and determine its automorphism group. This research is supported in part by a grant from Japan Society for the Promotion of Science.     

POINCARE-HILBERT SERIES,PI, AND NOETHERIANITY OF THE ENVELOPING OF THE RIT ALGEBRA

We investigate some properties, such as Poincaré–Hilbert series, growth, Noetherianity of the Relativistic Internal Time (RIT) Lie algebra which appears in non-equilibrium physics and its enveloping algebra U(RIT), using, in particular, the Groebner basis technique.     

A Lie algebra attached to a projective variety

Abstract. Each choice of a K?hler class on a compact complex manifold defines an action of the Lie algebra sl(2) on its total complex cohomology. If a nonempty set of such K?hler classes is given, then we prove that the corresponding sl(2)-copies generate a semisimple Lie algebra. We investigate the formal properties of the resulting representation and we work things out explicitly in the case of complex tori, hyperk?hler manifolds and flag varieties. We pay special attention to the cases where this leads to a Jordan algebra structure or a graded Frobenius algebra. Oblatum 21-V-1996 & 15-X-1996     

The structure of integrable Lax flows on nonlocal manifolds: Dynamic systems with sources

We consider a dynamic system on the extended phase space to the initial Lie algebra and study its generalized Hamiltonian and integrability in the cases when the initial Lie algebra coincides with the Grassmann algebra of pseudodifferential operators on the real line and on the centrally extended affine Lie algebra.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 106–115.     

Lie algebras admitting a unique quadratic structure

We prove that any Lie algebra g over a field K of characteristic zero admitting a unique up to a constant quadratic structure is necessarily a simple Lie algebra. If the field K is algebraically closed, such condition is also sufficient.

Further, a real Lie algebra g admits a unique quadratic structure if and only if its complexification g^C is a simple Lie algebra over C     

Generalised nilpotence conditions in n-engel lie algebras

A Lie algebra is said to satisfy the Baer condition provided each of its 1-dimensional subalgebras is a subideal; if the defect is bounded then the Lie algebra is bounded Baer. We first characterise restricted enveloping algebras (of odd characteristic p) that satisfy the bounded Baer condition. Using this characterisation, we are then able to construct a Lie algebra satisfying the bounded Engel condition, in each odd characteristic p, that does not satisfy the Baer condition. Such a bounded Engel Lie algebra must contain a non-nilpotent 1-generated ideal     

Simple left-symmetric algebras with solvable Lie algebra

Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie groupG correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we studysimple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special class of complete LSAs is studied.     

The q-Analogue of the Alternating Group and Its Representations

We define the new algebra. This algebra has a parameter q. The defining relations of this algebra at q = 1 coincide with the basic relations of the alternating group. We also give the new subalgebra of the Hecke algebra of type A which is isomorphic to this algebra. This algebra is free of rank half that of the Hecke algebra. Hence this algebra is regarded as a q-analogue of the alternating group.All the isomorphism classes of the irreducible representations of this algebra and the q-analogue of the branching rule between the symmetric group and the alternating group are obtained.     

An independent system of identities for a variety of mono-Leibniz algebras

We introduce the notion of a mono-Leibniz algebra generalizing the concept of a Leibniz algebra. Namely, an algebra A over a field K, charK ≠ 2, is mono-Leibniz if its one-generated subalgebras each is a Leibniz algebra. It is proved that a variety W of mono-Leibniz algebras over an infinite field K is defined by an independent system of identities such as x(xx) = 0 and x[(xx)x] = 0. Examples of mono-Leibniz algebras are given which show that W is not a Schreier variety.     

Wave Operators on Quantum Algebras via Noncanonical Quantization

We suggest a method to quantize basic wave operators of Relativistic Quantum Mechanics (Laplace, Maxwell, Dirac ones) without using canonical coordinates. We define two-parameter deformations of the Minkowski space algebra and its 3-dimensional reduction via the so-called Reflection Equation Algebra and its modified version. Wave operators on these algebras are introduced by means of quantized partial derivatives described in two ways. In particular, they are given in so-called pseudospherical form which makes use of a q-deformation of the Lie algebra sl(2) and quantum versions of the Cayley-Hamilton identity.     

The Magnus Embedding for Right-Symmetric Algebras

We construct a basis for the universal multiplicative enveloping algebra U(A) of a right-symmetric algebra A. We prove an analog of the Magnus embedding for right-symmetric algebras; i.e., we prove that a right-symmetric algebra A/R ², where A is a free right-symmetric algebra, is embedded into the algebra of triangular matrices of the second order.