Potentials for some tensor algebras

The comultiplication of the universal enveloping algebra of a Lie algebra is used to give modules produced from a subalgebra, an additional compatible structure of a module over an algebra of formal power series. When only the k \mathfrak {k} -finite elements of this algebra act on a module, conditions are given that insure that it is the Harish-Chandra module of a produced module. The results are then applied to Zuckerman derived functor modules for reductive Lie algebras. The main application describes a setting where the Zuckerman functors and production from a subalgebra commute.

Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

Norming the Algebra of Formal Power Series in n Indeterminates

A representation of solution of stochastic differential equations

In [ 2 ] Bieri and Strebel introduced a geometric invariant for finitely generated abstract metabelian groups that determines which groups are finitely presented. For a valuable survey of their results, see [ 6 ]; we recall the definition briefly in Section 4. We shall introduce a similar invariant for pro- p groups. Let [ ] be the algebraic closure of [ ] p and U be the formal power series algebra [ ][lobrk ] T [robrk ], with group of units U × . Let Q be a finitely generated abelian pro- p group. We write ℤ p [lobrk ] Q [robrk ] for the completed group algebra of Q over ℤ p . Let T ( Q ) be the abelian group Hom( Q , U × ) of continuous homomorphisms from Q to U × . We write 1 for the trivial homomorphism. Each v ∈ T ( Q ) extends to a unique continuous algebra homomorphism vmacr; from ℤ p [lobrk ] Q [robrk ] to U .

Using quantum differential operators, we construct a super representation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) -submodule isomorphic to the regular representation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) . In this way, we obtain a presentation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) by a basis together with explicit multiplication formulas of the basis elements by generators.

Representations of finite association schemes

Semiprime Graded Algebras of Dimension Two

Tensor, Clifford and Grassmann algebras belong to a wide class of non-commutative algebras that have a Poincare-Birkhoff-Witt (PBW) “monomial” basis. The necessary and sufficient condition for an algebra to have the PBW basis has been established by T. Mora and then V. Levandovskyy as the so called “non-degeneracy condition”. This has led V. Levandovskyy to a re-discovery of the so called G-algebras (previously introduced by J. Apel) and GR-algebras (Grobner-ready algebras). It was T. Mora who already in the 1990s considered a comprehensive and algorithmic approach to Grobner bases for commutative and non-commutative algebras. It was T. Stokes who eighteen years ago introduced Grobner left bases (GLB) and Grobner left ideal bases (GLIB) for left ideals in Grassmann algebras, with the GLIB bases solving an ideal membership problem. Thus, a natural question is to first seek Grobner bases with respect to a suitable admissible monomial order for ideals in tensor algebras T and then consider quotient algebras T/I. It was shown by Levandovskyy that these quotient algebras possess a PBW basis if and only if the ideal I has a Grobner basis. Of course, these quotient algebras are of great interest because, in particular, Grassmann and Clifford algebras of a quadratic form arise this way. Examples of G-algebras include the quantum plane, universal enveloping algebras of finite dimensional Lie algebras, some Ore extensions, Weyl algebras and their quantizations, etc. Examples of GR-algebras, which are either G algebras or are isomorphic to quotient algebras of a G-algebra modulo a proper two-sided ideal, include Grassmann and Clifford algebras. After recalling basic concepts behind the theory of commutative Grobner bases, a review of the Grobner bases in PBW algebras, G-,and GR-algebras will be given with a special emphasis on computation of such bases in Grassmann and Clifford algebras. GLB and GLIB bases will also be computed.

Skolem–Noether algebras

The groups of automorphisms of the Lie algebras of formally analytic vector fields with constant divergence

In this chapter we present some basic concepts from algebra which are of central importance in the development of algorithms and systems for symbolic mathematical computation. The main issues distinguishing various computer algebra systems arise out of the choice of algebraic structures to be manipulated and the choice of representations for the given algebraic structures.

The concept of modulation is generalized to pseudo-modulation and its subclasses including pre-modulation, generalized modulation and regular modulation. The motivation is to define the valued analogue of natural quiver, called {\em natural valued quiver}, of an artinian algebra so as to correspond to its valued Ext-quiver when this algebra is not $k$-splitting over the field $k$. Moreover, we illustrate the relation between the valued Ext-quiver and the natural valued quiver. The interesting fact we find is that the representation categories of a pseudo-modulation and of a pre-modulation are equivalent respectively to that of a tensor algebra of $\mathcal A$-path type and of a generalized path algebra. Their examples are given respectively from two kinds of artinian hereditary algebras. Furthermore, the isomorphism theorem is given for normal generalized path algebras with finite (acyclic) quivers and normal pre-modulations. Four examples of pseudo-modulations are given: (i) group species in mutation theory as a semi-normal generalized modulation; (ii) viewing a path algebra with loops as a pre-modulation with valued quiver which has not loops; (iii) differential pseudo-modulation and its relation with differential tensor algebras; (iv) a pseudo-modulation is considered as a free graded category.

It is well known that the complete norm topology on a Banach algebra is not unique in general, though semisimplicity is sufficient (but not necessary) for uniqueness. In this note we consider a class of topological algebras of formal power series which have unique Fréchet space topology. The structure of these algebras in the Banach algebra case will be considered in a later paper.

Let A and B be two algebras over the complex field C. A sequence {Fn}0g"gra (resp.{F"}0gn<a,) of linear operators from A into B is a higher derivation of rank m (resp.infinite rank) if, for each n=0,1, , m (resp.n=0,1, -) and any x,y eA, We consider the continuity of such {F"} when A and B are commutative topological algebras with complete metrizable topology.Some applications are given to algebras of formal power series.