Journal of Algebra and Its Applications

Let A be an integral domain, X an analytic indeterminate over A and I a proper ideal (not necessarily prime) of A. In this paper, we study the ring

Next, we study the Krull dimension of R. We give a necessary condition for R to be of finite Krull dimension. In particular, if R is of finite dimension then I must be an SFT ideal of A. Then we determine bounds for dim(R). Examples are given to indicate the sharpness of the results. In case I is a maximal ideal of A and A is either a Noetherian ring, SFT Prüfer domain or A[[X]] is catenarian and I SFT, we establish that dim(R) = dim(A[[X]]) = dim(A) + 1.

Finally, we examine the possible transfer of the LFD property and the catenarity between the rings A, A[[X]] and R in case I is a maximal ideal of A.

In this paper a *-algebra of regular functions on the Shilov boundary S(𝔻) of bounded symmetric domain 𝔻 is constructed. The algebras of regular functions on S(𝔻) are described in terms of generators and relations for two particular series of bounded symmetric domains. Also, the degenerate principal series of quantum Harish–Chandra modules related to S(𝔻) = U_n is investigated.

In this article, we explicitly calculated the values of the representations of the Hecke algebra , associated with a Gelfand–Graev character of GL₄(q), at some of the standard basis elements.

This paper surveys, and in some cases generalizes, many of the recent results on homomorphisms and the higher Ext groups for q-Schur algebras and for the Hecke algebra of type A. We review various results giving isomorphisms between Ext groups in the two categories, and discuss those cases where explicit results have been determined.

In this paper we find a description, up to equivalence, of all irreducible representations for a class of algebras generated by four linearly related idempotents.

Let H_n be the (2n + 1)-dimensional Heisenberg group. We decompose L²(H_n) as the closure of a direct sum of infinitely many left translation invariant eigenspaces (for certain systems of partial differential equations). The restriction of the left regular representation to each one of these eigenspaces disintegrates into a direct integral of unitary irreducible representations, such that each infinite dimensional unitary irreducible representation appears with multiplicity 0 or 1 in this disintegration.

Lie bialgebras occur as the principal objects in the infinitesimalization of the theory of quantum groups — the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In this paper, we study the simple complex Lie algebras using the double-bosonization construction of Majid. This construction expresses algebraically the induction process given by adding and removing nodes in Dynkin diagrams, which we call Lie induction.

We first analyze the deletion of nodes, corresponding to the restriction of adjoint representations to subalgebras. This uses a natural grading associated to each node. We give explicit calculations of the module and algebra structures in the case of the deletion of a single node from the Dynkin diagram for a simple Lie (bi-)algebra.

We next consider the inverse process, namely that of adding nodes, and give some necessary conditions for the simplicity of the induced algebra. Finally, we apply these to the exceptional series of simple Lie algebras, in the context of finding obstructions to the existence of finite-dimensional simple complex algebras of types E₉, F₅ and G₃. In particular, our methods give a new point of view on why there cannot exist such an algebra of type E₉.

In this paper we study the Auslander–Reiten quiver of the highest weight category of finite-dimensional G₁T-modules, associated to a smooth reductive algebraic group G. By relating properties of stable Auslander–Reiten components to those of their rank varieties we show that there are at most three isomorphism types of these components. For the Frobenius kernels G₁T_r the maximal ranks of tubes are determined.

We study the connection between *-representations of algebras associated with graphs, locally-scalar graph representations and the problem about the spectrum of a sum of two Hermitian operators. For algebras associated with Dynkin graphs we give an explicit description of the parameters for which there are irreducible representations and an algorithm for constructing these representations.