The tetrahedron algebra and its finite-dimensional irreducible modules

Recently Terwilliger and the present author found a presentation for the three-point sl₂ loop algebra via generators and relations. To obtain this presentation we defined a Lie algebra ? by generators and relations and displayed an isomorphism from ? to the three-point sl₂ loop algebra. In this paper we classify the finite-dimensional irreducible ?-modules.     

Graded automorphism group of TKK algebra

The classification of extended affine Lie algebras of type A_1 depends on the Tits-Kantor- Koecher (TKK) algebras constructed from semilattices of Euclidean spaces.One can define a unitary Jordan algebra J(S) from a semilattice S of R~v (v≥1),and then construct an extended affine Lie algebra of type A_1 from the TKK algebra T(J(S)) which is obtained from the Jordan algebra J(S) by the so-called Tits-Kantor-Koecher construction.In R~2 there are only two non-similar semilattices S and S′,where S is a lattice and S′is a non-lattice semilattice.In this paper we study the Z~2-graded automorphisms of the TKK algebra T(J(S)).     

Cyclic irreducible non-holonomic modules over the Weyl algebra: An algorithmic characterization

Let K be a field of characteristic zero, n≥1 an integer and A_n+1=K[X,Y₁,…,Y_n]〈∂_X,∂_Y1,…,∂_Yn〉 the (n+1)th Weyl algebra over K. Let S∈A_n+1 be an order-1 differential operator of the type with a_i,b_i∈K[X] and g_i∈K[X,Y_i] for every i=1,…,n. We construct an algorithm that allows one to recognize whether S generates a maximal left ideal of A_n+1, hence also whether A_n+1/A_n+1S is an irreducible non-holonomic A_n+1-module. The algorithm, which is a powerful instrument for producing concrete examples of cyclic maximal left ideals of A_n, is easy to implement and quite useful; we use it to solve several open questions.The algorithm also allows one to recognize whether certain families of algebraic differential equations have a solution in K[X,Y₁,…,Y_n] and, when they have one, to compute it.     

Classification of irreducible non-zero level quasifinite modules over twisted affine Nappi-Witten algebra

We obtain that every irreducible quasifinite module with non-zero level over the twisted affine Nappi-Witten algebra is either a highest weight module or a lowest one.     

The first occurrence for the irreducible modules of general linear groups in the polynomial algebra

Let be a prime number and let be the group of all invertible matrices over the prime field . It is known that every irreducible -module can occur as a submodule of , the polynomial algebra with variables over . Given an irreducible -module , the purpose of this paper is to find out the first value of the degree of which occurs as a submodule of , the subset of consisting of homogeneous polynomials of degree . This generalizes Schwartz-Tri's result to the case of any prime .

     

Graded-irreducible modules are irreducible

We show that if a graded submodule of a Noetherian module cannot be written as a proper intersection of graded submodules, then it cannot be written as a proper intersection of submodules at all. More generally, we show that a natural extension of the index of reducibility to the graded setting coincides with the ordinary index of reducibility. We also investigate the question of uniqueness of the components in a graded-irreducible decomposition, as well as the relation between the index of reducibility of a non-graded ideal and that of its largest graded subideal.     

Graded automorphism group of TKK Lie algebra over semilattice

Every extended affine Lie algebra of type A ₁ and nullity ν with extended affine root system R(A ₁, S), where S is a semilattice in ℝ ^ν , can be constructed from a TKK Lie algebra T (J (S)) which is obtained from the Jordan algebra J (S) by the so-called Tits-Kantor-Koecher construction. In this article we consider the ℤ ⁿ -graded automorphism group of the TKK Lie algebra T (J (S)), where S is the “smallest” semilattice in Euclidean space ℝ ⁿ .     

Partially integrable highest weight modules

We prove a more general version of a result announced without proof in [DP], claiming roughly that in a partially integrable highest weight module over a Kac-Moody algebra the integrable directions from a parabolic subalgebra.     

Symmetry algebra of nonlinear integrable equations

The complete symmetry algebras of the Veselov-Novikov equation and of the modified Kadomtsev-Petviashvili equation are constructed. Some related questions are discussed.P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 34–41, April, 1993.     

Verma modules for rank two Heisenberg-Virasoro algebra

Let■ be a compatible total order on the additive group Z~2,and L be the rank two HeisenbergVirasoro algebra.For any c=(c_1,c_2,c_3,c_4) ∈ C~4,we define a Z~2-graded Verma module M(c,■) for L.A necessary and sufficient condition for M(c,■) to be irreducible is provided.Moreover,the maximal Z~2-graded submodules of M(c,■) are characterized when M(c,■) is reducible.     

Convolution algebra for extended Feller convolution

Semigroup Forum - We apply the recently introduced framework of admissible homomorphisms in the form of a convolution algebra of $$mathbb{C}^2$$ -valued admissible homomorphisms to handle...     

Interpolation in nest algebra modules

Let be a nest algebra and its invariant projection (or subspace) lattice. In this paper, using order homomorphisms of , we give necessary and sufficient conditions on bounded linear operators and on a Hilbert space to guarantee the existence of an operator in a certain -module such that .