Periodic automorphisms of the free Lie algebra of rank 3

We prove that each automorphism of finite order of the free Lie algebra of rank 3 over an algebraically closed field is conjugate to a linear automorphism if the field characteristic fails to divide the automorphism order.     

Strictly nontame primitive elements of the free metabelian Lie algebra of rank 3

We prove that the free metabelian Lie algebra M ₃ of rank 3 over an arbitrary field K admits strictly nontame primitive elements.     

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.     

The derivation Lie algebra of the higher rank Virasoro-like algebra and its automorphism groups

In this paper, we study the derivation Lie Algebra of the higher rank Virasoro-like algebra. We prove that it isomorphic to the skew derivation Lie Algebra. We also characterize the automorphism groups of the higher rank Virasoro-like algebra and the skew derivation Lie Algebra. This generalizes the result of some related references.     

Computing the test rank of a free solvable Lie algebra

We computed the test rank of a free solvable Lie algebra of finite rank.     

Test rank of an abelian product of a free Lie algebra and a free abelian Lie algebra

Let F be a free Lie algebra of rank n ≥ 2 and A be a free abelian Lie algebra of rank m ≥ 2. We prove that the test rank of the abelian product F ×A is m. Morever we compute the test rank of the algebra F/g_k( F) ^￠F/\gamma _{k}\left( F\right) ^{^{\prime }}.     

The algebra of almost periodic functions has infinite topological stable rank

We show that if is the uniform algebra of almost periodic functions, then the set cannot be dense in for any positive integer .

     

Finite dimensional representations of the double affine Hecke algebra of rank 1

We classify the finite dimensional irreducible representations of the double affine Hecke algebra (DAHA) of type C^∨C₁ in the case when q is not a root of unity.     

First cohomology group of rank two Witt algebra to its Larsson modules

Let U = ℂ², Γ = ℤ², and let ℂ[x ₁^±1, x ₂^±1] be the ring of Laurent polynomials. The Witt algebra L is the Lie algebra of derivations over ℂ[x ₁^±1, x ₂^±1], which is spanned by elements of the form D(u, r) = x ^r (u ₁ d ₁ + u ₂ d ₂), u = (u ₁, u ₂) ∈ U, r ∈ Γ, where d ₁ and d ₂ are the degree derivations of ℂ[x ₁^±1, x ₂^±1]. The image of gl ₂-module V under Larsson functor F ^α , denoted by W = F ^α (V), gives a class of L-modules, often called the Larsson-modules of L. In this paper, we study the derivations from the Witt algebra L to its Larsson-modules W, and we determine the first cohomology group H ¹(L,W).     

A Plactic Algebra Action on Bosonic Particle Configurations

Algebras and Representation Theory - We study an action of the plactic algebra on bosonic particle configurations. These particle configurations together with the action of the plactic generators...     

On rank 3 projective designs

The finite projective designs with rank 3 collineation groupsG such thatG _p =G _l for some point-block pair (P, l) are divided into 4 classes. 2 of these classes are formed by the Paley designs and the designs complementary to the Paley designs. For the other 2 classes restrictions on the parameters are obtained. In particular it is shown that the only such design having =1 is the projective plane of order 2.With 2 Figures