A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2 : n;ω2) (of dimension one less than a power of p) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of H(2 : n;ω2) with a Block algebra. W...

Given a reductive Lie algebra g and choice of Cartan subalgebra h, the HarishChandra map gives an isomorphism between the center z of the universal enveloping algebra U(g) of g and the algebra of Weyl group invariant symmetric tensors of h, denoted S(h) . We define these objects and give a sketch of the proof, using sl(2,C) as a motivating example. The proofs and examples mirror those found in ...

This paper is the continuation of section 2 of the paper ”Floating bundles and their applications” [1]. Let (H, μ, η,∆, ε, S) be a commutative Hopf algebra over ring R without torsion and F(x ⊗ 1, 1⊗ x) be a formal group over Hopf algebra H. By HQ denote the Hopf algebra H⊗ Z Q over ring RQ = R⊗ Z Q. We shall write μ, η, . . . instead of μQ, ηQ, . . . . The aim of this paper is to prove the fol...

Let A be a finitary algebra over a finite field k, and A-mod the category of finite dimensional left A-modules. Let H(A) be the corresponding Hall algebra, and for a positive integer r let Dr(A) be the subspace of H(A) which has a basis consisting of isomorphism classes of modules in A-mod with at least r+1 indecomposable direct summands. If A is hereditary of type An, then Dr(A) is known to be...

The problems considered in this paper come as a natural continuation of our program to develop a free analogue of Sz.-Nagy–Foiaş theory, for row contractions. An n-tuple (T1, . . . , Tn) of operators acting on a Hilbert space is called row contraction if T1T ∗ 1 + · · ·+ TnT ∗ n ≤ I. In this study, the role of the unilateral shift is played by the left creation operators on the full Fock space ...

We prove that the normal cohomology groups H w(M, K(H)) of a von Neumann algebra M with coefficients in the algebra of compact operators are zero if M is atomic of type Ifin. In addition, the completely bounded normal cohomology groups H wcb(B(H), K(H)) are shown to be 0 as well.

Definition 1.2 The invariants of H in A is the set A of those a ∈ A, that ha = ε(h)a for each h ∈ H. Straightforvard computations shows, that A is the subalgebra of A. We refer reader to [5], [6] for the basic information concerning Hopf algebras and their actions on associative algebras. As a generalization of the well-known fact for group actions the following question raised in [5] ( Questio...

We show that, if there exists a realization of a Hopf algebra H in a H-module algebra A, then one can split their cross-product into the tensor product algebra of A itself with a subalgebra isomorphic to H and commuting with A. This result applies in particular to the algebra underlying inhomogeneous quantum groups like the Euclidean ones, which are obtained as cross-products of the quantum Euc...

The purpose of this paper is to give definitions of some new Hopf algebras analogous to the mod-p Steenrod algebras. For example, we produce a Hopf algebra over the integers whose reduction mod-p is the mod-p Steenrod algebra. Just for fun, we “quantize” these algebras too! Further study of the examples of this paper will be done in part II. 1 Deformations of Hopf Algebras Suppose that k is a c...

It is known that the vector space spanned by labeled rooted trees forms a Hopf algebra. Let k be a field and let R be a commutative kalgebra. Let H denote the Hopf algebra of rooted trees labeled using derivations in Der(R). In this paper, we introduce a construction that gives R a H-module algebra structure and show this induces a differential algebra structure of H acting on R. The work here ...