We study the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case. We prove that it admits a graded, finte-dimensional, connected gradation. Dually, the vector space R〈x0, x1〉 is both a pre-Lie algebra for the pre-Lie product dual of the coproduct of H, and an associative, commutative algebra for the shuffle product. These two structures admit a compatibilit...

II. LINEAR ALGEBRA Linear Algebra is the study of linear transformations. We used the following result on how to diagonalize an Hermitian matrix: II.1. Theorem: Diagonalization of Hermitian Matrix (Notes on Linear Algebra Theorem 6) Let H ∈ Cn×n be a Hermitian matrix, i.e., such that H = H∗. The eigenvalues λ1, . . . , λn of H are real (they are not necessarily distinct); H has n orthonormal ei...

Let g denote a Lie algebra, and let ĝ denote the tensor product of g with a ring of truncated polynomials. The Lie algebra ĝ is called a truncated current Lie algebra. The highest-weight theory of ĝ is investigated, and a reducibility criterion for the Verma modules is described. Let g be a Lie algebra over a field k of characteristic zero, and fix a positive integer N . The Lie algebra (1) ĝ =...

We provide a simplified version of a construction of Charles Read. For any n ≥ 2, there are n isometries with orthogonal ranges with the property that the nonself-adjoint weak-∗-closed algebra that they generate is all of B(H). A free semigroup algebra is the weak operator topology closed algebra generated by an n-tuple of isometries with pairwise orthogonal ranges. The C*-algebra generated by ...

The Heisenberg Lie Group is the most frequently used model for studying representation theory of groups. This group modular-noncompact and its algebra nilpotent. elements can be expressed in form matrices size 3×3. Another specialty also inherited by three-dimensional called algebra. whose Algebra extended to dimension 2n+1 generalized it denoted H h_n. In this study, surjectiveness exponential...

Starting with a given generalized boson algebra U〈q〉(h(1)) known as the bosonized version of the quantum super-Hopf Uq(osp(1|2)) algebra, we employ the Hopf duality arguments to provide the dually conjugate function algebra Fun〈q〉(H(1)). Both the Hopf algebras being finitely generated, we produce a closed form expression of the universal T matrix that caps the duality and generalizes the famili...

For every semi-simple Lie algebra g one can construct the DrinfeldJimbo algebra U h (g). This algebra is a deformation Hopf algebra defined by generators and relations. To study the representation theory of U h (g), Drinfeld used the KZ-equations to construct a quasi-Hopf algebra Ag . He proved that particular categories of modules over the algebras U h (g) and Ag are tensor equivalent. Analogo...

Commutative H *-algebra is characterized in terms of idempotents. Here we offer three characterizations. 1. Introduction. In the past, the author used commuting idempotents to characterize continuous functions defined on a certain space [3, 4]. For example , it was shown in [3] that a certain Banach algebra is isometrically isomorphic to the space C(S) of all continuous complex-valued functions...

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...