On the combinatorial rank of a graded braided bialgebra

Let B be a graded braided bialgebra. Let S(B) denote the algebra obtained dividing out B by the two sided ideal generated by homogeneous primitive elements in B of degree at least two. We prove that S(B) is indeed a graded braided bialgebra quotient of B. It is then natural to compute S(S(B)), S(S(S(B))) and so on. This process yields a direct system whose direct limit comes out to be a graded braided bialgebra which is strongly N-graded as a coalgebra. Following V.K. Kharchenko, if the direct system is stationary exactly after n steps, we say that B has combinatorial rank n and we write κ(B)=n. We investigate conditions guaranteeing that κ(B) is finite. In particular, we focus on the case when B is the braided tensor algebra T(V,c) associated to a braided vector space (V,c), providing meaningful examples such that κ(T(V,c))≤1.     

The structure of bialgebra with a Hopf module

Let H be a Hopf algebra, B a bialgebra, and (B, ?, ρ) a right H-Hopf module. Assume that (B, ρ) is a right H-comodule algebra, (B, ?) is a right H-module coalgebra, and let A = B ^{co H} = {a ∈ B | ρ(a) = a ? 1}. Then we prove that B has a factorization of A□ _ρ ^? (the underlying space is A ? H) as a bialgebra, which generalizes Radford’s factorization of bialgebras with projection [12].     

Formality theorem for Lie bialgebras and quantization of twists and coboundary r-matrices

Let (g,δ?) be a Lie bialgebra. Let (U?(g),Δ?) a quantization of (g,δ?) through Etingof-Kazhdan functor. We prove the existence of a L_∞-morphism between the Lie algebra C(g)=Λ(g) and the tensor algebra (without unit) T₊U=T₊(U?(g)[−1]) with Lie algebra structure given by the Gerstenhaber bracket. When s is a twist for (g,δ), we deduce from the formality morphism the existence of a quantum twist F. When (g,δ,r) is a coboundary Lie bialgebra, we get the existence of a quantization R of r.     

Twisted hopf comodule algebras

For k a commutative ring, H a k‐bialgebra and A a right H‐comodule k‐algebra, we define a new multiplication on the H‐comodule A to obtain a twisted algebra” A^T, T sumHom(H,End (A)). If ^T is convolution invertible, the categories of relative right Hopf modules over A and A^Tare isomorphic. Similarly a convolution invertible left twisting gives an isomorphism of the categories of relative left Hopf modules. We show that crossed products are invertible twistings of the tensor product, and obtain, as a corollary, a duality theorem for crossed products     

(H,R)-Lie coalgebras and (H,R)-Lie bialgebras

Given an (H,R)-Lie coalgebra Γ, we construct (H,R _T )-Lie coalgebra Γ^T through a right cocycle T, where (H,R) is a triangular Hopf algebra, and prove that there exists a bijection between the set of (H,R)-Lie coalgebras and the set of ordinary Lie coalgebras. We also show that if (L, [, ], Δ, R) is an (H,R)-Lie bialgebra of an ordinary Lie algebra then (L ^T , [, ], Δ_T, R _T ) is an (H,R _T )-Lie bialgebra of an ordinary Lie algebra.     

Hopf Bimodules, Coquasibialgebras, and an Exact Sequence of Kac

Based on the ideas of Tannaka-Kre?n reconstruction, we present a categorical construction that assigns to any cleft Hopf algebra inclusion K⊂H a coquasibialgebra having K* as a Hopf subalgebra. As a special case, the construction gives an intrinsic connection between the bismash product K#Q and the double cross- product Q?K* constructed from the same combinatorial data. A cocommutative coquasibialgebra is the same as a cocommutative bialgebra equipped with a Sweedler three-cocycle. Thus our construction assigns to every bicrossproduct (or Hopf algebra extension) of a commutative and a cocommutative factor a corresponding cocommutative double crossproduct equipped with a Sweedler three-cocycle. Based on this observation we use the construction to prove generalizations of Kac's exact sequence for the group of Hopf algebra extensions of a group algebra by a dual group algebra.     

Radford's biproduct for quasi-Hopf algebras and bosonization

Let B be a braided Hopf algebra (with bijective antipode) in the category of left Yetter-Drinfeld modules over a quasi-Hopf algebra H. As in the case of Hopf algebras (J. Algebra 92 (1985) 322), the smash product B#H defined in (Comm. Algebra 28(2) (2000) 631) and a kind of smash coproduct afford a quasi-Hopf algebra structure on B⊗H. Using this, we obtain the structure of quasi-Hopf algebras with a projection. Further we will use this biproduct to describe the Majid bosonization (J. Algebra 163 (1994) 165) for quasi-Hopf algebras.     

Compatible associative bialgebras

We introduce a non-symmetric operad 𝒩, whose dimension in degree n is given by the Catalan number c_n?1. It arises naturally in the study of coalgebra structures defined on compatible associative algebras. We prove that any free compatible associative algebra admits a compatible infinitesimal bialgebra structure, whose subspace of primitive elements is a 𝒩-algebra. The data (As,As²,𝒩) is a good triple of operads, in J.-L. Loday’s sense. Our construction induces another triple of operads (As,As₂,As), where As₂ is the operad of matching dialgebras. Motivated by A. Goncharov’s Hopf algebra of paths P(S), we introduce the notion of bi-matching dialgebras and show that the Hopf algebra P(S) is a bi-matching dialgebras.     

Classification of systems of Dyson-Schwinger equations in the Hopf algebra of decorated rooted trees

We consider systems of combinatorial Dyson-Schwinger equations (briefly, SDSE) , … , in the Connes-Kreimer Hopf algebra HI of rooted trees decorated by I={1,…,N}, where is the operator of grafting on a root decorated by i, and F₁,…,FN are non-constant formal series. The unique solution X=(X₁,…,XN) of this equation generates a graded subalgebra H(S) of HI. We characterise here all the families of formal series (F₁,…,FN) such that H(S) is a Hopf subalgebra. More precisely, we define three operations on SDSE (change of variables, dilatation and extension) and give two families of SDSE (cyclic and fundamental systems), and prove that any SDSE (S) such that H(S) is Hopf is the concatenation of several fundamental or cyclic systems after the application of a change of variables, a dilatation and iterated extensions.     

On the algebra of quasi-shuffles

For any commutative algebra R the shuffle product on the tensor module T(R) can be deformed to a new product. It is called the quasi-shuffle algebra, or stuffle algebra, and denoted T ^q (R). We show that if R is the polynomial algebra, then T ^q (R) is free for some algebraic structure called Commutative TriDendriform (CTD-algebras). This result is part of a structure theorem for CTD-bialgebras which are associative as coalgebras and whose primitive part is commutative. In other words, there is a good triple of operads (As, CTD, Com) analogous to (Com, As, Lie). In the last part we give a similar interpretation of the quasi-shuffle algebra in the noncommutative setting.     

The smooth Banach submanifold B*(E, F) in B(E, F)

Let E,F be two Banach spaces,B(E,F),B+(E,F),Φ(E,F),SΦ(E,F) and R(E,F) be bounded linear,double splitting,Fredholm,semi-Frdholm and finite rank operators from E into F,respectively. Let Σ be any one of the following sets:{T ∈Φ(E,F):Index T=constant and dim N(T)=constant},{T ∈ SΦ(E,F):either dim N(T)=constant< ∞ or codim R(T)=constant< ∞} and {T ∈ R(E,F):Rank T=constant< ∞}. Then it is known that Σ is a smooth submanifold of B(E,F) with the tangent space TAΣ={B ∈ B(E,F):BN(A)-R(A) } for any A ∈Σ. However,for ...     

DISTINGUISHED PRE-NICHOLS ALGEBRAS

We formally define and study the distinguished pre-Nichols algebra \( \tilde{B} \)(V) of a braided vector space of diagonal type V with finite-dimensional Nichols algebra B(V). The algebra \( \tilde{B} \)(V) is presented by fewer relations than B(V), so it is intermediate between the tensor algebra T(V) and B(V). Prominent examples of distinguished pre-Nichols algebras are the positive parts of quantized enveloping (super)algebras and their multiparametric versions. We prove that these algebras give rise to new examples of Noetherian pointed Hopf algebras of finite Gelfand-Kirillov dimension. We investigate the kernel (in the sense of Hopf algebras) of the projection from \( \tilde{B} \)(V) to B(V), generalizing results of De Concini and Procesi on quantum groups at roots of unity.     

Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations

We consider the combinatorial Dyson-Schwinger equation X=B⁺(P(X)) in the non-commutative Connes-Kreimer Hopf algebra of planar rooted trees HNCK, where B⁺ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra AN,P of HNCK. We describe all the formal series P such that AN,P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of HNCK, organized into three isomorphism classes: a first one, restricted to a polynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Faà di Bruno Hopf algebra. By taking the quotient, the last class gives an infinite set of embeddings of the Faà di Bruno algebra into the Connes-Kreimer Hopf algebra of rooted trees. Moreover, we give an embedding of the free Faà di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, together with a non-commutative version of this embedding.     

Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative

Let T be a surjective map from a unital semi-simple commutative Banach algebra A onto a unital commutative Banach algebra B. Suppose that T preserves the unit element and the spectrum σ(fg) of the product of any two elements f and g in A coincides with the spectrum σ(TfTg). Then B is semi-simple and T is an isomorphism. The condition that T is surjective is essential: An example of a non-linear and non-multiplicative unital map from a commutative C*-algebra into itself such that σ(TfTg)=σ(fg) holds for every f,g are given. We also show an example of a surjective unital map from a commutative C*-algebra onto itself which is neither linear nor multiplicative such that σ(TfTg)⊂σ(fg) holds for every f,g.     

All-derivable points of operator algebras

Let A be an operator subalgebra in B(H), where H is a Hilbert space. We say that an element Z∈A is an all-derivable point of A for the norm-topology (strongly operator topology, etc.) if, every norm-topology (strongly operator topology, etc.) continuous derivable linear mapping φ at Z (i.e. φ(ST)=φ(S)T+Sφ(T) for any S,T∈A with ST=Z) is a derivation. In this paper, we show that every invertible operator in the nest algebra is an all-derivable point of the nest algebra for the strongly operator topology. We also prove that every nonzero element of the algebra of all 2×2 upper triangular matrixes is an all-derivable point of the algebra.     

Lie algebras associated to systems of Dyson–Schwinger equations

We consider systems of combinatorial Dyson–Schwinger equations in the Connes–Kreimer Hopf algebra HI of rooted trees decorated by a set I. Let H(S) be the subalgebra of HI generated by the homogeneous components of the unique solution of this system. If it is a Hopf subalgebra, we describe it as the dual of the enveloping algebra of a Lie algebra g(S) of one of the following types:

• 1. 

  g(S) is an associative algebra of paths associated to a certain oriented graph.

• 2. 

  Or g(S) is an iterated extension of the Faà di Bruno Lie algebra.

• 3. 

  Or g(S) is an iterated extension of an infinite-dimensional abelian Lie algebra.

We also describe the character groups of H(S).     

Symplectic groups and the Klein-Gordon field

In this paper a Cohen factorization theorem x = a^t · x_t (t > 0) is proved for a Banach algebra A with a bounded approximate identity, where t ? a^t is a continuous one-parameter semigroup in A. This theorem is used to show that a separable Banach algebra B has a bounded approximate identity bounded by 1 if and only if there is a homomorphism θ from L¹($\text{R}$⁺) into B such that ∥ θ ∥ = 1 and θ(L¹($\text{R}$⁺)). B = B = B · θ(L¹($\text{R}$⁺)). Another corollary is that a separable Banach algebra with bounded approximate identity has a commutative bounded approximate identity, which is bounded by 1 in an equivalent algebra norm.     

Lie algebras generated by bounded linear operators on Hilbert spaces

It is proved that the operator Lie algebra ε(T,T^∗) generated by a bounded linear operator T on Hilbert space H is finite-dimensional if and only if T=N+Q, N is a normal operator, [N,Q]=0, and dimA(Q,Q^∗)<+∞, where ε(T,T^∗) denotes the smallest Lie algebra containing T,T^∗, and A(Q,Q^∗) denotes the associative subalgebra of B(H) generated by Q,Q^∗. Moreover, we also give a sufficient and necessary condition for operators to generate finite-dimensional semi-simple Lie algebras. Finally, we prove that if ε(T,T^∗) is an ad-compact E-solvable Lie algebra, then T is a normal operator.     

Kähler geometry of the universal Teichmüller space and coadjoint orbits of the Virasoro group

The Kähler geometry of the universal Teichmüller space and related infinite-dimensional Kähler manifolds is studied. The universal Teichmüller space T may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The classical Teichmüller spaces T(G), where G is a Fuchsian group, are contained in T as complex Kähler submanifolds. The homogeneous spaces Diff₊(S ¹)/Möb(S ¹) and Diff₊(S ¹)/S ¹ of the diffeomorphism group Diff₊(S ¹) of the unit circle are closely related to T. They are Kähler Frechet manifolds that can be realized as coadjoint orbits of the Virasoro group (and exhaust all coadjoint orbits of this group that have the Kähler structure).     

Hereditarily polaroid operators, SVEP and Weyl's theorem

A Banach space operator T∈B(X) is hereditarily polaroid, T∈HP, if every part of T is polaroid. HP operators have SVEP. It is proved that if T∈B(X) has SVEP and R∈B(X) is a Riesz operator which commutes with T, then T+R satisfies generalized a-Browder's theorem. If, in particular, R is a quasi-nilpotent operator Q, then both T+Q and T^∗+Q^∗ satisfy generalized a-Browder's theorem; furthermore, if Q is injective, then also T+Q satisfies Weyl's theorem. If A∈B(X) is an algebraic operator which commutes with the polynomially HP operator T, then T+N is polaroid and has SVEP, f(T+N) satisfies generalized Weyl's theorem for every function f which is analytic on a neighbourhood of σ(T+N), and f^∗(T+N) satisfies generalized a-Weyl's theorem for every function f which is analytic on, and constant on no component of, a neighbourhood of σ(T+N).