Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of $End_F M$, where $M$ is a Yetter-Drinfeld module over $B$ with dim $B< \infty $ . In particular, generalized classical braided m-Lie algebras $sl_{q, f}(GM_G(A), F)$ and $osp_{q, t} (GM_G(A), M, F)$ of generalized matrix algebra $GM_G(A)$ are constructed and their connection with special generalized matrix Lie superalgebra $sl_{s, f}(GM_{{\bf Z}_2}(A^s), F)$ and orthosymplectic generalized matrix Lie super algebra $osp_{s, t} (GM_{{\bf Z}_2}(A^s), M^s, F)$ are established. The relationship between representations of braided m-Lie algebras and their associated algebras are established.
We introduce the new concept of braided Hom-Lie bialgebras which is a generalization of Sommerhäuser-Majid's braided Lie bialgebras and Yau's Hom-Lie bialgebras. Using this concept we give the unified product construction for Hom-Lie bialgebras which can be seen as a Hom-Lie version of Bespalov-Drabant's cocycle cross product bialgebras. Some special cases of unified products such as crossed product and matched pair of braided Hom-Lie bialgebras are investigated. As an application, we solve the Agore-Militaru extending problem for Hom-Lie bialgebras by using some non-abelian cohomology theory.…
A braided bialgebra is called primitively generated if it is generated as an algebra by its space of primitive elements. We prove that any primitively generated braided bialgebra is isomorphic to the universal enveloping algebra of its infinitesimal braided Lie algebra, notions hereby introduced. This result can be regarded as a Milnor-Moore type theorem for primitively generated braided bialgebras and leads to the introduction of a concept of braided Lie algebra for an arbitrary braided vector space.
In this paper, the notion of universal enveloping algebra introduced in [A. Ardizzoni, \emph{A First Sight Towards Primitively Generated Connected Braided Bialgebras}, submitted. (arXiv:0805.3391v3)] is specialized to the case of braided vector spaces whose Nichols algebra is quadratic as an algebra. In this setting a classification of universal enveloping algebras for braided vector spaces of dimension not greater than 2 is handled. As an application, we investigate the structure of primitively generated connected braided bialgebras whose braided vector space of primitive elements forms a Nic…
Let $H$ be an infinite-dimensional braided Hopf algebra and assume that the braiding is symmetric on $H$ and its quasi-dual $H^d$. We prove the Blattner-Montgomery duality theorem, namely we prove $$ (R # H)# H^{d} \cong R \otimes (H # H^{d}) \hbox {as algebras in braided tensor category} {\cal C}.$$ In particular, we present two duality theorems for infinite braided Hopf algebras in the Yetter-Drinfeld module category.
We prove the existence of a basis of Poincare-Birkhoff-Witt type for braided Hopf algebras R generated by a braided subspace V of P(R) if the braiding on V fulfils a triangularity condition. We apply our result to pointed Hopf algebras with abelian coradical and to Nichols algebras of low dimensional simple U_q(sl_2) modules.
The notion of a coherent unit action on algebraic operads was first introduced by Loday for binary quadratic nonsymmetric operads and generalized by Holtkamp, to ensure that the free objects of the operads carry a Hopf algebra structure. There was also a classification of such operads in the binary quadratic nonsymmetric case. We generalize the notion of coherent unit action to braided operads and show that free objects of braided operads with such an action carries a braided Hopf algebra structure. Under the conditions of binary, quadratic and nonsymmetric, we give a characterization and clas…
In this letter, we use quantum quasi-shuffle algebras to construct Rota-Baxter algebras, as well as tridendriform algebras. We also propose the notion of braided Rota-Baxter algebras, which is the relevant object of Rota-Baxter algebras in a braided tensor category. Examples of such new algebras are provided by using quantum multi-brace algebras in a category of Yetter-Drinfeld modules.
We describe the groups of automorphisms of two generated free braided associative algebras with involutive diagonal braidings over a field of characteristic $\neq 2$. Depending on the form of the diagonal involutive braiding, five different automorphism groups arise as automorphism groups of two generated free braided associative algebras. This list covers all three automorphism groups that arise in the case of quantum planes \cite{AC}.
In this paper, we give a natural braiding on the universal central extension of a crossed module of Lie algebras with a given braiding and construct the universal central extension of a braided crossed module of Lie algebras, showing that, when one of the constructions exists, both exist and coincide.
Let $(H,α)$ be a monoidal Hom-Hopf algebra and $^{H}_{H}\mathcal{HYD}$ the Hom-Yetter-Drinfeld category over $(H,α)$. Then in this paper, we first introduce the definition of braided Hom-Lie algebras and show that each monoidal Hom-algebra in $^{H}_{H}\mathcal{HYD}$ gives rise to a braided Hom-Lie algebra. Second, we prove that if $(A,β)$ is a sum of two $H$-commutative monoidal Hom-subalgebras, then the commutator Hom-ideal $[A,A]$ of $A$ is nilpotent. Also, we study the central invariant of braided Hom-Lie algebras as a generalization of generalized Lie algebras. Finally, we obtain a constru…
The main aim of this paper is to investigate the structure of primitively generated connected braided bialgebras $A$ with respect to the braided vector space $P$ consisting of their primitive elements. When the Nichols algebra of $P$ is obtained dividing out the tensor algebra $T(P) $ by the two-sided ideal generated by its primitive elements of degree at least two, we show that $A$ can be recovered as a sort of universal enveloping algebra of $P$. One of the main applications of our construction is the description, in terms of universal enveloping algebras, of connected braided bialgebras who…
The faithful quasi-dual $H^d$ and strict quasi-dual $H^{d'}$ of an infinite braided Hopf algebra $H$ are introduced and it is proved that every strict quasi-dual $H^{d'}$ is an $H$-Hopf module. The connection between the integrals and the maximal rational $H^{d}$-submodule $H^{d rat}$ of $H^{d}$ is found. That is, $H^{d rat}\cong \int ^l_{H^d} \otimes H$ is proved. The existence and uniqueness of integrals for braided Hopf algebras in the Yetter-Drinfeld category $(^B_B{\cal YD},C)$ are given.
Rota-Baxter algebras and the closely related dendriform algebras have important physics applications, especially to renormalization of quantum field theory. Braided structures provide effective ways of quantization such as for quantum groups. Continuing recent study relating the two structures, this paper considers Rota-Baxter algebras and dendriform algebras in the braided contexts. Applying the quantum shuffle and quantum quasi-shuffle products, we construct free objects in the categories of braided Rota-Baxter algebras and braided dendriform algebras, under the commutativity condition. We f…
Based on a pairing of two regular multiplier Hopf algebras $A$ and $B$, Heisenberg double $\mathscr{H}$ is the smash product $A \# B$ with respect to the left regular action of $B$ on $A$. Let $\mathscr{D}=A\bowtie B$ be the Drinfel'd double, then Heisenberg double $\mathscr{H}$ is a Yetter-Drinfel'd $\mathscr{D}$-module algebra, and it is also braided commutative by the braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to some infinite dimensional cases.
We classify PBW-deformations of quadratic-constant type of certain quantizations of exterior algebras. These correspond to the fundamental modules of quantum $\mathfrak{sl}_N$, their duals, and their direct sums. We show that the first two cases do not admit any deformation, while in the third case we obtain an essentially unique algebra with good properties. We compare this algebra with other quantum Clifford algebras appearing in the literature.
We obtain the double factorization of braided bialgebras or braided Hopf algebras, give relation among integrals and semisimplicity of braided Hopf algebra and its factors.
In braided tensor categories we show the Maschke's theorem and give the necessary and sufficient conditions for double cross biproducts and crossbiproducts and biproducts to be bialgebras. We obtain the factorization theorem for braided Hopf algebras; In symmetric tensor categories, we show the duality theorem and construct quantum double; In ordinary vector space category with ordinary twist, we obtain the relation between the global dimension, the weak dimension, the Jacbson radical, Baer radical of algebra $R$ and its crossed product $R #_σH$. We also give the relation between the decomposi…
The aim of this paper is to construct a new braided $T$-category via the generalized Yetter-Drinfel'd modules and Drinfel'd codouble over Hopf algebra, an approach different from that proposed by Panaite and Staic \cite{PS}. Moreover, in the case of finite dimensional, we will show that this category coincides with the corepresentation of a certain coquasitriangular Turaev group algebra that we construct. Finally we apply our theory to the case of group algebra.
This paper introduces the braidings of dendriform algebras and tridendriform algebras. By studying free braided dendriform algebras, we obtain braidings of the Hopf algebras of Loday and Ronco of planar binary rooted trees. We also give a variation of the braiding of Foissy for the noncommutative Connes-Kreimer (a.k.a the Foissy-Holtkamp) Hopf algebra of planar rooted forests so that the well-known isomorphism between this Hopf algebra and the Loday-Ronco Hopf algebra is extended to the braided context. As free braided tridendriform algebras, we also give braided extension of the Hopf algebra …
We study the Yetter--Drinfeld D(B)-module algebra structure on the Heisenberg double H(B^*) endowed with a "heterotic" action of the Drinfeld double D(B). This action can be interpreted in the spirit of Lu's description of H(B^*) as a twist of D(B). In terms of the braiding of Yetter--Drinfeld modules, H(B^*) is braided commutative. By the Brzezinski--Militaru theorem, H(B^*)#D(B) is then a Hopf algebroid over H(B^*). For B a particular Taft Hopf algebra at a 2p-th root of unity, the construction is adapted to yield Yetter--Drinfeld module algebras over the 2p^3-dimensional quantum group U_q(s…
Let $\mathcal{C}$ be a finite braided multitensor category. Let $B$ be Majid's automorphism braided group of $\mathcal{C}$, then $B$ is a cocommutative Hopf algebra in $\mathcal{C}$. We show that the center of $\mathcal{C}$ is isomorphic to the category of left $B$-comodules in $\mathcal{C}$, and the decomposition of $B$ into a direct sum of indecomposable $\mathcal{C}$-subcoalgebras leads to a decomposition of $B$-$\operatorname*{Comod}_{\mathcal{C}}$ into a direct sum of indecomposable $\mathcal{C}$-module subcategories. As an application, we present an explicit characterization of the struc…
We introduce a family of braided Hopf algebras that generalizes the rank 1 Hopf algebras introduced by Krop anad Radford and we study its cleft extensions
Let $V$ be a linear space over a field ${\bf k}$ with a braiding $τ: V\otimes V\rightarrow V\otimes V.$ We prove that the braiding $τ$ has a unique extension on the free nonassociative algebra ${\bf k}\{V\}$ freely generated by $V$ so that ${\bf k}\{V\}$ is a braided algebra. Moreover, we prove that the free braided algebra ${\bf k}\{V\}$ has a natural structure of a braided nonassociative Hopf algebra such that every element of the space of generators $V$ is primitive. In the case of involutive braidings, $τ^2={\rm id}$, we describe braided analogues of Shestakov-Umirbaev operations and prove…
We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces…
We present new examples of deformations of smash product algebras that arise from Hopf algebra actions on pairs of module algebras. These examples involve module algebras that are Koszul, in which case a PBW theorem we established previously applies. Our construction generalizes several `double' constructions appearing in the literature, including Weyl algebras and some types of Cherednik algebras, and it complements the braided double construction of Bazlov and Berenstein. Many suggestions of further directions are provided at the end of the work.
This paper introduces group-cograded monoidal Hom-Hopf algebras, and shows that this kind of group-cograded monoidal Hom-Hopf algebras are monoidal Hom-Hopf algebras in the Turaev category $\mathcal{J}_{k}$ introduced by Canepeel and De Lombaerde. Then we define the $p$-Yetter-Drinfeld category over a group-cograded monoidal Hom-Hopf algebra, and construct a new kind of braided $T$-categories.
Let $ Aut_{mHH}(H)$ denote the set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel and Goyvaerts \cite{CG2011}. The main aim of this paper is to provide new examples of braided $T$-category in the sense of Turaev \cite{T2008}. For this, first we construct a monoidal Hom-Hopf $T$-coalgebra $\mathcal{MHD}(H)$ and prove that the $T$-category $Rep(\mathcal{MHD}(H))$ of representation of $\mathcal{MHD}(H)$ is isomorphic to $\mathcal {MHYD}(H)$ as braided $T$-categories, if $H$ is finite-dimensional. Then we construct a new braided $T$-category $…
Let $ Aut_{mHH}(H)$ denote a set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel S. and Goyvaerts I. (Commun. Algebra 39, 2216-2240, 2011) and let $G$ be a crossed product group $ Aut_{mHH}(H)\times Aut_{mHH}(H)$. The main aim of this paper is to provide further examples of braided $T$-category in the sense of Turaev (1994, 2008). For this purpose, we first introduce a class of new categories $_{H}\mathcal {MHYD}^{H}(A, B)$ of monoidal Hom $(A, B)$-Yetter-Drinfeld modules with $A, B \in Aut_{mHH}(H)$. Then we show that the category ${\cal M…
A definition of a quantum vertex algebra, which is a deformation of a vertex algebra, was proposed by Etingof and Kazhdan in 1998. In a nutshell, a quantum vertex algebra is a braided state-field correspondence which satisfies associativity and braided locality axioms. We develop a structure theory of quantum vertex algebras, parallel to that of vertex algebras. In particular, we introduce braided n-products for a braided state-field correspondence and prove for quantum vertex algebras a version of the Borcherds identity.
We study formal deformations of a crossed product $S(V)#_f G$, of a polynomial algebra with a group, induced from a universal deformation formula introduced by Witherspoon. These deformations arise from braided actions of Hopf algebras generated by automorphisms and skew derivations. We show that they are nontrivial in the characteristic free context, even if $G$ is infinite, by showing that their infinitesimals are not coboundaries. For this we construct a new complex which computes the Hochschild cohomology of $S(V)#_f G$.
Let $H$ be a finite Hopf algebra with $C_{H,H} = C_{H,H}^{-1}.$ The duality theorem is shown for $H$, i.e., $$ (R # H)# H^{\hat *} \cong R \otimes (H \bar \otimes H^{\hat *}) \hbox {as algebras in} {\cal C}.$$ Also, it is proved that the Drinfeld double $(D(H),[b])$ is a quasi-triangular Hopf algebra in ${\cal C}$.
We introduced a braided Sweedler cohomology, which is adequate to work with the H-braided cleft extensions studied in [J. A. Guccione and J. J. Guccione, Theory of braided Hopf crossed products, Journal of Algebra, Vol 261 (2003) 54-101]
