Narrow Results

An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommutative and one anti-cocommutative comultiplications generalizes and illustrates the polarization–depolarization process in the context of bialgebras. The study of a special class of adm-Poisson bialgebras which include the known coboundary Poisson bialgebras in [27] as a proper subclass in general, illustrating an advantage in terms of the presentation with one operation, leads to the introduction of adm-Poisson Yang–Baxter equation in an adm-Poisson algebra. It is an unexpected consequence that both the adm-Poisson Yang–Baxter equation and the associative Yang–Baxter equation have the same form and thus it motivates and simplifies the involved study from the study of the associative Yang–Baxter equation, which is another advantage in terms of the presentation with one operation. A skew-symmetric solution of adm-Poisson Yang–Baxter equation gives an adm-Poisson bialgebra. Finally, the notions of an ${𝒪}$-operator of an adm-Poisson algebra and a pre-adm-Poisson algebra are introduced to construct skew-symmetric solutions of adm-Poisson Yang–Baxter equation and hence adm-Poisson bialgebras. Note that a pre-adm-Poisson algebra gives an equivalent presentation for a pre-Poisson algebra introduced by Aguiar.

We introduce a notion of antipode for monoidal (complete) decomposition spaces, inducing a notion of weak antipode for their incidence bialgebras. In the connected case, this recovers the usual notion of antipode in Hopf algebras. In the non-connected case, it expresses an inversion principle of more limited scope, but still sufficient to compute the Möbius function as $\mu =\zeta \circ S$, just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors $S_{even}-S_{odd}$, and it is a refinement of the general Möbius inversion construction of Gálvez–Kock–Tonks, but exploiting the monoidal structure.

We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced [A. L. Agore, A. S. Gordienko and J. Vercruysse, On equivalences of (co)module algebra structures over Hopf algebras, J. Noncommut. Geom., doi: 10.4171/JNCG/428.] bi/Hopf-algebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear maps between tensor powers of A, called $\mathrm{\Omega }$-algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study V-universal measuring coalgebras and V-universal comeasuring algebras between $\mathrm{\Omega }$-algebras A and B, relative to a fixed subspace V of $Vect(A,B)$. By considering the case $A=B$, we derive the notion of a V-universal (co)acting bialgebra (and Hopf algebra) for a given algebra A. In particular, this leads to a refinement of the existence conditions for the Manin–Tambara universal coacting bi/Hopf algebras. We establish an isomorphism between the V-universal acting bi/Hopf algebra and the finite dual of the V-universal coacting bi/Hopf algebra under certain conditions on V in terms of the finite topology on $En{d}_F(A)$.

The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang–Baxter equation and ${𝒪}$-operators are studied.

We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

We set up an algebraic theory of multivariable integration, based on a hierarchy of Rota–Baxter operators and an action of the matrix monoid as linear substitutions. Given a suitable coefficient domain with a bialgebra structure, this allows us to build an operator ring that acts naturally on the given Rota–Baxter hierarchy. We conjecture that the operator relations are a noncommutative Gröbner–Shirshov basis for the ideal they generate.

We study the pre-Lie algebra of rooted trees $({𝒯},\to )$ and we define a pre-Lie structure on its doubling space $(V,⇝)$. Also, we find the enveloping algebras of the two pre-Lie algebras denoted, respectively, by $({{\mathscr{H}}}^{\prime },\star ,\mathrm{\Gamma })$ and $({{𝒟}}^{\prime },\star ,\chi )$. We prove that $({{𝒟}}^{\prime },\star ,\chi )$ is a module-bialgebra on $({{\mathscr{H}}}^{\prime },\star ,\mathrm{\Gamma })$ and we find some relations between the two pre-Lie structures.

This paper introduces the notion of a quadri-bialgebra, which leads to a bialgebra theory for Aguiar and Loday’s quadri-algebra (cf. [4]). We show that a quadri-bialgebra is equivalent to a Manin triple of dendriform algebras associated to a nondegenerate 2-cocycle, and to a Manin triple of quadri-algebras associated to a nondegenerate invariant bilinear form. Quadri-bialgebras also come from a variation of the classical Yang-Baxter equation, called the Q-equations. Moreover, quadri-bialgebras fit into the framework of construction of Rota-Baxter operators and Nijenhuis operators on the double spaces of quadri-algebras.

Let M be a k–vector space and R ∈ End_k(M ⊗ M). The Hopf equation: R²³R¹³R¹² = R¹²R²³ and the following FRT type theorem: if M is a finite dimensional vector space and R ∈ End_k(M ⊗ M) is a solution of the Hopf equation, then there exists a bialgebra B(R) such that (M, ·, ρ) ∈_B(R) M^B(R). were introduced and mentioned in the monograph of G. Militaru. In this paper, we will use the above to construct new examples of noncommutative and noncocommutative bialgebras which are different from the previous arising from quantum group theory.