[Paper Review] Partial bi(co)module algebras, globalizations, and partial (L,R)-smash products

Felipe Castro, Antonio Paques+2 more|arXiv (Cornell University)|May 19, 2015

Algebraic structures and combinatorial models6 references3 citations

TL;DR

This paper introduces partial bi(co)module algebras and constructs the partial $(L,R)$-smash product as a generalization of the global $(L,R)$-smash product in the context of partial Hopf actions. It establishes conditions for globalizations of partial $H$-bimodule and bicomodule algebras, proving that such globalizations are isomorphic as $H^0$-bimodules when $H^0$ separates points, and shows that the partial $(L,R)$-smash product forms an associative algebra, with unital structure possible under specific idempotent and action conditions.

ABSTRACT

In this paper we introduce the notions of partial bimodule algebra and partial bico- module algebra. We also deal with the existence of globalizations for these structures, generalizing related results appeared in [2, 4]. As an application we construct the partial (L;R)-smash product, extending the corresponding global notion appeared in [14] to the context of partial Hopf actions.

Motivation & Objective

• To generalize the notion of global $(L,R)$-smash products to the setting of partial Hopf actions.
• To define and study partial $H$-bimodule and bicomodule algebras as natural extensions of partial Hopf actions.
• To investigate the existence and structure of globalizations for partial $H$-bimodule and bicomodule algebras.
• To construct the partial $(L,R)$-smash product of a partial $H$-bimodule algebra and a partial $H$-bicomodule algebra as a new associative algebra.
• To identify conditions under which the partial $(L,R)$-smash product admits a unit, using idempotent elements and compatibility with the counit.

Proposed method

• Introduce the notion of a partial $H$-bimodule algebra via left and right partial actions satisfying axioms (LPMA1)-(LPMA3) and their right counterparts.
• Define partial $H$-bicomodule algebras via left and right partial coactions satisfying dual axioms, using Sweedler's notation and the finite dual $H^0$.
• Establish a correspondence between partial $H$-bicomodule algebras and partial $H^0$-bimodule algebras, showing that their globalizations are isomorphic as $H^0$-bimodules when $H^0$ separates points.
• Construct the partial $(L,R)$-smash product $A aturalar{A}$ as a new algebra with multiplication defined by $(a atural u)(b atural v) = (alat u^{+1})(u^{-1} ightharpoonup b) atural u^{-0}v^{+0}$, using the partial actions and coactions.
• Prove associativity of the partial $(L,R)$-smash product using the axioms of partial actions and coactions, and verify the multiplication rule via component-wise computation.
• Identify sufficient conditions—particularly involving idempotent elements $a \in A$ and $u \in \bar{A}$, and compatibility with the counit—under which $a\natural u$ becomes a unit in $A\natural\bar{A}$.

Experimental results

Research questions

• RQ1 Under what conditions does a partial $H$-bimodule algebra admit a globalization, and how does it relate to the globalization of a partial $H$-bicomodule algebra?
• RQ2 Is there a canonical isomorphism between the globalization of a partial $H$-bimodule algebra and that of a partial $H$-bicomodule algebra when $H^0$ separates points?
• RQ3 Can the partial $(L,R)$-smash product of a partial $H$-bimodule algebra and a partial $H$-bicomodule algebra be endowed with a unit, and what conditions ensure this?
• RQ4 How do the axioms of partial actions and coactions interact to ensure associativity of the partial $(L,R)$-smash product?
• RQ5 What role do idempotent elements in $A$ and $\bar{A}$ play in constructing a unital structure on the partial $(L,R)$-smash product?

Key findings

• The globalization of a partial $H$-bimodule algebra and the globalization of a partial $H$-bicomodule algebra are isomorphic as $H^0$-bimodules when $H^0$ separates points.
• The partial $(L,R)$-smash product $A\natural\bar{A}$ is an associative algebra under the defined multiplication rule, as verified by direct computation using the axioms of partial actions and coactions.
• If $a \in A$ and $u \in \bar{A}$ are nonzero idempotents satisfying $h\rightharpoonup a = \varepsilon(h)a$ and $a\leftharpoonup h = \varepsilon(h)a$, and $\rho(u) = u \otimes 1_H$, $\lambda(u) = 1_H \otimes u$, then $a\natural u$ is an idempotent in $A\natural\bar{A}$.
• When all four conditions (1)-(4) hold, $a\natural u$ is a two-sided identity in $A\natural\bar{A}$, making $A\underline{\natural}\bar{A}$ a unital algebra with identity $a\natural u$.
• The construction generalizes both the standard smash product and the global $(L,R)$-smash product, extending them to the partial setting.
• In the example with $H = \mathbb{H}_4$, $A = \bar{A} = \Bbbk$, and specific partial actions and coactions, $(1_\Bbbk \natural 1_\Bbbk)^2 = 0$ is possible, showing that the unit may fail to exist unless conditions on $r, s, t, u$ are met.