Smash Product Research Articles

Symmetric monoidal smash products in homotopy type theory

Abstract In homotopy type theory, few constructions have proved as troublesome as the smash product. While its definition is just as direct as in classical mathematics, one quickly realises that in order to define and reason about functions over iterations of it, one has to verify an exponentially growing number of coherences. This has led to crucial results concerning smash products remaining open. One particularly important such result is the fact that the smash product forms a (1-coherent) symmetric monoidal product on the universe of pointed types. This fact was used, without a complete proof, by, for example, Brunerie ((2016) PhD thesis, Université Nice Sophia Antipolis) to construct the cup product on integral cohomology and is, more generally, a fundamental result in traditional algebraic topology. In this paper, we salvage the situation by introducing a simple informal heuristic for reasoning about functions defined over iterated smash products. We then use the heuristic to verify, for example, the hexagon and pentagon identities, thereby obtaining a proof of symmetric monoidality. We also provide a formal statement of the heuristic in terms of an induction principle concerning the construction of homotopies of functions defined over iterated smash products. The key results presented here have been formalised in the proof assistant Cubical Agda.

A Note on the Smash Product and Regular Associativity

A Note on the Smash Product and Regular Associativity

Noncommutative differential geometry on crossed product algebras

We provide a differential structure on arbitrary cleft extensions B:=AcoH⊆A for an H-comodule algebra A. This is achieved by constructing a covariant calculus on the corresponding crossed product algebra B#σH from the data of a bicovariant calculus on the structure Hopf algebra H and a calculus on the base algebra B, which is compatible with the 2-cocycle and measure of the crossed product. The result is a quantum principal bundle with canonical strong connection and we describe the induced bimodule covariant derivatives on associated bundles of the crossed product. All results specialize to trivial extensions and smash product algebras B#H and we give a characterization of the smash product calculus in terms of the differentials of the cleaving map j:H→A and the inclusion B↪A. The construction is exemplified for pointed Hopf algebras. In particular, the case of Radford Hopf algebras H(r,n,q) is spelled out in detail.

Künneth formulas for Cotor

We investigate the question of how to compute the cotensor product, and more generally the derived cotensor (i.e. Cotor) groups, of a tensor product of comodules. In particular, we determine the conditions under which there is a Künneth formula for Cotor. We show that there is a simple Künneth theorem for Cotor groups if and only if an appropriate coefficient comodule has trivial coaction. This result is an application of a spectral sequence we construct for computing Cotor of a tensor product of comodules. Finally, for certain families of nontrivial comodules which are especially topologically natural, we work out necessary and sufficient conditions for the existence of a Künneth formula for the [Formula: see text]th Cotor group, i.e. the cotensor product. We give topological applications in the form of consequences for the [Formula: see text]-term of the Adams spectral sequence of a smash product of spectra, and the Hurewicz image of a smash product of spectra.

Morita equivalence and globalization for partial Hopf actions on nonunital algebras

In this work, we investigate partial actions of a Hopf algebra [Formula: see text] on nonunital algebras and the associated partial smash products, with the objective of providing a framework where one may obtain results for both [Formula: see text]-algebras with local units and [Formula: see text]-categories. We show that our partial actions correspond to nonunital algebras in the category of partial representations of [Formula: see text]. The central problem of existence of a globalization for a partial action is studied in detail, and we provide sufficient conditions for the existence (and uniqueness) of a minimal globalization for associative algebras in general. Extending previous results by Abadie, Dokuchaev, Exel and Simon, we define Morita equivalence for partial Hopf actions, and we show that if two symmetrical partial actions are Morita equivalent then their standard globalizations are also Morita equivalent. Particularizing to the case of a partial action on an algebra with local units, we obtain several strong results on equivalences of categories of modules of partial smash products of algebras and partial smash products of [Formula: see text]-categories.

(Co)homology of partial smash products

Given a cocommutative Hopf algebra H over a commutative ring K and a symmetric partial action of H on a K-algebra A, we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the smash product A#H, involving the Hochschild homology of A and the partial homology of H. An analogous third quadrant cohomological spectral sequence is also obtained. The definition of the partial (co)homology of H under consideration is based on the category of the partial representations of H. A specific partial representation of H on a subalgebra B of the partial “Hopf” algebra Hpar is involved in the definition and we construct a projective resolution of B.

Weak Hopf algebras, smash products and applications to adjoint-stable algebras

For a semisimple quasi-triangular Hopf algebra [Formula: see text] over a field [Formula: see text] of characteristic zero, and a strongly separable quantum commutative [Formula: see text]-module algebra [Formula: see text], we show that [Formula: see text] is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra [Formula: see text]. With these structures, [Formula: see text] is the monoidal category introduced by Cohen and Westreich, and [Formula: see text] is tensor equivalent to [Formula: see text]. If [Formula: see text] is in the Müger center of [Formula: see text], then the embedding is a quasi-triangular weak Hopf algebra morphism. This explains the presence of a subgroup inclusion in the characterization of irreducible Yetter–Drinfeld modules for a finite group algebra.

Some graded Hopf-module pseudoalgebras

In this paper, we define a G-graded Hopf module associative pseudoalgebra A for any grading group G and Hopf algebra H, and construct a smash product A#H of A with H, which is also a G-graded associative pseudoalgebra. Then we establish a Mortia context between A#H and AH, the fixed sub-pseudoalgebra under the action of H. In addition, we prove a similarity of the Schur-Weyl type duality between the group algebra k[Se(n)] of Sergeev group Se(n) and a super pseudoalgebra.

On BiHom-L-R Smash Products

Let [Formula: see text] be a BiHom-Hopf algebra and [Formula: see text] be an [Formula: see text]-BiHom-bimodule algebra, where the maps [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are bijective. We first prove the Maschke-type theorem for the BiHom-L-R smash product over a finite-dimensional semisimple BiHom-Hopf algebra. Next we give a Morita context between the BiHom-subalgebra [Formula: see text] and the BiHom-L-R smash product [Formula: see text].

Gorenstein Flat Modules of Hopf-Galois Extensions

Let A/B be a right H-Galois extension over a semisimple Hopf algebra H. The purpose of this paper is to give the relationship of Gorenstein flat dimensions between the algebra A and its subalgebra B, and obtain that the global Gorenstein flat dimension and the finitistic Gorenstein flat dimension of A is no more than that of B. Then the problem of preserving property of Gorenstein flat precovers for the Hopf-Galois extension will be studied. Finally, more relations for the crossed products and smash products will be obtained as applications.

External Spanier–Whitehead duality and homology representation theorems for diagram spaces

We construct a Spanier-Whitehead type duality functor relating finite $\mathcal{C}$-spectra to finite $\mathcal{C}^{\mathrm{op}}$-spectra and prove that every $\mathcal{C}$-homology theory is given by taking the homotopy groups of a balanced smash product with a fixed $\mathcal{C}^{\mathrm{op}}$-spectrum. We use this to construct Chern characters for certain rational $\mathcal{C}$-homology theories.

Scalar extension Hopf algebroids

Given a Hopf algebra [Formula: see text], Brzeziński and Militaru have shown that each braided commutative Yetter–Drinfeld [Formula: see text]-module algebra [Formula: see text] gives rise to an associative [Formula: see text]-bialgebroid structure on the smash product algebra [Formula: see text]. They also exhibited an antipode map making [Formula: see text] the total algebra of a Lu’s Hopf algebroid over [Formula: see text]. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair [Formula: see text] and [Formula: see text] of braided commutative Yetter–Drinfeld [Formula: see text]-module algebras, and output is a symmetric Hopf algebroid [Formula: see text] over [Formula: see text]. This construction does not require that the antipode of [Formula: see text] is invertible.

A Duality Theorem for Hopf Quasimodule Algebras

In this paper, we introduce and study two smash products A★H for a left H-quasimodule algebra A over a Hopf quasigroup H over a field K and B#U for a coquasi U-module algebra B over a Hopf coquasigroup U, respectively. Then, we prove our duality theorem (A★H)#H*≅A⊗(H#H*)≅A⊗Mn(K)≅Mn(A) in the setting of a Hopf quasigroup H of dimension n. As an application of our result, we consider a special case of a finite quasigroup.

A Morita-Takeuchi Context and Hopf Coquasigroup Galois Coextensions

For H a Hopf quasigroup and C, a left quasi H-comodule coalgebra, we show that the smash coproduct C⋊H (as a symmetry of smash product) is linked to some quotient coalgebra Q=C/CH*+ by a Morita-Takeuchi context (as a symmetry of Morita context). We use the Morita-Takeuchi setting to prove that for finite dimensional H, equivalent conditions for C/Q to be a Hopf quasigroup Galois coextension (as a symmetry of Galois extension). In particular, we consider a special case of quasigroup graded coalgebras as an application of our theory.

Harish-Chandra Modules over Hopf Galois Orders

Abstract The theory of Galois orders was introduced by Futorny and Ovsienko [9]. We introduce the notion of $\mathcal {H}$-Galois $\Lambda $-orders. These are certain noncommutative orders $F$ in a smash product of the fraction field of a noetherian integral domain $\Lambda $ by a Hopf algebra ${\mathcal {H}}$ (or, more generally, by a coideal subalgebra of a Hopf algebra). They are generalizations of Webster’s [25] principal flag orders. Examples include Cherednik algebras, as well as examples from Hopf Galois theory. We also define spherical Galois orders, which are the corresponding generalizations of principal Galois orders introduced by the author [12]. The main results are (1) for every maximal ideal $\mathfrak {m}$ of $\Lambda $ of finite codimension, there exists a simple Harish-Chandra $F$-module in the fiber of $\mathfrak {m}$; (2) for every character of $\Lambda $, we construct a canonical simple Harish-Chandra module as a subquotient of the module of local distributions; (3) if a certain stabilizer coalgebra is finite-dimensional, then the corresponding fiber of simple Harish-Chandra modules is finite; and (4) centralizers of symmetrizing idempotents are spherical Galois orders and every spherical Galois order appears that way.

Yoneda Ext-algebras of Takeuchi smash products

We prove that the Yoneda Ext-algebra of a Takeuchi smash product is the graded Takeuchi smash product of the Yoneda Ext-algebras of the two algebras or modules involved. As an application, we prove that graded Takeuchi smash products preserve Artin-Schelter regularity, and describe the Nakayama automorphism of the product.

Hopf Quasigroup Galois Extensions and a Morita Equivalence

For H, a Hopf coquasigroup, and A, a left quasi-H-module algebra, we show that the smash product A#H is linked to the algebra of H invariants AH by a Morita context. We use the Morita setting to prove that for finite dimensional H, there are equivalent conditions for A/AH to be Galois parallel in the case of H finite dimensional Hopf algebra.

Invariant properties of modules under smash products from finite dimensional algebras

<abstract><p>We give the relationship between indecomposable modules over the finite dimensional $ k $-algebra $ A $ and the smash product $ A\sharp G $ respectively, where $ G $ is a finite abelian group satisfying $ G\subseteq Aut(A) $, and $ k $ is an algebraically closed field with the characteristic not dividing the order of $ G $. More precisely, we construct all indecomposable $ A\sharp G $-modules from indecomposable $ A $-modules and prove that an $ A\sharp G $-module is indecomposable if and only if it is an indecomposable $ G $-stable module over $ A $. Besides, we give the relationship between simple, projective and injective modules in $ modA $ and those in $ modA\sharp G $.</p></abstract>

On the derived Lusztig correspondence

Abstract Let G be a connected reductive group, T a maximal torus of G, N the normalizer of T and $W=N/T$ the Weyl group of G. Let ${\mathfrak {g}}$ and ${\mathfrak {t}}$ be the Lie algebras of G and T. The affine variety $\mathfrak {car}={\mathfrak {t}}/\!/W$ of semisimple G-orbits of ${\mathfrak {g}}$ has a natural stratification $$ \begin{align*} \mathfrak{car}=\coprod_{\lambda}\mathfrak{car}_{\lambda} \end{align*} $$ indexed by the set of G-conjugacy classes of Levi subgroups: the open stratum is the set of regular semisimple orbits and the closed stratum is the set of central orbits. In [17], Rider considered the triangulated subcategory $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\mathrm {nil}}/G])^{\mathrm {Spr}}$ of $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\mathrm {nil}}/G])$ generated by the direct summand of the Springer sheaf. She proved that it is equivalent to the derived category of finitely generated dg modules over the smash product algebra ${\overline {\mathbb {Q}}_{\ell }}[W]\# H^{\bullet }_G(G/B)$ where $H^{\bullet }_G(G/B)$ is the G-equivariant cohomology of the flag variety. Notice that the later derived category is $D_{\mathrm {c}}^{\mathrm {b}}(\mathrm {B}(N))$ where $\mathrm {B}(N)=[\mathrm {Spec}(k)/N]$ is the classifying stack of N-torsors. The aim of this paper is to understand geometrically and generalise Rider’s equivalence of categories: For each $\lambda $ we construct a cohomological correspondence inducing an equivalence of categories between $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {t}}_{\lambda }/N])$ and $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\lambda }/G])^{\mathrm {Spr}}$ .

Superpotentials and Quiver Algebras for Semisimple Hopf Actions

We consider the action of a semisimple Hopf algebra H on an m-Koszul Artin–Schelter regular algebra A. Such an algebra A is a derivation-quotient algebra for some twisted superpotential w, and we show that the homological determinant of the action of H on A can be easily calculated using w. Using this, we show that the smash product A#H is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra Λ to which A#H is Morita equivalent, generalising a result of Bocklandt–Schedler–Wemyss. We also show how Λ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan–Kirkman–Walton–Zhang follow using our techniques.