[Paper Review] Mini-course on Hopf algebras--Hopf crossed products--

Akira Masuoka|arXiv (Cornell University)|Jul 6, 2012

Algebraic structures and combinatorial models14 references3 citations

TL;DR

This paper presents a comprehensive mini-course on Hopf crossed products, generalizing group crossed products to Hopf algebras, and establishes their characterization as Hopf-Galois extensions with a normal basis. The key contribution is a structural decomposition theorem for super-commutative Hopf superalgebras, showing they are isomorphic to a tensor product of an exterior algebra and a group-like Hopf algebra via Hopf crossed product construction.

ABSTRACT

Hopf crossed products, or in other words, cleft comodule algebras form a special but important class in Hopf-Galois extensions. To discuss this interesting subject, we will start with the more familiar group crossed products, and then see that they are naturally generalized by Hopf crossed products; these Hopf crossed products are characterized as Hopf-Galois extensions with normal basis. After showing this characterization due to Doi and Takeuchi, we will proceed to two applications of Hopf crossed products---the equivariant smoothness of Hopf algebras, and the tensor product decomposition in super-commutative Hopf superalgebras.

Motivation & Objective

• To generalize group crossed products to Hopf crossed products using Hopf algebra structures.
• To characterize Hopf crossed products as Hopf-Galois extensions with a normal basis, extending classical results by Doi and Takeuchi.
• To apply Hopf crossed products to prove equivariant smoothness of Hopf algebras and to decompose super-commutative Hopf superalgebras.
• To establish a canonical isomorphism between a super-commutative Hopf superalgebra and a tensor product of an exterior algebra and a group-like Hopf algebra.

Proposed method

• Define group crossed products using a group action and a 2-cocycle satisfying associativity and normalization conditions.
• Generalize group crossed products to Hopf crossed products using a Hopf algebra H and a right H-comodule algebra B with a normalized cocycle σ.
• Use the Hopf-Galois property and the existence of a normal basis to characterize Hopf crossed products via the Doi-Takeuchi theorem.
• Construct a canonical isomorphism α: A → ∧(W) ⊗ H, where A is a super-commutative Hopf superalgebra, W is the space of odd primitives in the dual coalgebra, and H is a group-like Hopf algebra.
• Utilize the dual coalgebra A° and the space of odd primitives U = W* to induce an isomorphism γ: B → ∧(W) on the neutral component.
• Verify that the map α is an augmented, right H-colinear, left B-linear isomorphism by checking its behavior modulo the nilpotent ideal B⁺.

Experimental results

Research questions

• RQ1 How can group crossed products be generalized to Hopf crossed products in the context of Hopf algebras?
• RQ2 What characterizes Hopf crossed products as Hopf-Galois extensions with a normal basis?
• RQ3 Under what conditions does a super-commutative Hopf superalgebra admit a tensor product decomposition via Hopf crossed products?
• RQ4 How does the dual coalgebra and the space of odd primitives relate to the structure of a super-commutative Hopf superalgebra?
• RQ5 What is the role of the cocycle σ in ensuring the associativity and algebra structure of the Hopf crossed product?

Key findings

• Hopf crossed products are characterized as Hopf-Galois extensions with a normal basis, generalizing the classical Doi-Takeuchi theorem.
• Every strongly graded algebra with invertible elements in each component arises as a group crossed product B ⋊σ Γ.
• For a super-commutative Hopf superalgebra A, there exists a canonical isomorphism α: A → ∧(W) ⊗ H, where W is the dual of the space of odd primitives in the dual coalgebra.
• The neutral component B of A is isomorphic to the exterior algebra ∧(W), and B⁺ is nilpotent, with B⁺ = B₁B.
• The decomposition A ≅ ∧(W) ⊗ H holds as superalgebras, with H being a group-like Hopf algebra, and the isomorphism α is both augmented and right H-colinear.
• The isomorphism α is an isomorphism modulo the nilpotent ideal B⁺, and since α is left B-linear and an isomorphism modulo B⁺, it is an isomorphism overall.