[Paper Review] Primitive Cohomology of Hopf algebras

D.-G. Wang, J.J. Zhang+1 more|arXiv (Cornell University)|Nov 17, 2014

Algebraic structures and combinatorial models30 references3 citations

TL;DR

This paper introduces primitive cohomology of Hopf algebras via a modified cobar construction on the underlying coalgebra, using it to classify non-locally PI, pointed Hopf algebra domains of Gelfand-Kirillov dimension two and all pointed Hopf algebras of rank one. The key contribution is a complete classification of these classes, extending prior results and identifying new families of Hopf algebras through cohomological invariants derived from coalgebraic structure.

ABSTRACT

Primitive cohomology of a Hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra. Among many of its applications, two classifications are presented. Firstly we classify all non locally PI, pointed Hopf algebra domains of Gelfand-Kirillov dimension two; and secondly we classify all pointed Hopf algebras of rank one. The first classification extends some results of Brown, Goodearl and others in an ongoing project to understand all Hopf algebras of low Gelfand-Kirillov dimension. The second generalizes results of Krop-Radford and Wang-You-Chen which classified Hopf algebras of rank one under extra hypothesis. Properties and algebraic structures of the primitive cohomology are discussed.

Motivation & Objective

• To develop a cohomological invariant—primitive cohomology—based on the coalgebraic structure of Hopf algebras for studying infinite-dimensional Hopf algebras.
• To classify all non-locally PI, pointed Hopf algebra domains of Gelfand-Kirillov dimension two.
• To classify all pointed Hopf algebras of rank one, generalizing prior results that required additional generation hypotheses.
• To compute primitive cohomology explicitly for key families of Hopf algebras, using which the main classifications are proven.

Proposed method

• Primitive cohomology is defined as the Hochschild cohomology of the coalgebra with coefficients in a special 1-dimensional bicomodule, constructed via a generalized cobar complex.
• The construction uses the cobar complex of the underlying coalgebra, modified to incorporate grouplike and skew-primitive elements.
• The paper employs a left adjoint action of a cocommutative Hopf subalgebra $ K $ on the cohomology ring $ rak{P}_D(H) $, induced by its action on tensor complexes.
• Key computations are performed for families of Hopf algebras, including $ U(rak{g}) $, $ A(n,q) $, $ C(n) $, and new families like $ E_G $, $ F_G $, $ L_G $, etc.
• The classification theorems rely on detailed computation of $ rak{P}^1_{g,h}(C) $, particularly its vanishing or non-vanishing, to distinguish algebraic structures.
• The theory is applied to classify Hopf algebras of low GK-dimension and rank, using cohomological obstructions to finitely generated or PI structures.

Experimental results

Research questions

• RQ1 Which non-locally PI, pointed Hopf algebra domains of GK-dimension two exist, and how can they be completely classified using coalgebraic cohomological invariants?
• RQ2 How can the class of pointed Hopf algebras of rank one be fully classified without assuming generation by grouplike and skew-primitive elements?
• RQ3 What is the role of primitive cohomology in distinguishing Hopf algebras of low GK-dimension and low rank?
• RQ4 How does the primitive cohomology detect structural features such as non-PI or non-locally finite generation?
• RQ5 What new families of Hopf algebras emerge from the generalized rank-one classification, and what are their defining cohomological properties?

Key findings

• All non-locally PI, pointed Hopf algebra domains of GK-dimension less than three are isomorphic to one of: the universal enveloping algebra $ U(rak{g}) $ of the 2-dimensional solvable Lie algebra, $ A(n,q) $ for $ n>0 $, $ q $ not a root of unity, or $ C(n) $ for $ n eq 1 $.
• The classification of pointed Hopf algebras of rank one includes nine isomorphism classes, including new families such as $ E_G $, $ F_G $, $ L_G $, $ N_G $, $ O_G $, $ P_G $, and $ Q_G $, which were not previously classified.
• The primitive cohomology $ rak{P}^1_{g,h}(C) $ is isomorphic to the quotient $ P_{h,g}/k(g-h) $, providing a cohomological obstruction to certain algebraic structures.
• The left adjoint action of a cocommutative Hopf subalgebra $ K $ on the cohomology ring $ rak{P}_D(H) $ is compatible with the differential, ensuring $ ext{ad}_l(a)( ext{d}(x)) = ext{d}( ext{ad}_l(a)(x)) $, which is essential for the cohomological structure.
• The computation of $ rak{P}^n_{g,h}(C) $ for key families is central to proving the classification theorems, particularly for detecting non-PI and non-locally finite structures.
• The paper establishes that the primitive cohomology is a powerful invariant for classifying Hopf algebras of low GK-dimension and low rank, especially when combined with homological and algebraic constraints.