Polynomial identities of Hopf algebras

Kotchetov, Mikhail V. (2002) Polynomial identities of Hopf algebras. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

In this dissertation we consider Hopf algebras that satisfy a polynomial identity as algebras or coalgebras. The notion of a polynomial identity for an algebra is classical. The dual notion of an identity for a coalgebra is new. -- In Chapter 0 we give basic definitions and facts that are used throughout the rest of this work. -- Chapter 1 is devoted to coalgebras with a polynomial identity. First we introduce the notion of identity of a coalgebra and discuss its general properties. Then we study what classes of coalgebras are varieties, i.e. can be defined by a set of identities. In the case of algebras, varieties are characterized by the classical Theorem of Birkhoff. Somewhat unexpectedly, the dual statement for coalgebras does not hold. Further, we give two realizations of a relatively (co)free coalgebra of a variety: one via the so called finite dual of a relatively free algebra and the other a direct construction using some kind of symmetric functions. -- In Chapter 2 we give necessary and sufficient conditions for a cocommutative Hopf algebra (with additional restrictions in the case of prime characteristic) to satisfy a polynomial identity as an algebra. These results generalize the well-known Passman's Theorem on group algebras with a polynomial identity and Bahturin-Latysev's Theorem on universal enveloping algebras with a polynomial identity. The proofs for the case of prime characteristic are given in Chapter 4. -- In Chapter 3 we dualize the results of Chapter 2 to obtain some criteria for a commutative Hopf algebra (assumed reduced in the case of prime characteristic) to satisfy an identity as a coalgebra. We also extend our result in charecteristic zero to a certain class of nearly commutative Hopf algebras (pseudoinvolutive Hopf algebras of Etingof-Gelaki). -- Finally, in Chapter 4 we use the interpretation of cocommutative Hopf algebras as formal groups to prove the results of Chapter 2. Our method also demonstrates that Bahturin-Latysev's Theorem in characteristic zero is in fact a corollary of Passman's Theorem. -- For the most part, this dissertation is based on my papers [19], [20], and [21].

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