This is a slight extension of an expository paper I wrote a while ago as a supplement to my joint work with Declan Quinn on Burnside's theorem for Hopf algebras. It was never published, but may still ...be of interest to students and beginning researchers.
Let K be a field, and let A be an algebra over K. Then the tensor product A ⊗ A = A ⊗
K
A is also a K-algebra, and it is quite possible that there exists an algebra homomorphism Δ: A → A ⊗ A. Such a map Δ is called a comultiplication, and the seemingly innocuous assumption on its existence provides A with a good deal of additional structure. For example, using Δ, one can define a tensor product on the collection of A-modules, and when A and Δ satisfy some rather mild axioms, then A is called a bialgebra. Classical examples of bialgebras include group rings KG and Lie algebra enveloping rings U(L). Indeed, most of this paper is devoted to a relatively self-contained study of some elementary bialgebra properties of these examples. Furthermore, Δ determines a convolution product on Hom
K
(A, A), and this leads quite naturally to the definition of a Hopf algebra.
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2.
Permutational and rewritable groups Passman, D. S.
Proceedings of the American Mathematical Society,
03/2019, Volume:
147, Issue:
3
Journal Article
Peer reviewed
Permutational groups and rewritable groups were introduced in 1985 and 1988, respectively. In this note, we briefly survey key properties of these groups. In addition, we consider certain related ...parameters and we compute a number of examples of interest.
We classify two types of finite groups with certain normality conditions, namely SSN groups and groups with all noncyclic subgroups normal. These conditions are key ingredients in the study of the ...multiplicative Jordan decomposition problem for group rings.
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In this paper, we complete the classification of those finite 3-groups G whose integral group rings have the multiplicative Jordan decomposition property. If G is abelian, then it is clear that ℤG ...satisfies the multiplicative Jordan decomposition (MJD). In the nonabelian case, we show that ℤG satisfies MJD if and only if G is one of the two nonabelian groups of order 3
3
= 27.
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5.
Explicit free groups in division rings Gonçalves, J. Z.; Passman, D. S.
Proceedings of the American Mathematical Society,
02/2015, Volume:
143, Issue:
2
Journal Article
Peer reviewed
Open access
Let D be a division ring of characteristic ≠ 2 and suppose that the multiplicative group D• = D \ {0} has a subgroup G isomorphic to the Heisenberg group. Then we use the generators of G to construct ...an explicit noncyclic free subgroup of D•. The main difficulty occurs here when D has characteristic 0 and the commutators in G are algebraic over ℚ.
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In this paper, we continue our study of the maximal bounded \mathbb{Z}-filtrations of a complex semisimple Lie algebra L. Specifically, we discuss the functionals which give rise to such filtrations, ...and we show that they are related to certain semisimple subalgebras of L of full rank. In this way, we determine the ``order'' of these functionals and count them without the aid of computer computations. The main results here involve the Lie algebras of type E_6, E_7 and E_8, since we already know a good deal about the functionals for the remaining types. Nevertheless, we reinterpret our previous results into the new context considered here. Finally, we describe the associated graded Lie algebras of all of the maximal filtrations obtained in this manner.
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Noetherian down-up algebras Kirkman, Ellen; Musson, Ian M.; Passman, D. S.
Proceedings of the American Mathematical Society,
11/1999, Volume:
127, Issue:
11
Journal Article
Peer reviewed
Open access
Down-up algebras A= A(\alpha ,\beta ,\gamma ) were introduced by G. Benkart and T. Roby to better understand the structure of certain posets. In this paper, we prove that \beta \neq 0 is equivalent ...to A being right (or left) Noetherian, and also to A being a domain. Furthermore, when this occurs, we show that A is Auslander-regular and has global dimension 3.
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Filtrations in semisimple rings Passman, D. S.
Transactions of the American Mathematical Society,
12/2005, Volume:
357, Issue:
12
Journal Article
Peer reviewed
Open access
In this paper, we describe the maximal bounded \mathbb{Z}-filtrations of Artinian semisimple rings. These turn out to be the filtrations associated to finite \mathbb{Z}-gradings. We also consider ...simple Artinian rings with involution, in characteristic \neq 2, and we determine those bounded \mathbb{Z}-filtrations that are maximal subject to being stable under the action of the involution. Finally, we briefly discuss the analogous questions for filtrations with respect to other Archimedean ordered groups.
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