Fundamentals of Hopf Algebras

Underwood, Robert G.

doi:10.1007/978-3-319-18991-8

Cited by 40 publications

(8 citation statements)

References 25 publications

(32 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…KG is an algebra with multiplication given by µ(g ⊗ h) = gh and a unit map given by λ(r) = r1 for all g, h ∈ G and r ∈ K [25]. KG is a coalgebra with comultiplication given by ∆(g) = g ⊗ g and a counit map given by (g) = 1 for all g ∈ G. Moreover, KG is a Hopf algebra with an antipode map given by S(g) = g −1 for all g ∈ G [9].…”

Section: The Group Algebra H = Rcmentioning

confidence: 99%

An Application of Finite Groups to Hopf algebras

Almutairi

Al-Shomrani

2021

Eur. J. Pure Appl. Math.

View full text Add to dashboard Cite

Kaplanskyâ€™s famous conjectures about generalizing results from groups to Hopf al-gebras inspired many mathematicians to try to find solusions for them. Recently, Cohen and Westreich in [8] and [10] have generalized the concepts of nilpotency and solvability of groups to Hopf algebras under certain conditions and proved interesting results. In this article, we follow their work and give a detailed example by considering a finite group G and an algebraically closed field K. In more details, we construct the group Hopf algebra H = KG and examine its properties to see what of the properties of the original finite group can be carried out in the case of H.

show abstract

Section: The Group Algebra H = Rcmentioning

confidence: 99%

An Application of Finite Groups to Hopf algebras

Almutairi

Al-Shomrani

2021

Eur. J. Pure Appl. Math.

View full text Add to dashboard Cite

show abstract

“…Algebras and tensors. In general algebra [24,23], a bialgebra in some vector space V over some field K is something that satifies all axioms of Definition 3 when arrows from n → m are linear maps from V ⊗n to V ⊗m .…”

Section: Consequences and Applicationsmentioning

confidence: 99%

“…These are the first examples in[24] and the first commutative examples in[23]. In[23] the binomial algebra is seen as a bialgebra over the symmetric algebra 6.…”

mentioning

confidence: 99%

Strong shift equivalence as a category notion

Jeandel

2021

Preprint

View full text Add to dashboard Cite

In this paper, we present a completely radical way to investigate the main problem of symbolic dynamics, the conjugacy problem, by proving that this problem actually relates to a natural question in category theory regarding the theory of traced bialgebras.As a consequence of this theory, we obtain a systematic way of obtaining new invariants for the conjugacy problem by looking at existing bialgebras in the literature.

show abstract

“…Since the algebra C[X] is a principal ideal domain, it turns out that the space of linearly recursive sequences Lin(C) can be identified with C[X] • via the isomorphism Φ, whence it becomes an augmented subalgebra of HC N and a Hopf algebra. For further reading, we refer the interested reader to [3,6,10] and [11,Chapter 2].…”

Section: The Space Of Linearly Recursive Sequences and Hurwitz's Productmentioning

confidence: 99%