Hopf subalgebras and tensor powers of generalized permutation modules

Kadison, Lars

doi:10.1016/j.jpaa.2013.06.008

Cited by 13 publications

(51 citation statements)

References 30 publications

(88 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…by the exact functor − ⊗ R H (as R H is a free module). This proof demonstrates that a fourth equivalent condition one may add to [35,Theorem 3.5], which characterizes the semisimplicity of R, is that R + H is a projective H-module (and see the sufficient condition below in Prop. 3.1).…”

Section: Existence Of Rightmentioning

confidence: 83%

“…It follows from Eq. (7) that Q ⊗2 [35,24]. Note that d(Q R ) = 0 since hr = hr = hε(r) for each h ∈ H, r ∈ R. Then d(R, H) ≤ 2.)…”

Section: Introductionmentioning

confidence: 99%

“…where Q ⊗n := Q ⊗ · · · ⊗ Q (n times Q, ⊗ the tensor product in mod-H) (see Eq. (6) below and [35,Prop. 3.6]).…”

Section: Introductionmentioning

confidence: 99%

“…Example 1.2. Note that d(H, G) = 1 if the corresponding group algebras satisfy [35], where it is noted that [24]), that from the…”

Section: Introductionmentioning

confidence: 99%

“…One computes that the quotient module coalgebra Q H R in simplified notation Q G The main problem in the area of Hopf algebra depth is whether in general any one of d(Q H R ), d h (R, H), or d(R, H) is finite [5] (which would imply the finiteness of the other two depths). In order to emphasize the point that Q H R is an R-relative projective H-module, we may make the following definition, and prove the next proposition, which formally addresses this problem and extends [35,Cor. 5.8(i)].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An In-Depth Look at Quotient Modules

Kadison

2018

Algebr Represent Theor

Self Cite

View full text Add to dashboard Cite

The coset G-space of a finite group and a subgroup is a fundamental module of study of Schur and others around 1930; for example, its endomorphism algebra is a Hecke algebra of double cosets. We study and review its generalization Q to Hopf subalgebras, especially the tensor powers and similarity as modules over a Hopf algebra, or what's the same, Morita equivalence of the endomorphism algebras. We prove that Q has a nonzero integral if and only if the modular function restricts to the modular function of the Hopf subalgebra. We also study and organize knowledge of Q and its tensor powers in terms of annihilator ideals, sigma categories, trace ideals, Burnside ring formulas, and when dealing with semisimple Hopf algebras, the depth of Q in terms of the McKay quiver and the Green ring.1991 Mathematics Subject Classification. 16D20, 16D90, 16T05, 18D10, 20C05.

show abstract

Section: Existence Of Rightmentioning

confidence: 83%

“…It follows from Eq. (7) that Q ⊗2 [35,24]. Note that d(Q R ) = 0 since hr = hr = hε(r) for each h ∈ H, r ∈ R. Then d(R, H) ≤ 2.)…”

Section: Introductionmentioning

confidence: 99%

“…where Q ⊗n := Q ⊗ · · · ⊗ Q (n times Q, ⊗ the tensor product in mod-H) (see Eq. (6) below and [35,Prop. 3.6]).…”

Section: Introductionmentioning

confidence: 99%

“…Example 1.2. Note that d(H, G) = 1 if the corresponding group algebras satisfy [35], where it is noted that [24]), that from the…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An In-Depth Look at Quotient Modules

Kadison

2018

Algebr Represent Theor

Self Cite

View full text Add to dashboard Cite

show abstract

A quantum subgroup depth

2017

View full text Add to dashboard Cite

The Green ring of the half quantum group H = U n (q) is computed in [9]. The tensor product formulas between indecomposables may be used for a generalized subgroup depth computation in the setting of quantum groups -to compute depth of the Hopf subalgebra H in its Drinfeld double D(H). In this paper the Hopf subalgebra quotient module Q (a generalization of the permutation module of cosets for a group extension) is computed and, as H-modules, Q and its second tensor power are decomposed into a direct sum of indecomposables. We note that the least power n, referred to as depth, for which Q ⊗(n) has the same indecomposable constituents as Q ⊗(n+1) is n = 2, since Q ⊗(2) contains all H-module indecomposables, which determines the minimum even depth d ev (H, D(H)) = 6. √ 8n+1−1 2 + 1, as computed in 1991 Mathematics Subject Classification. 16D20, 16D90.

show abstract

Algebraic quotient modules and subgroup depth

Hernández

Kadison

Young

2014

Abh. Math. Semin. Univ. Hambg.

Self Cite

View full text Add to dashboard Cite

Abstract. In [28] it was shown that subgroup depth may be computed from the permutation module of the left or right cosets: this holds more generally for a Hopf subalgebra, from which we note in this paper that finite depth of a Hopf subalgebra R ⊆ H is equivalent to the H-module coalgebra Q = H/R + H representing an algebraic element in the Green ring of H or R. This approach shows that subgroup depth and the subgroup depth of the corefree quotient lie in the same closed interval of length one. We also establish a previous claim that the problem of determining if R has finite depth in H is equivalent to determining if H has finite depth in its smash product Q * #H. A necessary condition is obtained for finite depth from stabilization of a descending chain of annihilator ideals of tensor powers of Q. As an application of these topics to a centerless finite group G, we prove that the minimum depth of its group C -algebra in the Drinfeld double D(G) is an odd integer, which determines the least tensor power of the adjoint representation Q that is a faithful C G-module.

show abstract

Hopf subalgebras and tensor powers of generalized permutation modules

Cited by 13 publications

References 30 publications

An In-Depth Look at Quotient Modules

An In-Depth Look at Quotient Modules

A quantum subgroup depth

Algebraic quotient modules and subgroup depth

Contact Info

Product

Resources

About