Actions of quantum linear spaces on quantum algebras

Jason Gaddis; Zachary Cline

REVIEW 2 major objections 3 minor 27 references

Actions of quantum linear spaces on quantum algebras

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Inner faithful linear actions of bosonized quantum linear spaces have rank capped by the number of generators of the algebra: at most 2(t−1), 3, or 2N−2.

desk verdict Sharp rank bounds for bosonization actions; the quantum-matrix classification is conditional on an automorphism hypothesis not known at root-of-unity q, so the abstract overstates the scope. read the letter →

arxiv 1908.05248 v1 pith:RCBZ7LNC submitted 2019-08-14 math.RA math.QA

Zachary Cline , Jason Gaddis This is my paper · ORCID

classification math.RAmath.QA MSC 16T0516S3616W5016W70

keywords pointedHopfalgebrasactionsquantumaffinespacesmatrixbosonizationsoflineargeneralizedTaftinnerfaithfulrankbounds

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper classifies the quantum symmetries—actions by pointed Hopf algebras—that a bosonization of a quantum linear space can have on two basic families of quantum algebras. It shows that when such an action is linear, inner faithful (no nonzero Hopf ideal kills the algebra), and all parameters are roots of unity of order at least three, the rank θ of the bosonization is severely limited: θ ≤ 2(t−1) for a quantum affine space in t variables, θ ≤ 3 for the 2×2 quantum matrix algebra $O_q(M_2(k))$, and θ ≤ 2N−2 for $O_q(M_N(k))$ with N ≥ 3. These bounds are sharp, with explicit examples reaching them. The rank-one case is classified first: every generalized Taft action on a quantum affine space is a trivial extension of an action on a two- or three-variable subalgebra. The interest is that these are non-semisimple Hopf actions, a much less charted territory than group actions on quantum algebras.

What carries the argument

The load-bearing mechanism is the skew-primitive relation $gx = \lambda xg$ together with the linearity assumption that both $g$ and $x$ act on the degree-one component. Writing $g \cdot u_i = \alpha_i u_i$ and $x \cdot u_j = \sum_i \eta_{ij} u_i$, the relation $(gx - \lambda xg) \cdot u_j = 0$ gives $\eta_{ij}(\alpha_i - \lambda \alpha_j) = 0$ for every pair. This single identity rules out diagonal entries and cross pairs of nonzero entries, forcing the action matrix of $x$ to be a single shift along a row or column. The rank bound for quantum affine spaces comes from drawing a directed graph whose arrows are the nonzero entries of the $x_k$ and counting how many arrows the compatibility lemmas allow; the quantum matrix bounds come from a compatibility table of the eight possible rank-one actions.

What would settle it

Find a linear inner faithful action on $k_p[u_1,\dots,u_t]$ with all parameter orders at least 3 and rank θ > 2(t−1), or on $O_q(M_2(k))$ with $m_i \ge 3$, each $g_i$ diagonal or transpose, and θ = 4; the paper's Theorems 3.13 and 4.8 would then be wrong. A more targeted check is to search the compatibility table of the eight rank-one $O_q(M_2)$ actions for four pairwise compatible types with all scaling parameters nonzero.

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Extended reading notes

Core claim

The paper's central discovery is that the size of a bosonization acting on a quantum algebra is controlled by the number of generators of the algebra, through a rank bound that is attained. Concretely, for inner faithful linear actions of $B(G,g,\chi)$ on $k_p[u_1,\dots,u_t]$ with all $m_i$ and the orders of the $p_{ij}$ at least 3, the paper proves θ ≤ 2(t−1); for $O_q(M_2(k))$ it proves θ ≤ 3; for $O_q(M_N(k))$ with N ≥ 3 it proves θ ≤ 2N−2. These statements rest on a full classification of the rank-one building blocks: generalized Taft algebra actions on the same algebras, given by explicit matrices. The paper then shows which of these rank-one actions can be patched together inside one bosonization, using the relations $g_i x_j = \chi_j(g_i) x_j g_i$ and $x_i x_j = \chi_j(g_i) x_j x_i$ as compatibility conditions. Sharpness is established by explicit rank θ = 2(t−1), θ = 3, and θ = 2N−2 examples.

Load-bearing premise

The quantum-matrix theorems assume every group-like element $g_i$ acts by a diagonal scaling or the transpose automorphism of $O_q(M_N(k))$—an identification of the automorphism group that the cited source proves only when q is not a root of unity, although the paper's setting includes root-of-unity q; all rank bounds also assume the action is linear, meaning it preserves the degree-one component.

Editorial extensions

If this is right

Every linear inner faithful action of a generalized Taft algebra on a quantum affine space is a trivial extension of an action on a quantum plane $A_{ij}$ or a quantum 3-space $A_{ijk}$ (Theorem 3.9).
A bosonization of rank θ acting linearly and inner faithfully on $k_p[u_1,\dots,u_t]$ satisfies θ ≤ 2(t−1), and this bound is achieved by explicit examples (Theorem 3.13, Example 3.14).
On $O_q(M_2(k))$ the rank of a bosonization is at most 3 under the hypotheses, with an example attaining it (Theorem 4.8, Example 4.6).
On $O_q(M_N(k))$ with N ≥ 3 the rank is at most 2N−2, again attained (Theorem 4.18, Example 4.17).
The rank-one classifications imply that the possible shifting parameters λ for a generalized Taft action on $O_q(M_2(k))$ are exactly $q^{\pm 2}$ and $q^{\pm 4}$, and that the actions descend to $O_q(SL_N)$ and lift to $O_q(GL_N)$ in the listed cases (Propositions 4.4, 4.11, 5.11).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the automorphism-group identification for $O_q(M_N)$ extends to roots of unity, the same rank bounds should hold without the restrictive hypothesis on the $g_i$; a test is to compute $\operatorname{Aut}(O_q(M_N))$ at root-of-unity q and check whether any new automorphisms allow a fourth mutually compatible action when N = 2.
Dropping linearity may open the door to higher-rank actions, since the vanishing pattern in Lemma 2.7 is what forces the shift structure; constructing a non-linear inner faithful action would directly probe whether the bounds are a feature of Hopf actions or of linear ones.
The directed-graph counting argument for quantum affine spaces is likely reusable: any connected graded algebra whose automorphisms are monomial and whose relations have parameters of order at least 3 should admit a similar rank bound in terms of its number of generators.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This paper proves sharp bounds on the rank of bosonizations of quantum linear spaces acting on quantum affine spaces (θ ≤ 2(t−1)) and on single-parameter quantum matrix algebras (θ ≤ 3 for N=2, θ ≤ 2N−2 for N≥3), with explicit examples showing the bounds are attained. It also shows that generalized Taft algebra actions on quantum affine spaces are trivial extensions of actions on quantum planes or certain 3-spaces. That is a genuine extension of the Gaddis–Won–Yee Taft classification, and the authors are careful about inner faithfulness and the parameter hypotheses (all orders at least 3).

The proofs are computational but coherent. Lemma 2.7 is the engine, and Lemmas 3.6, 3.12, 4.9, and 4.11 build the classifications. The sharpness examples 3.14 and 4.17 are explicit and convincing at a skim; I did not independently verify every coefficient in Tables 4.2 and 4.5. Several compatibility checks are delegated to the reader with no derivation. That is a minor-to-moderate completeness concern, not a fatal one.

The stress-test note is right about the main soft spot: the quantum matrix classification is conditional on each gi acting as an element of H ⋊ ⟨τ⟩. The cited Yakimov result identifying Aut(O_q(M_N)) with H ⋊ ⟨τ⟩ is stated for q not a root of unity. The paper explicitly allows root-of-unity q (Example 4.6 uses q a fifth root of unity), and for such q the automorphism group identification is not known. So Theorems 4.8 and 4.18 are logically sound conditional statements, but the abstract's phrase "we classify actions on quantum matrix algebras" understates the hypothesis. The paper also assumes linear actions (preserving the degree-one component), which is stated in Section 2.3 but is not automatic for Hopf actions and should appear in the abstract too.

The rank bounds themselves look solid under the stated hypotheses. I found no circularity: the bounds come from the algebra relations, not from assuming the target result. The citation pattern is normal, and self-citations to [14] are appropriate.

This paper is for people working on Hopf algebra actions, pointed Hopf algebras, and automorphisms of quantum algebras. It deserves a serious referee; the main referee tasks are checking the tables and pressing on the root-of-unity automorphism gap. My recommendation: send to peer review, with the expectation that the authors clarify the scope in the abstract and either justify the H ⋊ ⟨τ⟩ hypothesis at roots of unity or restrict the main theorems accordingly.

Referee Report

2 major / 3 minor

Summary. The paper studies linear, inner faithful actions of bosonizations B(G,g,χ) of quantum linear spaces on quantum affine spaces k_p[u_1,...,u_t] and on quantum matrix algebras O_q(M_N(k)). The main results are rank bounds: θ ≤ 2(t−1) for quantum affine spaces (Theorem 3.13), θ ≤ 3 for O_q(M_2) (Theorem 4.8), and θ ≤ 2N−2 for O_q(M_N) with N≥3 (Theorem 4.18), under hypotheses including parameter orders at least 3 and, for the matrix algebras, each g_i acting as an element of H⋊⟨τ⟩. The paper also classifies generalized Taft algebra actions in the relevant settings (Propositions 3.1, 4.4, 4.11), gives sharpness examples (Examples 3.14, 4.6, 4.17), and studies invariants, quantum exterior algebras, quantized Weyl algebras, and O_q(SL_N)/O_q(GL_N).

Significance. If the results stand, they represent a substantial advance in the understanding of pointed Hopf algebra actions on quantum algebras, extending the earlier Taft algebra classifications to higher-rank bosonizations and higher-dimensional algebras. The rank bounds are explicit and sharp, with concrete examples attaining them, and the paper gives a useful inner-faithfulness criterion (Proposition 2.6). The overall proof strategy is coherent: the key lemmas build systematically to the main theorems, and the conditional statements are clearly formulated. However, the proof of the main quantum affine space bound has a technical gap in its graph counting argument, and the advertised scope of the quantum matrix algebra classification is broader than what the hypotheses actually support for root-of-unity q. These issues are fixable but require attention.

major comments (2)

[Theorem 3.13 proof, Section 3] In the first paragraph of the proof, the graph Γ is defined by drawing an arrow v_j → v_i if the (i,j) entry of some x_k is nonzero, and Γ_1 is declared to be the number of arrows. The assertion 'It is clear that θ ≤ Γ_1' is false under this definition. In Example 3.14, for each k = 1,...,t−1, both x_k and x'_k have the unique nonzero entry (1, k+1), so the graph contains Γ_1 = t−1 arrows while θ = 2(t−1). Thus θ ≤ Γ_1 fails. The proof must either count arrows with multiplicity (one per nonzero entry per x_k) or supply a different argument; if multiplicity counting is intended, the subsequent steps involving 'the target of any arrow is the source of at least one other' must be reworked to handle parallel arrows. This gap affects the proof of the main quantum affine space bound.
[Abstract and Section 4, Theorems 4.8 and 4.18] The classification of actions on quantum matrix algebras is conditional on the hypothesis that each g_i acts as an element of H ⋊ ⟨τ⟩. This hypothesis is known to be automatic only for q not a root of unity, by the cited automorphism theorem [27]. The paper explicitly allows and features root-of-unity q (e.g., Examples 4.6 and 4.17 use a fifth root of unity), and for such q it does not prove that every automorphism lies in H ⋊ ⟨τ⟩. Consequently, the abstract's statement that the paper classifies actions on quantum matrix algebras under 'mild conditions' overstates the scope. The hypothesis should be stated in the abstract and introduction, or the scope of the matrix algebra claims should be restricted to the q not a root of unity regime.

minor comments (3)

[Abstract] The abstract says 'all actions of generalized Taft algebras are trivial extensions of actions on quantum planes,' but Theorem 3.9 states that every action is a trivial extension of an action on some A_{ij} or A_{ijk}, i.e., a quantum plane or a quantum 3-space. Please align the abstract with the theorem.
[Tables 4.2 and 4.5, Section 4] The compatibility tables (Table 4.2 and Table 4.5) are asserted to follow from 'basic computations' with no sample derivation or verification method. Since these tables are load-bearing for Theorems 4.8 and 4.18, please include at least one representative computation or provide a reproducible verification file.
[Section 5, Proposition 5.9] The bound in Proposition 5.9 is stated as 'rank B ≤ 2(2t − 1)' after applying Theorem 3.13 to the associated graded algebra, which has 2t generators; this is correct but worth a brief parenthetical explanation for readability.

Circularity Check

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No circular derivation: rank bounds are computed from algebra relations; the quantum-matrix theorems are conditional on an automorphism hypothesis that is unproved for root-of-unity q, a scope gap rather than circularity.

full rationale

The central results do not reduce to their inputs by construction. Theorem 3.13 is obtained from Lemma 2.7, Lemma 3.6, and a graph-counting argument that counts nonzero entries of the matrices representing the x_i; no parameter is fitted and the bound is not assumed. The quantum matrix algebra results (Theorems 4.8 and 4.18) are explicitly conditional on the hypothesis that each g_i lies in H⋊⟨τ⟩, so the automorphism classification is used as a stated assumption rather than smuggled in as a conclusion. The only load-bearing external input is Yakimov's theorem [27], which the paper itself describes as proven only for q not a root of unity, while Examples 4.6 and 4.17 use q a fifth root of unity; this is an overstatement of scope in the abstract, not a circular step. The authors cite their own [14] as motivation and for a similar argument ('the proof of this follows similarly to [14, Proposition 2.1]' in Proposition 3.1), but the main bounds are independently computed and do not depend on assuming the target theorem. Krop-Radford and Yakimov are used as external tools, no fitted constants appear, and no prediction is renamed input.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The theorems are conditional on three explicit restrictions: linearity of actions, all parameters having order at least 3, and for Oq(MN) the hypothesis gi ∈ H ⋊ ⟨τ⟩. The first and third are not consequences of the algebraic setup, and the third is not justified for root-of-unity q by the cited literature. No free parameters are fitted; the statements are uniform over all parameter values satisfying the hypotheses.

assumptions (7)

standard math The base field k is algebraically closed of characteristic zero
Stated in Section 2; used throughout for roots of unity and characters.
domain assumption Krop-Radford classification of rank-one finite-dimensional pointed Hopf algebras
Quoted as Theorem 1 to parametrize all rank-one actions via H(G,g,χ,γ).
domain assumption Automorphisms of a quantum affine space are monomial matrices when pij ≠ 1
Cited from [18, Lemma 3.5(e)] and used to represent g as a permutation matrix with scalar entries.
domain assumption Aut(Oq(MN)) = H ⋊ ⟨τ⟩ when q is not a root of unity
Cited from [27] and used to motivate the hypothesis on gi.
ad hoc to paper Each gi acting on Oq(MN) lies in H ⋊ ⟨τ⟩ even when q is a root of unity
Adopted as a hypothesis in Theorems 4.8 and 4.18; not established for root-of-unity q, where automorphism groups are generally larger.
ad hoc to paper All actions are linear, i.e., gi and xi preserve A(1)
Assumed in Section 2.3; without it the matrix representation used throughout is unavailable.
ad hoc to paper All parameters (mi, ord(pij), q) have order at least 3
Assumed throughout to avoid special cases; the authors note in Section 6 that this restriction may be artificial.

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Pith. "Pith review of Actions of quantum linear spaces on quantum algebras." pith.science (2026). https://pith.science/paper/RCBZ7LNC

@misc{pith2026190805248,
  author       = {Pith},
  title        = {Pith review of: Actions of quantum linear spaces on quantum algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RCBZ7LNC}},
  note         = {Machine review of arXiv:1908.05248}
}

read the original abstract

We study actions of bosonizations of quantum linear spaces on quantum algebras. Under mild conditions, we classify actions on quantum affine spaces and quantum matrix algebras. In the former case, it is shown that all actions of generalized Taft algebras are trivial extensions of actions on quantum planes. In both cases we achieve bounds on the rank of the bosonization acting on the algebra.

Figures

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**Figure 4.4.** 16 [PITH_FULL_IMAGE:figures/full_fig_p016_4_4.png] view at source ↗

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Works this paper leans on

27 extracted references · 27 canonical work pages

[14]

Discriminant s of Taft Algebra Smash Products and Applications

Jason Gaddis, Robert W on, and Daniel Yee. Discriminant s of Taft Algebra Smash Products and Applications. Algebr. Represent. Theory, 22(4):785–799, 2019

work page 2019
[27]

The Launois-Lenagan conjecture

Milen Yakimov. The Launois-Lenagan conjecture. J. Algebra, 392:1–9, 2013. 26 Bucknell University, Department of Mathematics, Lewisburg , Pennsylvania 17837 E-mail address : z.cline@bucknell.edu Miami University, Department of Mathematics, 301 S. Patters on A ve., Oxford, Ohio 45056 E-mail address : gaddisj@miamioh.edu 27

work page 2013
[1]

Alev and M

J. Alev and M. Chamarie. D´ erivations et automorphismes de quelques alg` ebres quantiques. Comm. Algebra , 20(6):1787– 1802, 1992

work page 1992
[2]

Alev and F

J. Alev and F. Dumas. Rigidit´ e des plongements des quoti ents primitifs minimaux de Uq(sl(2)) dans l’alg` ebre quantique de Weyl-Hayashi. Nagoya Math. J. , 143:119–146, 1996

work page 1996
[3]

Andruskiewitsch and H.-J

N. Andruskiewitsch and H.-J. Schneider. Lifting of quan tum linear spaces and pointed Hopf algebras of order p3. J. Algebra, 209(2):658–691, 1998

work page 1998
[4]

On the classiﬁcation of ﬁnite-dimensional pointed Hopf alg ebras

Nicol´ as Andruskiewitsch and Hans-J¨ urgen Schneider. On the classiﬁcation of ﬁnite-dimensional pointed Hopf alg ebras. Ann. of Math. (2) , 171(1):375–417, 2010

work page 2010
[5]

Brown and Ken R

Ken A. Brown and Ken R. Goodearl. Lectures on algebraic quantum groups . Advanced Courses in Mathematics. CRM Barcelona. Birkh¨ auser Verlag, Basel, 2002

work page 2002
[6]

Ceken, J

S. Ceken, J. H. Palmieri, Y.-H. W ang, and J.J. Zhang. The d iscriminant controls automorphism groups of noncommutati ve algebras. Adv. Math. , 269:551–584, 2015

work page 2015

Show all 27 references

[7]

Ceken, J

S. Ceken, J. H. Palmieri, Y.-H. W ang, and J.J. Zhang. The d iscriminant criterion and automorphism groups of quantize d algebras. Adv. Math. , 286:754–801, 2016

2016
[8]

K. Chan, C. W alton, Y. H. W ang, and J. J. Zhang. Hopf action s on ﬁltered regular algebras. J. Algebra , 397:68–90, 2014

2014
[9]

Kenneth Chan, Ellen Kirkman, Chelsea W alton, and James J . Zhang. Quantum binary polyhedral groups and their actions on quantum planes. J. Reine Angew. Math. , 719:211–252, 2016

2016
[10]

The division ring of quotients of the coord inate ring of the quantum general linear group

Gerald Cliﬀ. The division ring of quotients of the coord inate ring of the quantum general linear group. J. London Math. Soc. (2) , 51(3):503–513, 1995

1995
[11]

On actions of Drinfel’d doubles on ﬁnite dimensional algebras

Zachary Cline. On actions of Drinfel’d doubles on ﬁnite dimensional algebras. J. Pure Appl. Algebra , 223(8):3635–3664, 2019

2019
[12]

Finite dimensional H opf actions on algebraic quantizations

Pavel Etingof and Chelsea W alton. Finite dimensional H opf actions on algebraic quantizations. Algebra Number Theory , 10(10):2287–2310, 2016

2016
[13]

Pointed Hopf actions on ﬁelds, II

Pavel Etingof and Chelsea W alton. Pointed Hopf actions on ﬁelds, II. J. Algebra , 460:253–283, 2016

2016
[15]

Jing and J

N. Jing and J. J. Zhang. Gorensteinness of invariant sub rings of quantum algebras. J. Algebra , 221(2):669–691, 1999

1999
[16]

Naihuan Jing and James J. Zhang. On the trace of graded au tomorphisms. J. Algebra , 189(2):353–376, 1997

1997
[17]

Actions of some pointed Hopf algebras on path algebras of quivers

Ryan Kinser and Chelsea W alton. Actions of some pointed Hopf algebras on path algebras of quivers. Algebra Number Theory, 10(1):117–154, 2016

2016
[18]

Kirkman, J

E. Kirkman, J. Kuzmanovich, and J. J. Zhang. Shephard-T odd-Chevalley theorem for skew polynomial rings. Algebr. Represent. Theory, 13(2):127–158, 2010

2010
[19]

Kirkman, J

E. Kirkman, J. Kuzmanovich, and J. J. Zhang. Invariant t heory of ﬁnite group actions on down-up algebras. Transform. Groups, 20(1):113–165, 2015

2015
[20]

Leonid Krop and David E. Radford. Finite-dimensional H opf algebras of rank one in characteristic zero. J. Algebra , 302(1):214–230, 2006

2006
[21]

Launois and T

S. Launois and T. H. Lenagan. Automorphisms of quantum m atrices. Glasg. Math. J. , 55(A):89–100, 2013

2013
[22]

Quantized Weyl algebra s at roots of unity

Jesse Levitt and Milen Yakimov. Quantized Weyl algebra s at roots of unity. Israel J. Math. , 225(2):681–719, 2018

2018
[23]

Montgomery and H.-J

S. Montgomery and H.-J. Schneider. Hopf crossed produc ts, rings of quotients, and prime ideals. Adv. Math. , 112(1):1–55, 1995

1995
[24]

Quantum linear groups

Brian Parshall and Jian Pan W ang. Quantum linear groups . Mem. Amer. Math. Soc. , 89(439):vi+157, 1991

1991
[25]

Spectre de l’alg` ebre de Weyl quantique

Laurent Rigal. Spectre de l’alg` ebre de Weyl quantique . Beitr¨ age Algebra Geom., 37(1):119–148, 1996

1996
[26]

Earl J. Taft. The order of the antipode of ﬁnite-dimensi onal Hopf algebra. Proc. Nat. Acad. Sci. U.S.A. , 68:2631–2633, 1971

1971

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [14]

Discriminant s of Taft Algebra Smash Products and Applications

Jason Gaddis, Robert W on, and Daniel Yee. Discriminant s of Taft Algebra Smash Products and Applications. Algebr. Represent. Theory, 22(4):785–799, 2019

work page 2019

[2] [27]

The Launois-Lenagan conjecture

Milen Yakimov. The Launois-Lenagan conjecture. J. Algebra, 392:1–9, 2013. 26 Bucknell University, Department of Mathematics, Lewisburg , Pennsylvania 17837 E-mail address : z.cline@bucknell.edu Miami University, Department of Mathematics, 301 S. Patters on A ve., Oxford, Ohio 45056 E-mail address : gaddisj@miamioh.edu 27

work page 2013

[3] [1]

Alev and M

J. Alev and M. Chamarie. D´ erivations et automorphismes de quelques alg` ebres quantiques. Comm. Algebra , 20(6):1787– 1802, 1992

work page 1992

[4] [2]

Alev and F

J. Alev and F. Dumas. Rigidit´ e des plongements des quoti ents primitifs minimaux de Uq(sl(2)) dans l’alg` ebre quantique de Weyl-Hayashi. Nagoya Math. J. , 143:119–146, 1996

work page 1996

[5] [3]

Andruskiewitsch and H.-J

N. Andruskiewitsch and H.-J. Schneider. Lifting of quan tum linear spaces and pointed Hopf algebras of order p3. J. Algebra, 209(2):658–691, 1998

work page 1998

[6] [4]

On the classiﬁcation of ﬁnite-dimensional pointed Hopf alg ebras

Nicol´ as Andruskiewitsch and Hans-J¨ urgen Schneider. On the classiﬁcation of ﬁnite-dimensional pointed Hopf alg ebras. Ann. of Math. (2) , 171(1):375–417, 2010

work page 2010

[7] [5]

Brown and Ken R

Ken A. Brown and Ken R. Goodearl. Lectures on algebraic quantum groups . Advanced Courses in Mathematics. CRM Barcelona. Birkh¨ auser Verlag, Basel, 2002

work page 2002

[8] [6]

Ceken, J

S. Ceken, J. H. Palmieri, Y.-H. W ang, and J.J. Zhang. The d iscriminant controls automorphism groups of noncommutati ve algebras. Adv. Math. , 269:551–584, 2015

work page 2015