Monomorphisms of coalgebras

A. L. Agore

Displaying similar documents to “Monomorphisms of coalgebras”

The bicrossed products of $H_{4}$ and $H_{8}$

Daowei Lu, Yan Ning, Dingguo Wang (2020)

Czechoslovak Mathematical Journal

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Let $H_{4}$ and $H_{8}$ be the Sweedler’s and Kac-Paljutkin Hopf algebras, respectively. We prove that any Hopf algebra which factorizes through $H_{8}$ and $H_{4}$ (equivalently, any bicrossed product between the Hopf algebras $H_{8}$ and $H_{4}$ ) must be isomorphic to one of the following four Hopf algebras: $H_{8} \otimes H_{4}, H_{32, 1}, H_{32, 2}, H_{32, 3}$ . The set of all matched pairs $(H_{8}, H_{4}, ▹, ◃)$ is explicitly described, and then the associated bicrossed product is given by generators and relations.

The structures of Hopf $*$ -algebra on Radford algebras

Hassan Suleman Esmael Mohammed, Hui-Xiang Chen (2019)

Czechoslovak Mathematical Journal

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We investigate the structures of Hopf $*$ -algebra on the Radford algebras over $ℂ$ . All the $*$ -structures on $H$ are explicitly given. Moreover, these Hopf $*$ -algebra structures are classified up to equivalence.

Right coideal subalgebras of $U_{q}^{+} ({𝔰𝔬}_{2 n + 1})$

V. K. Kharchenko (2011)

Journal of the European Mathematical Society

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We give a complete classification of right coideal subalgebras that contain all grouplike elements for the quantum group $U_{q}^{+} ({𝔰𝔬}_{2 n + 1})$ provided that $q$ is not a root of 1. If $q$ has a finite multiplicative order $t > 4$ ; this classification remains valid for homogeneous right coideal subalgebras of the Frobenius–Lusztig kernel $u_{q}^{+} ({𝔰𝔬}_{2 n + 1})$ . In particular, the total number of right coideal subalgebras that contain the coradical equals $(2 n)!!$ ; the order of the Weyl group defined by the root system of type $B_{n}$ .

Automorphism group of green algebra of weak Hopf algebra corresponding to Sweedler Hopf algebra

Liufeng Cao, Dong Su, Hua Yao (2023)

Czechoslovak Mathematical Journal

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Let $r (𝔴_{2}^{0})$ be the Green ring of the weak Hopf algebra $𝔴_{2}^{0}$ corresponding to Sweedler’s 4-dimensional Hopf algebra $H_{2}$ , and let $Aut (R (𝔴_{2}^{0}))$ be the automorphism group of the Green algebra $R (𝔴_{2}^{0}) = r (𝔴_{2}^{0}) \otimes_{ℤ} ℂ$ . We show that the quotient group $Aut (R (𝔴_{2}^{0})) / C_{2} ≅ S_{3}$ , where $C_{2}$ contains the identity map and is isomorphic to the infinite group $(ℂ^{*}, \times)$ and $S_{3}$ is the symmetric group of order 6.

The duality theorem for twisted smash products of Hopf algebras and its applications

Zhongwei Wang, Liangyun Zhang (2015)

Colloquium Mathematicae

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Let $A_{T} H$ denote the twisted smash product of an arbitrary algebra A and a Hopf algebra H over a field. We present an analogue of the celebrated Blattner-Montgomery duality theorem for $A_{T} H$ , and as an application we establish the relationship between the homological dimensions of $A_{T} H$ and A if H and its dual H* are both semisimple.

Classification of ideals of $8$ -dimensional Radford Hopf algebra

Yu Wang (2022)

Czechoslovak Mathematical Journal

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Let $H_{m, n}$ be the $m n^{2}$ -dimensional Radford Hopf algebra over an algebraically closed field of characteristic zero. We give the classification of all ideals of $8$ -dimensional Radford Hopf algebra $H_{2, 2}$ by generators.

A class of quantum doubles of pointed Hopf algebras of rank one

Hua Sun, Yueming Li (2023)

Czechoslovak Mathematical Journal

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We construct a class of quantum doubles $D (H_{D_{n}})$ of pointed Hopf algebras of rank one $H_{𝒟}$ . We describe the algebra structures of $D (H_{D_{n}})$ by generators with relations. Moreover, we give the comultiplication $Δ_{D}$ , counit $ε_{D}$ and the antipode $S_{D}$ , respectively.

Yetter-Drinfeld-Long bimodules are modules

Daowei Lu, Shuan Hong Wang (2017)

Czechoslovak Mathematical Journal

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Let $H$ be a finite-dimensional bialgebra. In this paper, we prove that the category $ℒℛ (H)$ of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category $_{H \otimes H^{*}}^{H \otimes H^{*}} 𝒴𝒟$ over the tensor product bialgebra $H \otimes H^{*}$ as monoidal categories. Moreover if $H$ is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.

Troesch complexes and extensions of strict polynomial functors

Antoine Touzé (2012)

Annales scientifiques de l'École Normale Supérieure

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We develop a new approach of extension calculus in the category of strict polynomial functors, based on Troesch complexes. We obtain new short elementary proofs of numerous classical $Ext$ -computations as well as new results. In particular, we get a cohomological version of the “fundamental theorems” from classical invariant theory for $G L_{n}$ for $n$ big enough (and we give a conjecture for smaller values of $n$ ). We also study the “twisting spectral sequence” $E^{s, t} (F, G, r)$ converging to the extension groups...

$A F$ -algebras and topology of mapping tori

Igor Nikolaev (2015)

Czechoslovak Mathematical Journal

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The paper studies applications of $C^{*}$ -algebras in geometric topology. Namely, a covariant functor from the category of mapping tori to a category of $A F$ -algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding $A F$ -algebras. We use this functor to develop an obstruction theory for the torus bundles of dimension $2$ , $3$ and $4$ . In conclusion, we consider two numerical examples illustrating our main results.

Automorphism group of representation ring of the weak Hopf algebra $\tilde{H_{8}}$

Dong Su, Shilin Yang (2018)

Czechoslovak Mathematical Journal

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Let $H_{8}$ be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra $\tilde{H_{8}}$ based on $H_{8}$ , then we investigate the structure of the representation ring of $\tilde{H_{8}}$ . Finally, we prove that the automorphism group of $r (\tilde{H_{8}})$ is just isomorphic to $D_{6}$ , where $D_{6}$ is the dihedral group with order 12.

Singularity categories of skewed-gentle algebras

Xinhong Chen, Ming Lu (2015)

Colloquium Mathematicae

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Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let $(Q^{s g}, I^{s g})$ and $(Q^{g}, I^{g})$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $K Q^{s g} / ⟨ I^{s g} ⟩$ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of $K Q^{g} / ⟨ I^{g} ⟩$ . As a corollary, we find that $g l d i m K Q^{s g} / ⟨ I^{s g} ⟩ < \infty$ if and only if $g l d i m K Q / ⟨ I ⟩ < \infty$ if and only if $g l d i m K Q^{g} / ⟨ I^{g} ⟩ < \infty$ .

The unit groups of semisimple group algebras of some non-metabelian groups of order $144$

Gaurav Mittal, Rajendra K. Sharma (2023)

Mathematica Bohemica

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We consider all the non-metabelian groups $G$ of order $144$ that have exponent either $36$ or $72$ and deduce the unit group $U (𝔽_{q} G)$ of semisimple group algebra $𝔽_{q} G$ . Here, $q$ denotes the power of a prime, i.e., $q = p^{r}$ for $p$ prime and a positive integer $r$ . Up to isomorphism, there are $6$ groups of order $144$ that have exponent either $36$ or $72$ . Additionally, we also discuss how to simply obtain the unit groups of the semisimple group algebras of those non-metabelian groups of order $144$ that are a direct product of two...

On bifurcation and uniqueness results for some semilinear elliptic equations involving a singular potential

Manuela Chaves, Jesús García-Azorero (2006)

Journal of the European Mathematical Society

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We present some results concerning the problem $- Δ u = λ \frac{u}{{| x |}^{2}} + u^{q}$ , $u > 0$ in $Ω$ , ${u |}_{\partial Ω} = 0$ , where $0 < q < (N + 2) / (N - 2)$ , $q \neq 1$ , $λ \geq 0$ and $Ω$ is a smooth bounded domain containing the origin. In particular, bifurcation and uniqueness results are discussed.

Hall algebras of two equivalent extriangulated categories

Shiquan Ruan, Li Wang, Haicheng Zhang (2024)

Czechoslovak Mathematical Journal

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For any positive integer $n$ , let $A_{n}$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories $ℳ_{n + 1}$ and $ℱ_{n}$ , where $ℳ_{n + 1}$ and $ℱ_{n}$ are the two extriangulated categories corresponding to the representation category of $A_{n + 1}$ and the morphism category of projective representations of $A_{n}$ , respectively. As a...