S. Burciu
In this note the notion of kernel of a representation of a semisimple Hopf algebra is introduced. Similar properties to the kernel of a group representation are proved in some special cases. In particular, every normal Hopf subalgebra of a semisimple Hopf algebra H is the kernel of a representation of H.
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Algorithms are described that help with obtaining a classification of the semisimple subalgebras of a given semisimple Lie algebra, up to linear equivalence. The algorithms have been used to obtain classifications of the semisimple subalgebras of the simple Lie algebras of ranks <= 8. These have been made available as a database inside the SLA package of GAP4. The subalgebras in this database are explicitly given, as well as the inclusion relations among them.
We consider the problem of constructing semisimple subalgebras of real (semi-) simple Lie algebras. We develop computational methods that help to deal with this problem. Our methods boil down to solving a set of polynomial equations. In many cases the equations turn out to be sufficiently "pleasant" to be able to solve them. In particular this is the case for S-subalgebras.
A description of all normal Hopf subalgebras of a semisimple Drinfeld double is given. This is obtained by considering an analogue of Goursat's lemma concerning fusion subcategories of Deligne products of two fusion categories. As an application we show that the Drinfeld double of any abelian extension is also an abelian extension.
On $4n$-dimensional neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras
本文研究了 $4n$ 维非指针非半单霍普夫代数及其相关弱霍普夫代数的结构，揭示了其准三角性、 $R$-矩阵描述及绿色环的非可交换性。
Let H be a finite dimensional semisimple Hopf algebra over an algebraically closed field of characteristic zero. In this note we give a short proof of the fact that a Hopf subalgebra of H is a depth two subalgebra if and only if it is normal Hopf subalgebra.
Subalgebras of the rank two semisimple Lie algebras
本文综述秩2半单李代数的子代数分类研究，结合作者团队对经典秩2半单李代数子代数的分类成果与Mayanskiy对例外李代数$G_2$子代数的分类，用低维李代数统一分类方案识别子代数，分类基于内自同构与复数基域。
Let $k$ be an algebraically closed field of characteristic 0. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $9q^2$ over $k$, where $q$ is a prime number. We also prove that odd-dimensional semisimple Hopf algebras over $k$ of dimension less than 600 are of Frobenius type.
A criterion for subcoalgebras to be invariant under the adjoint action is given generalizing Masuoka's criterion for normal Hopf subalgebras. At the level of characters, the image of the induction functor from a normal Hopf subalgebra is isomorphic to the image of the restriction functor.
We show that there is a family of complex semisimple Hopf algebras that do not admit a Hopf order over any number ring. They are Drinfel'd twists of certain group algebras. The twist contains a scalar fraction which makes impossible the definability of such Hopf algebras over number rings. We also prove that a complex semisimple Hopf algebra satisfies Kaplansky's sixth conjecture if and only if it admits a weak order, in the sense of Rumynin and Lorenz, over the integers.
We study {\em disemisimple} Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra $\mathfrak{g}$ is disemisimple if and only if its solvable radical coincides with its nilradical and is a prehomogeneous $\mathfrak{s}$-module for a Levi subalgebra $\mathfrak{s}$ of $\mathfrak{g}$. We use the classification of prehomogeneous $\mathfrak{s}$-modules for simple Lie algebras $\mathfrak{s}$ given by Vinberg to show that the solvable radical of a disemisimple Lie algebra with simple Levi subalgebra is abelian. We extend th…
A general Mackey type decomposition for representations of semisimple Hopf algebras is investigated. We show that such a decomposition occurs in the case that the module is induced from an arbitrary Hopf subalgebra and it is restricted back to a group subalgebra. Some other examples when such a decomposition occurs are also constructed. They arise from gradings on the category of corepresentations of a semisimple Hopf algebra and provide new examples of Green functors in the literature.
We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension.
The notion of double coset for semisimple finite dimensional Hopf algebras is introduced. This is done by considering an equivalence relation on the set of irreducible characters of the dual Hopf algebra. As an application formulae for the restriction of the irreducible characters to normal Hopf subalgebras are given.
We consider the action of a semisimple Hopf algebra $H$ on an $m$-Koszul Artin-Schelter regular algebra $A$. Such an algebra $A$ is a derivation-quotient algebra for some twisted superpotential $\mathsf{w}$, and we show that the homological determinant of the action of $H$ on $A$ can be easily calculated using $\mathsf{w}$. Using this, we show that the smash product $A\,\#\,H$ is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra $Λ$ to which $A\,\#\,H$ is Morita equivalent, generalising a result of Bocklandt-Schedler-Wemyss. We also show how $Λ$ can be u…
In this paper we explore minimum odd and minimum even depth sub algebra pairs in the context of double cross products of finite dimensional Hopf algebras. We start by defining factorization algebras and outline how subring depth in this context relates with the module depth of the regular left module representation of the given subalgebra. Next we study minimum odd depth for double cross product Hopf subalgebras and determine their value in terms of their related module depth, we conclude that minimum odd depth of Drinfeld double Hopf subalgebras is 3. Finaly we produce a necessary and suffici…
Some new results on the module structure of Hopf algebras over a certain class of Hopf subalgebras and right coideal subalgebras are proved.
Let $\mathfrak{g}$ be a real semisimple Lie algebra with Iwasawa decomposition $\mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$. We show that, except for some explicit exceptional cases, every derivation of the nilpotent subalgebra $\mathfrak{n}$ that preserves its restricted root space decomposition is of the form $\text{ad}( W)$, where $W \in \mathfrak{m}\oplus .
In this note we prove a generalization of the Frobenius-Schur theorem for finite groups for the case of semisimple Hopf algebra over an algebraically closed field of characteristic 0. A similar result holds in characteristic $p > 2$ if the Hopf algebra is also cosemisimple. In fact we show a more general version for any finite-dimensional semisimple algebra with an involution.
In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple non-commutative Hopf algebras of dimension 16. Moreover, we prove that non-commutative semisimple Hopf algebras of dimension p^n, p is prime, cannot have a cyclic group of grouplikes.
An extension of $k$-algebras $B \subset A$ is said to have depth one if there exists a positive integer $n$ such that $ A$ is a direct summand of $ B^n$ in $_B\mtr{Mod}_B$. Depth one extensions of semisimple algebras are completely characterized in terms of their centers. For extensions of semisimple Hopf algebras our results are similar to those obtained for finite group algebra extensions in \cite{BKone}.
Masuoka proved (2009) that a finite-dimensional irreducible Hopf algebra $H$ in positive characteristic is semisimple if and only if it is commutative semisimple if and only if the Hopf subalgebra generated by all primitives is semisimple. In this paper, we give another proof of this result by using Hochschild cohomology of coalgebras.
We construct a class of non-commutative, non-cocommutative, semisimple Hopf algebras of dimension $2n^2$ and present conditions to define an inner faithful action of these Hopf algebras on quantum polynomial algebras, providing, in this way, more examples of semisimple Hopf actions which do not factor through group actions. Also, under certain condition, we classify the inner faithful Hopf actions of the Kac-Paljutkin Hopf algebra of dimension $8$, $H_8$, on the quantum plane.
We define left and right kernels of representations of Hopf algebras. In the case of group algebras, left and right kernels coincide and they are the usual kernels of modules. In the general case we show that these kernels coincide with the categorical left and right Hopf kernels of morphisms of Hopf algebras defined in \cite{AD}. Brauer's theorem for kernels over group algebras is generalized to Hopf algebras.
We study the flatness and the projectivity of Hopf algebras, defined over a Dedekind ring, over their Hopf subalgebras. We give a criterion for the faithful flatness and use it to show the faithful flatness of an arbitrary flat Hopf algebra upon its finite normal Hopf subalgebra. For the projectivity of a projective Hopf algebras we need some finiteness condition in terms of the module of integral. In particular we show the the module of integral has rank one.
In one of his last papers, Boris Weisfeiler proved that if modular semisimple Lie algebra possesses a solvable maximal subalgebra which defines in it a long filtration, then associated graded algebra is isomorphic to one constructed from the Zassenhaus algebra tensored with the divided powers algebra. We completely determine such class of algebras, calculating in process low-dimensional cohomology groups of Zassenhaus algebra tensored with any associative commutative algebra.
It is proved in the paper that a Noetherian residually finite dimensional Hopf algebra is a flat module over any right Noetherian right coideal subalgebra. In the case of Hopf subalgebras we get faithful flatness. These results are obtained by verifying the existence of the classical quotient rings of those algebras. It is also proved that the antipode of either right or left Noetherian residually finite dimensional Hopf algebra is bijective. As a consequence, such a Hopf algebra is right and left Noetherian simultaneously.
For each nontrivial semisimple Hopf algebra $H$ of dimension sixteen over $\mathbb{C}$, the smallest dimension inner-faithful representation of $H$ acting on a quadratic AS regular algebra $A$ of dimension 2 or 3, homogeneously and preserving the grading, is determined. Each invariant subring $A^H$ is determined. When $A^H$ is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that $H$ is a reflection Hopf algebra for $A$.
Let $H$ be a pointed Hopf algebra over an algebraically closed field of characteristic zero. If $H$ is a domain with finite Gelfand-Kirillov dimension greater than or equal to two, then $H$ contains a Hopf subalgebra of Gelfand-Kirillov dimension two.
Let $k$ be an algebraically closed field of characteristic $0$. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $p^2q^2$ over $k$, where $p,q$ are prime numbers with $p^2<q$. As an application, we also obtain the structure theorems for semisimple Hopf algebras of dimension $9p^2$ and $25q^2$ for all primes $3\neq p$ and $5\neq q$.
Let $p,q$ be prime numbers with $p>q^3$, and $k$ an algebraically closed field of characteristic 0. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $pq^3$.
Failure of flatness over finite-dimensional Hopf subalgebras
本文证明任意有限维非半单 Hopf 代数作为子代数时存在非平坦的包含代数，并指出一类无限维 Hopf 代数保证所有 Hopf 代数在其子代数上忠实平坦。
We study actions of semisimple Hopf algebras H on Weyl algebras A over a field of characteristic zero. We show that the action of H on A must factor through a group algebra; in other words, if H acts inner faithfully on A, then H is cocommutative. The techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.
In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension $2^{2n+1}$ for $n\geq 2$ over an algebraically closed field of characteristic $0$ with the group of group-like elements isomorphic to $\mathbb{Z}_{2^{n}}\times \mathbb{Z}_{2^{n}}$. Moreover we classify all such nonisomorphic Hopf algebras of dimension $32$ and show that they are not twist-equivalent to each other. More generally, given an abelian group of order $2^{m-1}$ we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension $2^{m}$ which have this group as t…
In this thesis we consider the maximal subalgebras of the exceptional Lie algebras in algebraically closed fields of positive characteristic. This begins with a quick recap of the article by Herpel and Stewart which considered the Cartan type maximal subalgebras in the exceptional Lie algebras for good characteristic, and then the article by Premet considering non-semisimple maximal subalgebras in good characteristic. For $p=5$ we give an example of what appears to be a new maximal subalgebra in the exceptional Lie algebra of type $E_8$. We show that this maximal subalgebra is isomorphic to th…
Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a non-trivial semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ can be constructed either from group algebras and their duals by means of extensions, or from Radford's biproduct $H\cong R#kG$, where $kG$ is the group algebra of $G$ of order 2, $R$ is a semisimple Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^3$.
Under the assumption that a residually finite dimensional Hopf algebra H has an Artinian ring of fractions it is proved that H is a flat module over any right coideal subalgebra satisfying a polynomial identity and is faithfully flat over any polynomial identity Hopf subalgebra. As a consequence we find a large class of Hopf algebras which are flat over all coideal subalgebras and are faithfully flat over all Hopf subalgebras.
Let $\Bbbk$ be a base field of characteristic $p>0$ and let $U$ be the restricted enveloping algebra of a 2-dimensional nonabelian restricted Lie algebra. We classify all inner-faithful $U$-actions on noetherian Koszul Artin-Schelter regular algebras of global dimension up to three.
Let $p,q$ be prime numbers with $p^4<q$, and $k$ an algebraically closed field of characteristic 0. We show that semisimple Hopf algebras of dimension $p^2q^2$ can be constructed either from group algebras and their duals by means of extensions, or from Radford biproduct $R#kG$, where $kG$ is the group algebra of group $G$ of order $p^2$, $R$ is a semisimple Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^2$. As an application, the special case that the structure of semisimple Hopf algebras of dimension $4q^2$ is given.
We develop a mechanism for classication of isomorphism types of non-trivial semisimple Hopf algebras whose group of grouplikes $G(H)$ is abelian of prime index $p$ which is the smallest prime divisor of $|G(H)|$. We describe structure of the second cohomology group of extensions of $\k C_p$ by $\k^G$ where $C_p$ is a cyclic group of order $p$ and $G$ a finite abelian group. We carry out an explicit classification for Hopf algebras of this kind of dimension $p^4$ for any odd prime $p$. The ground field is algebraically closed of characteristic $0$.
One-dimensional partial actions of 8-dimensional Hopf algebras and right coideal subalgebras
本文完成8维Hopf代数一维部分作用的分类，证明其对称性，并证明有限维余半单Hopf代数的右余理想子代数均可实现为部分smash积。
We describe the Galois objects and biGalois groups of monomial nonsemisimple Hopf algebras. The main feature of our description is the use of modified versions of the second cohomology group of the grouplike elements. These computations generalize the previous ones of Masuoka and Schauenburg for the Taft algebras.
We develop the theory of semisimple weak Hopf algebras and obtain analogues of a number of classical results for ordinary semisimple Hopf algebras. We prove a criterion for semisimplicity and analyze the square of the antipode S^2 of a semisimple weak Hopf algebra A. We explain how the Frobenius-Perron dimensions of irreducible A-modules and eigenvalues of S^2 can be computed using the inclusion matrix associated to A. A trace formula of Larson and Radford is extended to a relation between the global and Frobenius-Perron dimensions of A. Finally, an analogue of the Class Equation of Kac and Zh…
In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra pl(1,1), provides a noetherian prime counterexample.
The semisimple subalgebras of the rank $2$ symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{C})$ are well-known, and we recently classified its Levi decomposable subalgebras. In this article, we classify the solvable subalgebras of $\mathfrak{sp}(4,\mathbb{C})$, up to inner automorphism. This completes the classification of the subalgebras of $\mathfrak{sp}(4,\mathbb{C})$. More broadly speaking, in completing the classification of the subalgebras of $\mathfrak{sp}(4,\mathbb{C})$ we have completed the classification of the subalgebras of the rank $2$ semisimple Lie algebras.
We consider the combinatorial Dyson-Schwinger equation X=B^+(P(X)) in the non-commutative Connes-KreimerHopf algebra of planar rooted trees H, where B^+ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra A_P of\H. We describe all the formal series P such that A_P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of H, organized into three isomorphism classes: a first one, restricted to a olynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, i…
Let $q>2$ be a prime number, $d$ be an odd square-free natural number, and $n$ be a non-negative integer. We prove that a semisimple quasitriangular Hopf algebra of dimension $dq^n$ is solvable in the sense of Etingof, Nikshych and Ostrik. In particular, if $n\leq 3$ then it is either isomorphic to $k^G$ for some abelian group $G$, or twist equivalent to a Hopf algebra which fits into a cocentral abelian exact sequence.
We continue our study of the McKay Correspondence for grading preserving actions of semisimple Hopf algebras H on (noncommutative) Artin-Schelter regular algebras A. Here, we establish correspondences between module categories over A^H, over A\#H, and over \End_{A^H} A. We also study homological properties of (endomorphism rings of) maximal Cohen-Macaulay modules over A^H.
We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is normal (where the ground field $k$ is algebraically closed of characteristic zero). This means on the one hand that a Hopf subalgebra is normal when inducing (then restricting) modules several times as opposed to one time creates no new simple constituents. This point of view was taken in the paper http://arxiv.org/abs/math/0409346 which established a normality result in case H and K are finite group algebras. On the other hand this means that K is normal in H when H | K is a Galois extension with respect to action o…
Let H be a Hopf algebra of dimension pq over an algebraically closed field of characteristic 0, where p <= q are odd primes. Suppose that S is the antipode of H. If H is not semisimple, then S^{4p}=id_H and Tr(S^{2p}) is an integer divisible by p^2. In particular, if dim H = p^2, we prove that H is isomorphic to a Taft algebra. We then complete the classification for the Hopf algebras of dimension p^2.
In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin-Schelter regular algebras A of global dimension two, where A is a graded H-module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring A^H under such an action arises an analogue of a coordinate ring of a Kleinian singularity.
The question of whether or not a Hopf algebra $H$ is faithfully flat over a Hopf subalgebra $A$ has received positive answers in several particular cases: when $H$ (or more generally, just $A$) is commutative, or cocommutative, or pointed, or when $K$ contains the coradical of $H$. We prove the statement in the title, adding the class of cosemisimple Hopf algebras to those known to be faithfully flat over all Hopf subalgebras. We also show that the third term of the resulting "exact sequence" $A\to H\to C$ is always a cosemisimple coalgebra, and that the expectation $H\to A$ is positive when $…
Let $G$ be a finite group and let $π: G \to G'$ be a surjective group homomorphism. Consider the cocycle deformation $L = H^σ$ of the Hopf algebra $H = k^G$ of $k$-valued linear functions on $G$, with respect to some convolution invertible 2-cocycle $σ$. The (normal) Hopf subalgebra $k^{G'} \subseteq k^G$ corresponds to a Hopf subalgebra $L' \subseteq L$. Our main result is an explicit necessary and sufficient condition for the normality of $L'$ in $L$.
Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ is always semisolvable. That is, such Hopf algebras can be obtained by (a number of) extensions from group algebras or duals of group algebras.
We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability under extension of the base field. Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of A, and recognize the difference of the representation category and the representation ring of A. Actually, by computing and comparing polynomial invariants, we find new examples of pairs of Hopf algebras whose r…
Let H be a semisimple (so, finite dimensional) Hopf algebra over an algebraically closed field k of characteristic zero and let A be a commutative domain over k. We show that if A arises as an H-module algebra via an inner faithful H-action, then H must be a group algebra. This answers a question of E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen. The main results of this article extend to working over k of positive characteristic. On the other hand, we obtain results on Hopf actions on Weyl algebras as a consequence of the main theorem.
S. Montgomery and S. Witherspoon proved that upper and lower semisolvable, semisimple, finite dimensional Hopf algebras are of Froebenius type when their dimensions are not divisible by the characteristic of the base field. In this note we show that a finite dimensional, semisimple, lower solvable Hopf algebra is always of Froebenius type, in arbitrary characteristic.
In this paper, we prove that a non-semisimple Hopf algebra H of dimension 4p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided H contains more than two group-like elements. In particular, we prove that non-semisimple Hopf algebras of dimensions 20, 28 and 44 are pointed or their duals are pointed, and this completes the classification of Hopf algebras in these dimensions.
In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules $V$ of a semisimple quasi-Hopf algebra $H$ via the categorical counterpart developed in \cite{NS05}. We prove that this definition of higher FS-indicators coincides with the higher indicators introduced by Kashina, Sommerhäuser, and Zhu when $H$ is a Hopf algebra. We also obtain a sequence of canonical central elements of $H$, which is invariant under gauge transformations, whose values, when evaluated by the character of an $H$-module $V$, are the higher Frobenius-Schur indicators of $V$. As an …
We develop a theory of multigraded (i.e., $N^l$-graded) combinatorial Hopf algebras modeled on the theory of graded combinatorial Hopf algebras developed by Aguiar, Bergeron, and Sottile [Compos. Math. 142 (2006), 1--30]. In particular we introduce the notion of canonical $k$-odd and $k$-even subalgebras associated with any multigraded combinatorial Hopf algebra, extending simultaneously the work of Aguiar et al. and Ehrenborg. Among our results are specific categorical results for higher level quasisymmetric functions, several basis change formulas, and a generalization of the descents-to-pea…
We call a monoidal category ${\mathcal C}$ a Serre category if for any $C$, $D \in {\mathcal C}$ such that $C\ot D$ is semisimple, $C$ and $D$ are semisimple objects in ${\mathcal C}$. Let $H$ be an involutory Hopf algebra, $M$, $N$ two $H$-(co)modules such that $M \otimes N$ is (co)semisimple as a $H$-(co)module. If $N$ (resp. $M$) is a finitely generated projective $k$-module with invertible Hattory-Stallings rank in $k$ then $M$ (resp. $N$) is (co)semisimple as a $H$-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel'd modules over…
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