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Register a new userNon-existence of integral Hopf orders for twists of several simple groups of Lie type
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Let $p$ be a prime number and $q=p^m$, with $m geq 1$ if $p eq 2,3$ and $m>1$ otherwise. Let $Omega$ be any non-trivial twist for the complex group algebra of $mathbf{PSL}_2(q)$ arising from a $2$-cocycle on an abelian subgroup of $mathbf{PSL}_2(q)$. We show that the twisted Hopf algebra $(mathbb{C} mathbf{PSL}_2(q))_{Omega}$ does not admit a Hopf order over any number ring. The same conclusion is proved for the Suzuki group $^2!B_2(q)$ and $mathbf{SL}_3(p)$ when the twist stems from an abelian $p$-subgroup. This supplies new families of complex semisimple (and simple) Hopf algebras that do not admit a Hopf order over any number ring. The strategy of the proof is formulated in a general framework that includes the finite simple groups of Lie type. As an application, we combine our results with two theorems of Thompson and Barry and Ward on minimal simple groups to establish that for any finite non-abelian simple group $G$ there is a twist $Omega$ for $mathbb{C} G$, arising from a $2$-cocycle on an abelian subgroup of $G$, such that $(mathbb{C} G)_{Omega}$ does not admit a Hopf order over any number ring. This partially answers in the negative a question posed by Meir and the second author.
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We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfeld twists of group algebras for the following groups: $A_n$, the alternating group on $n$ elements, with $n geq 5$; and $S_{2m}$, the symmetric group on $2m$ elements, with $m geq 4$ even. The twist for $A_n$ arises from a $2$-cocycle on the Klein four-group contained in $A_4$. The twist for $S_{2m}$ arises from a $2$-cocycle on a subgroup generated by certain transpositions which is isomorphic to $mathbb{Z}_2^m$. This provides more examples of complex semisimple Hopf algebras that can not be defined over number rings. As in the previous family known, these Hopf algebras are simple.
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 Drinfeld 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 Kaplanskys sixth conjecture if and only if it admits a weak order, in the sense of Rumynin and Lorenz, over the integers.
A Hopf algebra is co-Frobenius when it has a nonzero integral. It is proved that the composition length of the indecomposable injective comodules over a co-Frobenius Hopf algebra is bounded. As a consequence, the coradical filtration of a co-Frobenius Hopf algebra is finite; this confirms a conjecture by Sorin Du{a}scu{a}lescu and the first author. The proof is of categorical nature and the same result is obtained for Frobenius tensor categories of subexponential growth. A family of co-Frobenius Hopf algebras that are not of finite type over their Hopf socles is constructed, answering so in the negative another question by the same authors.
Serious difficulties arise in the construction of chains of twists for symplectic Lie algebras. Applying the canonical chains of extended twists to deform the Hopf algebras U(sp(N)) one is forced to deal only with improper chains (induced by the U(sl(N)) subalgebras). In the present paper this problem is solved. For chains of regular injections the sets of maximal extended jordanian twists F_{E,k} are considered. We prove that there exists for U(sp(N)) the twist F_{B,k} composed of the factors F_{E,k}. It is demonstrated that the twisting procedure deforms the space of the primitive subalgebra sp(N-1). The recursive algorithm for such deformation is found. This construction generalizes the results obtained for orthogonal classical Lie algebras and demonstrates the universality of primitivization effect for regular chains of subalgebras. For the chain of maximal length the twists F_{B,k,max} become full, their carriers contain the Borel subalgebra B(sp(N)). Using such twisting procedures one can obtain the explicit quantizations for a wide class of classical r-matrices. As an example the full chain of extended twists for U(sp(3)) is considered.
Let $p$ be an odd prime number and $K$ a number field having a primitive $p$-th root of unity $zeta.$ We prove that Nikshychs non-group theoretical Hopf algebra $H_p$, which is defined over $mathbb{Q}(zeta)$, admits a Hopf order over the ring of integers $mathcal{O}_K$ if and only if there is an ideal $I$ of $mathcal{O}_K$ such that $I^{2(p-1)} = (p)$. This condition does not hold in a cyclotomic field. Hence this gives an example of a semisimple Hopf algebra over a number field not admitting a Hopf order over any cyclotomic ring of integers. Moreover, we show that, when a Hopf order over $mathcal{O}_K$ exists, it is unique and we describe it explicitly.
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