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We describe the structure of the quotient $\mathfrak{G}/\mathfrak{H}$ of a formal supergroup $\mathfrak{G}$ by its formal sub-supergroup $\mathfrak{H}$. This is a consequence which arises as a continuation of the authors' work (partly with M. Hashi) on algebraic/analytic supergoups.The results are presented and proved in terms of super-cocommutative Hopf superalgebras. The notion of co-free super-coalgebras plays a role, in particular., Comment: 21 pages; to appear in Comm. Algebra

M. Takeuchi (1989) proposed a Hopf-algebraic approach to Picard-Vessiot (or PV) theory, giving a new definition of PV extensions by which such extensions become more smoothly connected, through Hopf-Galois extensions, to the associated affine group schemes. This paper extends PV theory to the supersymmetric (or SUSY) context, following Takeuchi's approach. The notion of SUSY fields is defined. Differential fields in the existing PV theory are replaced by $D$-SUSY fields, where $D$ is an acting super-cocommutative Hopf superalgebra. Hopf-Galis extensions here play a more and more crucial role., Comment: Added table of contents; corrected typos and minor errors

Torsors under affine groups are generalized in the super context by super-torsors under affine super-groups. We investigate those super-torsors by using Hopf-algebra language and techniques. It is explicitly shown, under suitable assumptions, that every super-torsor arises from an ordinary torsor. Especially, the objects with affinity restriction, or namely, the affine super-torsors and the affine ordinary torsors are shown to be precisely in one-to-one correspondence. The results play substantial roles in ongoing construction of super-symmetric Picard-Vessiot theory., Comment: 32 pages; as major changes we added Proposition 3.19, and strengthened Proposition 5.1, removing the assumption that the relevant Hopf super-algebra was finitely generated

We generalize to the super context, the known fact that if an affine algebraic group $G$ over a commutative ring $k$ acts freely (in an appropriate sense) on an affine scheme $X$ over $k$, then the dur sheaf $X\tilde{\tilde{/}}G$ of $G$-orbits is an affine scheme in the following two cases: (I) $G$ is finite; (II) $k$ is a field, and $G$ is linearly reductive. An emphasize is put on the more difficult generalization in the second case; the replaced assumption then is that an affine algebraic super-group $G$ over an arbitrary field has an integral. Those super-groups which satisfy the assumption are characterized, and are seen to form a large class if $\operatorname{char}k=0$. Hopf-algebraic techniques including bosonization are applied to prove the results., Comment: Made very minor changes and corrections; final version to appear in Comm. Algebra

Discussed here is descent theory in the differential context where everything is equipped with a differential operator. To answer a question personally posed by A. Pianzola, we determine all twisted forms of the differential Lie algebras over $\mathbb{C}(t)$ associated with complex simple Lie algebras. Hopf-Galois Theory, a ring-theoretic counterpart of theory of torsors for group schemes, plays a role when we grasp the above-mentioned twisted forms from torsors., Comment: Largely revised; in particular, changed the title and added several articles to the References, see the added Remark 2.2

It was proved by the first-named author and Zubkov [13] that given an affine algebraic supergroup $\mathbb{G}$ and a closed sub-supergroup $\mathbb{H}$ over an arbitrary field of characteristic $\ne 2$, the faisceau $\mathbb{G} \tilde{/} \mathbb{H}$ (in the fppf topology) is a superscheme, and is, therefore, the quotient superscheme $\mathbb{G}/\mathbb{H}$, which has desirable properties, in fact. We reprove this, by constructing directly the latter superscheme $\mathbb{G}/\mathbb{H}$. Our proof describes explicitly the structure sheaf of $\mathbb{G}/\mathbb{H}$, and reveals some new geometric features of the quotient, that include one which was desired by Brundan [2], and is shown in general, here for the first time., Comment: 28 Pages; final version to appear in Transform. Groups

We show basic results on super-manifolds and super Lie groups over a complete field of characteristic $\ne 2$, extensively using Hopf-algebraic techniques. The main results are two theorems. The first main theorem shows a category equivalence between super Lie groups and Harish-Chandra pairs, which is applied especially to construct the Hopf super-algebra of all analytic representative functions on a super Lie group. The second constructs homogeneous super-manifolds by a new Hopf-algebraic method, showing their remarkable property., Comment: Added an article, [4], and two remarks, Remarks 3.8 and 7.3

By finite quantum groups we mean Lusztig's finite-dimensional pointed Hopf algebras called quantum Frobenius Kernels [9, 10], and their natural generalizations due to Andruskiewitsch and Schneider [2, 3]. For a Hopf algebra $H$ in a special class of the latter generalizations, which arises from a pair of quantum linear spaces, Krop and Radford [8] described the simple modules over $H$ and over the Drinfel'd double $D(H)$, showing that they fall into a simple pattern of parametrization. We extend the description to a wider class of Hopf algebras which includes the quantum Frobenius Kernels, renewing the parametrization pattern so as to connect directly to the so-called triangular decomposition., Comment: Essentially the same as the last version, v.3; to appear in: H. Yamane et al. eds., The proceedings of Meeting for Study of Number theory, Hopf algebras and Related Topics, Toyama, 12-15 February 2017

To construct an affine supergroup from a Harish-Chandra pair, Gavarini [2] invented a natural method, which first constructs a group functor and then proves that it is representable. We give a simpler and more conceptual presentation of his construction in a generalized situation, using Hopf superalgebras over a superalgebra. As an application of the construction, given a closed super-subgroup of an algebraic supergroup, we describe the normalizer and the centralizer, using Harish-Chandra pairs. We also prove a tensor product decomposition theorem for Hopf superalgebras, and describe explicitly by cocycle deformation, the difference which results from the two choices of dualities found in literature., Comment: Added "Note" to the end of the text

We study solvability, nilpotency and splitting property for algebraic supergroups over an arbitrary field $K$ of characteristic $\mathrm{char}\, K \ne 2$. Our first main theorem tells us that an algebraic supergroup $\mathbb{G}$ is solvable if the associated algebraic group $\mathbb{G}_{ev}$ is trigonalizable. To prove it we determine the algebraic supergroups $\mathbb{G}$ such that $\dim \mathrm{Lie}(\mathbb{G})_1=1$; their representations are studied when $\mathbb{G}_{ev}$ is diagonalizable. The second main theorem characterizes nilpotent connected algebraic supergroups. A super-analogue of the Chevalley Decomposition Theorem is proved, though it must be in a weak form. An appendix is given to characterize smooth Noetherian superalgebras as well as smooth Hopf superalgebras., Comment: Secondary revised; added Section 3.3 and Corollary 5.3

This note is a development of our two previous papers, arXiv:1212.3392v1 and 1306.3660v1. The fundamental question is whether there exists a Galois theory, in which the Galois group is a quantum group. For a linear equations with respect to a Hopf algebra, we arrived at a final form if the base field consists of constants. In this case, we have non-commutative Picard-Vessiot rings and asymmetric Tannaka theory. For non-linear equations there are examples that might make us optimistic., Comment: 101 pages, we added Part III, arXiv admin note: text overlap with arXiv:1212.3392, arXiv:1312.4682

In previous work, Eli Aljadeff and the first-named author attached an algebra B_H of rational fractions to each Hopf algebra H. The generalized Noether problem is the following: for which finite-dimensional Hopf algebra H is B_H the localization of a polynomial algebra? A positive answer to this question when H is the algebra of functions on a finite group implies a positive answer for the classical Noether problem for the group. We show that the generalized Noether problem has a positive answer for all pointed finite-dimensional Hopf algebras over a field of characteristic zero. We actually give a precise description of B_H for such a Hopf algebra, including a bound on the degrees of the generators. A theory of polynomial identities for comodule algebras over a Hopf algebra H gives rise to a universal comodule algebra whose subalgebra of coinvariants V_H maps injectively into B_H. In the second half of this paper, we show that B_H is a localization of V_H when again H is a pointed finite-dimensional Hopf algebra in characteristic zero. We also report on a result by Uma Iyer showing that the same localization result holds when H is the algebra of functions on a finite group., Comment: 19 pages. Section 4.3 and three references have been added to Version 2

Reproducing my talk at Algebra Symposium held at Hiroshima University, August 26--29, 2013, I review recent results on super algebraic groups, emphasizing results obtained by myself and my coauthors using Hopf algebraic techniques. The results are all basic, and I intend to make this report into a somewhat informal introduction to the subject., Comment: Corrected typos on Page 2, line 13 and on Page 9, line 14, adding Footnote 5 for the latter; corrected Condition 2 of Lemma 7.3; Up-dated the References

A handlebody-knot is a handlebody embedded in the 3-sphere. We establish a uniform method to construct invariants for handlebody-links. We introduce the category $\mathcal{T}$ of handlebody-tangles and present it by generators and relations. The result tells us that every functor on $\mathcal{T}$ that gives rise to invariants is derived from what we call a quantum-commutative quantum-symmetric algebra in the target category. The example of such algebras of our main concern is finite-dimensional unimodular Hopf algebras. We investigate how those Hopf algebras give rise to handlebody-knot invariants., Comment: 25 Pages

We prove a category equivalence between algebraic supergroups and Harish-Chandra pairs over a commutative ring which is $2$-torsion free. The result is applied to re-construct the Chevalley $\mathbb{Z}$-supergroups constructed by Fioresi and Gavarini [8] and by Gavarini [9, 10]. For a wide class of algebraic supergroups we describe their representations by using their super-hyperalgebras., Comment: Added "Note" to the end of the text

Realizing the possibility suggested by Hardouin [6], we show that her own Picard-Vessiot Theory for iterative $q$-difference rings is covered by the (consequently, more general) framework, settled by Amano and Masuoka [2], of artinian simple module algebras over a cocommutative pointed Hopf algebra. An essential point is to represent iterative $q$-difference modules over an iterative $q$-difference ring $R$, by modules over a certain cocommutative $\times_R$-bialgebra. Recall that the notion of $\times_R$-bialgebras was defined by Sweedler [17], as a generalization of bialgebras., Comment: 24 Pages; the final version accepted by Internat. J. Math

We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such lifting., Comment: 32 pages

Hopf crossed products, or in other words, cleft comodule algebras form a special but important class in Hopf-Galois extensions. To discuss this interesting subject, we will start with the more familiar group crossed products, and then see that they are naturally generalized by Hopf crossed products; these Hopf crossed products are characterized as Hopf-Galois extensions with normal basis. After showing this characterization due to Doi and Takeuchi, we will proceed to two applications of Hopf crossed products---the equivariant smoothness of Hopf algebras, and the tensor product decomposition in super-commutative Hopf superalgebras., Comment: The notes based on the author's talks at the Conference on Ring Theory held at the Sun Yat-Sen University, Guangzhou in P. R. China, June 24--30, 2012

Over an arbitrary field of characteristic $\ne 2$, we define the notion of Harish-Chandra pairs, and prove that the category of those pairs is anti-equivalent to the category of algebraic affine supergroup schemes. The result is applied to characterize some classes of affine supergroup schemes such as those which are (a) simply connected, (b) unipotent or (c) linearly reductive in positive characteristic., Comment: 35 Pages; Revised following the referees' suggestions

To generalize some fundamental results on group schemes to the super context, we study the quotient sheaf $G \tilde{/} H$ of an algebraic supergroup $G$ by its closed supersubgroup $H$, in arbitrary characteristic $\neq$ 2. Our main theorem states that $G \tilde{/} H$ is a Noetherian superscheme. This together with derived results give positive answers to interesting questions posed by J. Brundan., Comment: Revised the manuscript, improving the exposition and correcting typos, 40 pages; the final version accepted for publication

We prove that given a super affine closed subgroup $H$ of a super affine group $G$ over a field $k$ of charctersitic $\mathrm{ch} k \ne 2$, the dur $k$-sheaf $G\tilde{\tilde{/}} H$ of right cosets is affine if the affine $k$-group $\bar{H}$ assocoiated to $H$ is (a) reductive or (b) pro-finite. Especially when $G$ is algebraic, the result in Case (a) gives rise to a positive answer to Brundan's question which was recently discussed by Zubkov \cite{Z}., Comment: 11 Pages; Added afterthought and erratum

Non-commutative torsors (equivalently, two-cocycles) for a Hopf algebra can be used to twist comodule algebras. After surveying and extending the literature on the subject, we prove a theorem that affords a presentation by generators and relations for the algebras obtained by such twisting. We give a number of examples, including new constructions of the quantum affine spaces and the quantum tori., Comment: 27 pages. Masuoka is a new coauthor. Introduction was revised. Sections 1 and 2 were thoroughly restructured. The presentation theorem in Section 3 is now put in a more general framework and has a more general formulation. Section 4 was shortened. All examples (quantum affine spaces and tori, twisting of SL(2), twisting of the enveloping algebra of sl(2)) are left unchanged

In previous work, to each Hopf algebra H and each invertible right two-cocycle on H, Eli Aljadeff and the first-named author attached a subalgebra B of the free commutative Hopf algebra S generated by the coalgebra underlying H; the algebra B is the subalgebra of coinvariants of a generic Hopf Galois extension. In this paper we give conditions under which S is faithfully flat, or even free, as a B-module. We also show that B is generated as an algebra by certain elements arising from the theory of polynomial identities for comodule algebras developped jointly with Aljadeff., Comment: 15 pages. Corollary 3.7, Theorem 3.8, missing hypothesis in Theorem 3.13, and one reference added. Misprints corrected.

The quantized enveloping algebra $U_q$ is constructed as a quotient of the generalized quantum double $ U^{\leq 0}_q \cmdbicross_{\tau} U^{\geq 0}_q $ associated to a natural skew pairing $ \tau : U^{\leq 0}_q \otimes U^{\geq 0}_q \to k $. This double is generalized by $\cmdcalD = (\cmdfrakB (V) \cmddotrtimes F) \cmdbicross_{\tau} (\cmdfrakB (W) \cmddotrtimes G)$, where $F$, $G$ are abelian groups, $ V \in {}^F_F \cmdcalYD $, $ W \in {}^G_G \cmdcalYD $ are Yetter-Drinfeld modules and $ \cmdfrakB (V) $, $ \cmdfrakB (W) $ are their Nichols algebras. We prove some results on Hopf ideals of $ \cmdcalD $, including a characterization of what we call thin Hopf ideals. As an application we give an explicit description of those minimal quasitriangular pointed Hopf algebras in characteristic zero which are generated by skew primitives., Comment: 2 pages. v4 (final version); Changed the title, refomulated Remark 7.3 as Corollary 7.3, and added Corollary 7.4

We define the generalized $q$-boson algebra $ \cmdB$ associated to a pair of Nichols algebras and a skew pairing. We study integrable $ \cmdB$-modules, generalizing results by M. Kashiwara and T. Nakashima on integrable modules over a $q$-boson (Kashiwara) algebra., Comment: 18 pages

We prove that a finite-dimensional irreducible Hopf algebra $H$ in positive characteristic is semisimple, if and only if it is commutative and semisimple, if and only if the restricted Lie algebra $P(H)$ of the primitives is a torus. This generalizes Hochschild's theorem on restricted Lie algebras, and also generalizes Demazure and Gabriel's and Sweedler's results on group schemes, in the special but essential situation with finiteness assumption added., Comment: 8 pages; to appear in the Proc. Amer. Math. Soc

By using cocycle deformation, we construct a certain class of Hopf algebras, containing the quantized enveloping algebras and their analogues, from what we call pre-Nichols algebras. Our construction generalizes in some sense the known construction by (generalized) quantum doubles, but unlike in the known situation, it saves us from difficulties in checking complicated defining relations., Comment: To appear in a theme issue of the Arabian Journal for Science and Engineering. The published version will renumber the sections so that for example, Theorem 3.10 in this version will be called Theorem 4.10

For a class of pointed Hopf algebras including the quantized enveloping algebras, we discuss cleft extensions, cocycle deformations and the second cohomology. We present such a non-standard method of computing the abelian second cohomology that derives information from the non-abelian second cohomology classifying cleft extensions. As a sample computation, a quantum analogue of Whitehead's second lemma for Lie-algebra cohomology is proved., Comment: 40 Pages, the final version to appear in J. Algebra; mainly, typos corrected