Let $H$ be a finite-dimensional Hopf algebra. We study the behaviou r of primitive and maximal ideals in certain types of ring extensions determined by $H$. The main focus is on the class of faithfully flat Galois extensions, which includes includes smash and crossed products. It is shown how analogous results can be obtained for the larger class of extensions possessing a total integral, which includes extensions $A^H\subseteq A $ when $H$ is semisimple. We use Passman's "primitivity machine" to reduce the whole theory of Kr ull relations for prime ideals to the case of primitive ideals. The concept of strongly semiprimitive Hopf algebra is introduced and investigated. Several examples and open problems are discussed.
We identify a collection of primitive elements generating the Hopf algebra NCSym of symmetric functions in noncommuting variables and give a combinatorial formula for the antipode.
Primitive cohomology of a Hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra. Among many of its applications, two classifications are presented. Firstly we classify all non locally PI, pointed Hopf algebra domains of Gelfand-Kirillov dimension two; and secondly we classify all pointed Hopf algebras of rank one. The first classification extends some results of Brown, Goodearl and others in an ongoing project to understand all Hopf algebras of low Gelfand-Kirillov dimension. The second generalizes results of Krop-Radford and Wang-You-Chen which …
We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra over the primitives). The choice of such an isomorphism implies the existence a finer algebraic structure on the Hopf algebra and on the indecomposables (resp. the primitives). For instance a cofree-cocommutative right-sided combinatorial Hopf algebra is completely determined by its primitive part which is a pre-Lie algebra. The key example is the Connes-Kreimer Hopf algebra. The study of all these co…
We extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the `Irreducibility theorem' and `Duflo theorem', to much wider classes of algebras. Our general version of Irreducibility theorem says that if A is a positively filtered associative algebra such that gr(A) is a commutative Poisson algebra with finitely many symplectic leaves, then the associated variety of any primitive ideal in A is the closure of a single connected symplectic leaf. Our general version of Duflo theorem says that if A is an algebra with a `triangular structure' (see sect. 2…
For finite-dimensional Hopf algebras, their classification in characteristic $0$ (e.g. over $\mathbb{C}$) has been investigated for decades with many fruitful results, but their structures in positive characteristic have remained elusive. In this paper, working over an algebraically closed field $\mathbf{k}$ of prime characteristic $p$, we introduce the concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional connected Hopf algebras which are almost primitively generated; that is, these connected Hopf algebras are $p^{n+1}$-dimensional, wh…
Necessary and sufficient conditions are obtained for an ambiskew polynomial algebra A over a Hopf k-algebra R to possess the structure of a Hopf algebra extending that of R, in which the added variables X+ and X- are skew primitive. The coradical filtration is calculated, many examples are described, and properties determined.
Motivated by the orthogonality relations for irreducible characters of a finite group, we evaluate the sum of a finite group of linear characters of a Hopf algebra, at all grouplike and skew-primitive elements. We then discuss results for products of skew-primitive elements. Examples include groups, (quantum groups over) Lie algebras, the small quantum groups of Lusztig, and their variations (by Andruskiewitsch and Schneider).
A Poisson algebra $\Bbb C[G]$ considered as a Poisson version of the twisted quantized coordinate ring $\Bbb C_{q,p}[G]$, constructed by Hodges, Levasseur and Toro in \cite{HoLeT}, is obtained and its Poisson structure is investigated. This establishes that all Poisson prime and primitive ideals of $\Bbb C[G]$ are characterized. Further it is shown that $\Bbb C[G]$ satisfies the Poisson Dixmier-Moeglin equivalence and that Zariski topology on the space of Poisson primitive ideals of $\Bbb C[G]$ agrees with the quotient topology induced by the natural surjection from the maximal ideal space of …
This note discusses a framework for the investigation of the prime spectrum of an associative algebra A that is equipped with an action of a Hopf algebra H. In particular, we study a notion of H-rationality for ideals of A and comment on a possible Dixmier-Moeglin equivalence for H-prime ideals of A.
We provide an explicit description of the primitive ideals of the enveloping algebra $\operatorname{U}(\frak{sl}(\infty))$ of the infinite-dimensional finitary Lie algebra $\frak{sl}(\infty)$ over an uncountable algebraically closed field of characteristic 0. Our main new result is that any primitive ideal of $\operatorname{U}(\frak{sl}(\infty))$ is integrable. A classification of integrable primitive ideals of $\operatorname{U}(\frak{sl}(\infty))$ has been known previously, and relies on the pioneering work of A. Zhilinskii.
The quantized enveloping algebra $U_q$ is constructed as a quotient of the generalized quantum double $ U^{\leq 0}_q \cmdbicross_τ U^{\geq 0}_q $ associated to a natural skew pairing $ τ: U^{\leq 0}_q \otimes U^{\geq 0}_q \to k $. This double is generalized by $\cmdcalD = (\cmdfrakB (V) \cmddotrtimes F) \cmdbicross_τ (\cmdfrakB (W) \cmddotrtimes G)$, 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 $, includ…
We first prove that a graded, connected, free and cofree Hopf algebra is always self-dual; then that two graded, connected, free and cofree Hopf algebras are isomorphic if, and only if, they have the same Poincaré-Hilbert formal series. If the characteristic of the base field is zero, we prove that the Lie algebra of the primitive elements of such an object is free, and we deduce a characterization of the formal series of free and cofree Hopf algebras by a condition of growth of the coefficients. We finally show that two graded, connected, free and cofree Hopf algebras are isomorphic as (non g…
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.
For a semisimple Lie algebra defined over a discrete valuation ring with field of fractions $K$, we prove that any primitive ideal with rational central character in the affinoid enveloping algebra, $\widehat{U(\mathfrak{g})_{K}},$ is the annihilator of an affinoid highest weight module.
Let $p$ be a prime, $k$ be an algebraically closed field of characteristic $p$. In this paper, we provide the classification of connected Hopf algebras of dimension $p^3$, except the case when the primitive space of the Hopf algebra is two dimensional and abelian. Each isomorphism class is presented by generators $x, y, z$ with relations and Hopf algebra structures. Let $μ$ be the multiplicative group of $(p^2+p-1)$-th roots of unity. When the primitive space is one-dimensional and $p$ is odd, there is an infinite family of isomorphism classes, which is naturally parameterized by $A_{k}^1/μ$.
We provide necessary and sufficient conditions to extend the Hopf-Galois algebra structure on an algebra R to a generalized ambiskew ring based on R, in a way such that the added variables for the extension are skew-primitive in an appropriate sense. We show that the associated Hopf algebra is again a a generalized ambiskew ring, based on a suitable Hopf algebra H(R). Several examples are examined, including the Hopf-Galois objects over Uq(sl2).
Let $V$ be a linear space over a field ${\bf k}$ with a braiding $τ: V\otimes V\rightarrow V\otimes V.$ We prove that the braiding $τ$ has a unique extension on the free nonassociative algebra ${\bf k}\{V\}$ freely generated by $V$ so that ${\bf k}\{V\}$ is a braided algebra. Moreover, we prove that the free braided algebra ${\bf k}\{V\}$ has a natural structure of a braided nonassociative Hopf algebra such that every element of the space of generators $V$ is primitive. In the case of involutive braidings, $τ^2={\rm id}$, we describe braided analogues of Shestakov-Umirbaev operations and prove…
The aim of this paper is to study the representation theory of quantum Schubert cells. Let $\g$ be a simple complex Lie algebra. To each element $w$ of the Weyl group $W$ of $\g$, De Concini, Kac and Procesi have attached a subalgebra $U_q[w]$ of the quantised enveloping algebra $U_q(\g)$. Recently, Yakimov showed that these algebras can be interpreted as the quantum Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of $U_q[w]$. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter that the primitive spectrum of $U_q[w]$ admits …
We prove that a finite-dimensional cocommutative Hopf algebra $H$ is local, if and only if the subalgebra generated by the first term of its coradical filtration $H_1$ is local. In particular if $H$ is connected, $H$ is local if and only if all the primitive elements of $H$ are nilpotent.
Let p be a prime, K a p-adic field, G a nilpotent, uniform pro-p group. We prove that all faithful, primitive ideals in the Iwasawa algebra KG are controlled by the centraliser of the second term in the upper central series for G.
We prove that any finite W-algebra U(g,e) admits a one-dimensional representation fixed by the action of the component group of the centraliser of e. As a consequence, for any nilpotent orbit O in g there exists a multiplicity-free (and hence completely prime) primitive ideal of the universal enveloping algebra U(g) whose associated variety coincides with the Zariski closure of O.
Given an affine algebraic variety V and a quantization A of its coordinate ring, it is conjectured that the primitive ideal space of A can be expressed as a topological quotient of V. Evidence in favor of this conjecture is discussed, and positive solutions for several types of varieties (obtained in joint work with E. S. Letzter) are described. In particular, explicit topological quotient maps are given in the case of quantum toric varieties.
We show that any finite-dimensional pointed Hopf algebra over an abelian group $Γ$ such that its infinitesimal braiding is of standard type is generated by group-like and skew-primitive elements. This fact agrees with the long-standing conjecture by Andruskiewitsch and Schneider. We also show that the quantum Serre relations hold in any coradically graded pointed Hopf algebra over $Γ$ of finite dimension and determine how these relations are lifted in the standard case.
Let $p$ be an odd prime number and $K$ a number field having a primitive $p$-th root of unity $ζ.$ We prove that Nikshych's non-group theoretical Hopf algebra $H_p$, which is defined over $\mathbb{Q}(ζ)$, 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 $\math…
In this paper, we study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a Theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of various quantum algebras is always even. Next we give a combinatorial criterion for a prime ideal that is invariant under the torus action to be primitive. We use this criterion to obtain a formula for the number of primitive ideals in the algebra of $2\times n$ quantum matrices that are invariant under the action of the torus. Roughly speaking, this can be thou…
Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I[x] for some ideals I of R.
We study inclusions between primitive ideals in the universal enveloping algebra of general linear superalgebras. For classical Lie superalgebras, any primitive ideal is the annihilator of a simple highest weight module. It therefore suffices to study the quasi-order on highest weights determined by the relation of inclusion between primitive ideals. For the specific case of reductive Lie algebras, this quasi-order is essentially the left Kazhdan-Lusztig quasi-order. For Lie superalgebras, the classification is unknown in general, safe from some low dimensional specific cases. We derive an alt…
Explicit generating sets are found for all primitive ideals in the generic quantized coordinate rings of the 3x3 special and general linear groups over an arbitrary algebraically closed field. (Previously, generators were only known up to certain localizations.) The generating sets form polynormal regular sequences, from which it follows that all primitive factor algebras of these quantized coordinate rings are Auslander-Gorenstein and Cohen-Macaulay.
Let $\mathfrak{g}$ be a semisimple Lie algebra. We establish a new relation between the Goldie rank of a primitive ideal $\mathcal{J}\subset U(\mathfrak{g})$ and the dimension of the corresponding irreducible representation $V$ of an appropriate finite W-algebra. Namely, we show that $\operatorname{Grk}(\mathcal{J}) \leqslant \dim V/d_V$, where $d_V$ is the index of a suitable equivariant Azumaya algebra on a homogeneous space. We also compute $d_V$ in representation theoretic terms.
We introduce bidendriform bialgebras, which are bialgebras such that both product and coproduct can be split into two parts satisfying good compatibilities. For example, the Malvenuto-Reutenauer Hopf algebra and the non-commutative Connes-Kreimer Hopf algebras of planar decorated rooted trees are bidendriform bialgebras. We prove that all connected bidendriform bialgebras are generated by their primitive elements as a dendriform algebra bidendriform Milnor-Moore theorem) and then is isomorphic to a Connes-Kreimer Hopf algebra. As a corollary, the Hopf algebra of Malvenuto-Reutenauer is isomorp…
Let $H$ be a connected Hopf algebra acting on an algebra $A$. Working over a base field having characteristic $0$, we show that for a given prime (semi-prime, completely prime) ideal $I$ of $A$, the largest $H$-stable ideal of A contained in $I$ is also prime (semi-prime, completely prime). We also prove a similar result for certain subrings of convolution algebras.
Let $\mathcal G$ be an ultragraph and let $K$ be a field. We describe prime and primitive ideals in the ultragraph Leavitt path algebra $L_K(\mathcal G)$. We identify the graded prime ideals in terms of downward directed sets and then we characterize the non-graded prime ideals. We show that the non-graded prime ideals of $L_K(\mathcal G)$ are always primitive.
The main object of this note is to study the conormal module $M$ and the computation of the second symbolic power $\bar I^{(2)}$ of an ideal $\bar I$ in the residue ring $R/H$ of a polynomial ring $R$ over a field of characteristic zero. The torsion part $T(M)$ of $M$ and the torsion free module $M/T(M)$ are expressed by the primitive ideal of $I$ relative to $H$. Two characterizations for $M/T(M)$ to be free are proved. Some immediate applications are worked out.
Panov proved necessary and sufficient conditions to extend the Hopf algebra structure of an algebra $R$ to an Ore extension $R[x;σ,δ]$ with $x$ being a skew-primitive element. In this paper we extend Panov's result to Ore extensions over weak Hopf algebras. As an application we study Ore extensions of connected groupoid algebras.
We present an algorithm which computes the annihilator in ${\rm U}(\frak{sl}(\infty))$ of any simple highest weight $\frak{sl}(\infty)$-module $L_\frak b(λ)$. This algorithm is based on an infinite version of the Robinson-Schensted algorithm.
Given a $p$-adic field $K$ and a nilpotent uniform pro-$p$ group $G$, we prove that all primitive ideals in the $K$-rational Iwasawa algebra $KG$ are maximal, and can be reduced to a particular standard form. Setting $\mathcal{L}$ as the associated $\mathbb{Z}_p$-Lie algebra of $G$, our approach is to study the action of $KG$ on a Dixmier module $\widehat{D(λ)}$ over the affinoid envelope $\widehat{U(\mathcal{L})}_K$, and to prove that all primitive ideals can be reduced to annihilators of modules of this form.
We examine actions of finite-dimensional pointed Hopf algebras on central simple division algebras in characteristic 0. (By a Hopf action we mean a Hopf module algebra structure.) In all examples considered, we show that the given Hopf algebra does admit a faithful action on a central simple division algebra, and we construct such a division algebra. This is in contrast to earlier work of Etingof and Walton, in which it was shown that most pointed Hopf algebras do not admit faithful actions on fields. We consider all bosonizations of Nichols algebras of finite Cartan type, small quantum groups…
The celebrated Milnor-Moore theorem and the more general Cartier-Kostant-Milnor-Moore theorem establish close relationships of a connected and a pointed cocommutative Hopf algebra with its Lie algebra of primitive elements and its group of group-like elements. Crossed homomorphisms for Lie algebras, groups and Hopf algebras have been studied extensively, first from a cohomological perspective and then more broadly, with an important case given by difference operators. This paper shows that the relationship among the different algebraic structures captured in the Milnor-Moore theorem can be str…
We show that the invariants of a free associative algebra of finite rank under a linear action of a finite-dimensional Hopf algebra generated by group-like and skew-primitive elements form a finitely generated algebra exactly when the action is scalar. This generalizes an analogous result for group actions by automorphisms obtained by Dicks and Formanek, and Kharchenko.
Let q be a nonzero complex number that is not a root of unity. We give a criterion for (0) to be a primitive ideal of the algebra O_q(M_{m,n}) of quantum matrices. Next, we describe all height one primes of O_q(M_{m,n}); these two problems are actually interlinked since it turns out that (0) is a primitive ideal of O_q(M_{m,n}) whenever O_q(M_{m,n}) has only finitely many height one primes. Finally, we compute the automorphism group of O_q(M_{m,n}) in the case where m is not equal to n. In order to do this, we first study the action of this group on the prime spectrum of O_q(M_{m,n}). Then, by…
We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. We also describe the structure constants of the multiplication as a certain number of facets of the permutahedron. Our results reveal a close relationship between the structure of this Hopf algebra and the weak order on the symmetric groups.
Let $A$ be a $d\times n$ integer matrix whose column vectors generate the lattice $\Z^d$, and let $D(R_A)$ be the ring of differential operators on the affine toric variety defined by $A$. We show that the classification of $A$-hypergeometric systems and that of $\Z^d$-graded simple $D(R_A)$-modules (up to shift) are the same. We then show that the set of $\Z^d$-homogeneous primitive ideals of $D(R_A)$ is finite. Furthermore, we give conditions for the algebra $D(R_A)$ being simple.
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.
We consider the following question, concerning associative algebras R over an algebraically closed field k: When can the space of (equivalence classes of) finite dimensional irreducible representations of R be topologically embedded into a classical affine space? We provide an affirmative answer for algebraic quantum groups at roots of unity. More generally, we give an affirmative answer for k-affine maximal orders satisfying a polynomial identity, when k has characteristic zero. Our approach closely follows the foundational studies by Artin and Procesi on finite dimensional representations. O…
Anti-cocommutative elements were introduced by Wang, Zhang, Zhuang (2013) in their paper Coassociative Lie Algebras. We use this notion to extend universal enveloping algebras of Lie algebras with regards to their Hopf structure, and see if these connected Hopf algebras are enveloping algebras. Furthermore, we apply these results to compare global dimension of connected Hopf algebras and the dimension of their corresponding Lie algebras of primitive elements. Title has been renamed to "GK-Dimension of some Connected Hopf Algebras".
For a given Hopf algebra $A$ we classify all Hopf algebras $E$ that are coalgebra split extensions of $A$ by $H_4$, where $H_4$ is the Sweedler's 4-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras $A # H_4$ by computing explicitly two classifying objects: the cohomological 'group' ${\mathcal H}^{2} (H_4, A)$ and $\textsc{C}\textsc{r}\textsc{p} (H_4, A) :=$ the set of types of isomorphisms of all crossed products $A # H_4$. All crossed products $A #H_4$ are described by generators and relations and classified: they are parameterized by the set ${\mathcal…
Let $\mathfrak{n}$ be a locally nilpotent infinite-dimensional Lie algebra over $\mathbb{C}$. Let $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$ be its universal enveloping algebra and its symmetric algebra respectively. Consider the Jacobson topology on the primitive spectrum of $\mathrm{U}(\mathfrak{n})$ and the Poisson topology on the primitive Poisson spectrum of $\mathrm{S}(\mathfrak{n})$. We provide a homeomorphism between the corresponding topological spaces (on the level of points, it gives a bijection between the primitive ideals of $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S…
