Let g be a complex simple Lie algebra of type B_2 and q be a non-zero complex number which is not a root of unity. In the classical case, a theorem of Dixmier asserts that the simple factor algebras of the positive part U^+(g) of the enveloping algebra of g, whose Gelfand-Kirillov dimension is equal to 2, are isomorphic to the first Weyl algebra. In order to obtain some new quantized analogues of the first Weyl algebra, we explicitly describe the prime and primitive spectra of the positive part U_q^+(g) of the quantized enveloping algebra of g and then we study the simple factor algebras of U_q^+(g) whose Gelfand-Kirillov dimension is equal to 2. In particular, we show that the centers of such simple factor algebras are reduced to the ground field C and we compute their group of invertible elements. These computations suggest that we distinguish between two families of such simple factor algebras of U_q^+(g). The first family consists of those simple factor algebras whose group of units is non-trivial; algebras in this first family are the so-called Weyl-Hayashi algebras and turn out to be examples of generalized Weyl algebras over a Laurent polynomial ring in one indeterminate. In contrast, the second family consists of those simple factor algebras whose group of units is trivial; algebras in this second class can not be presented as generalized Weyl algebras over a (Laurent) polynomial ring in one indeterminate. Finally we use these results to describe the structure of the automorphism group of U_q^+(g). More precisely, we prove that this group is isomorphic to the torus (C^*)^2, as conjectured by Andruskiewitsch and Dumas. As a corollary, we obtain that the action of this group on the set of all (left) primitive ideals of U_q^+(g) has exactly 8 orbits that we describe explicitly.
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 establish a Primitive Element Theorem for fields equipped with several commuting operators such that each of the operators is either a derivation or an automorphism. More precisely, we show that for every extension $F \subset E$ of such fields of zero characteristic such that $\bullet$ $E$ is generated over $F$ by finitely many elements using the field operations and the operators, $\bullet$ every element of $E$ satisfies a nontrivial equation with coefficient in $F$ involving the field operations and the operators, $\bullet$ the action of the operators on $E$ is irredundant there exists an…
The main goal of the paper is to establish the existence of tensor product decompositions for those prime ideals P of the generic algebra A of quantum n by n matrices which are invariant under winding automorphisms of A. More specifically, every such P is the kernel of a map from A to (A^+/P^+) tensor (A^-/P^-) obtained by composing comultiplication, localization, and quotient maps, where A^+ and A^- are special localized quotients of A while P^+ and P^- are prime ideals invariant under winding automorphisms. Further, the algebras A^+ and A^-, which vary with P, can be chosen so that the corre…
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.
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.
We consider rings whose one-sided ideals are close to automorphism-invariant modules. We study rings in which every (finitely generated) right ideal is automorphism invariant and rings in which every right ideal is a finite direct sum of automorphism invariant ideals. Connections between these classes of rings, $q$-ring and $Σ$-$q$-rings are also considered
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.
Motivated by Weyl algebra analogues of the Jacobian conjecture and the Tame Generators problem, we prove quantum versions of these problems for a family of analogues to the Weyl algebras. In particular, our results cover the Weyl-Hayashi algebras and tensor powers of a quantization of the first Weyl algebra which arises as a primitive factor algebra of $U_q^+(\mathfrak{so}_5)$.
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…
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.
Let ${\mathfrak g}$ be a finite dimensional complex simple Lie algebra, and let $r,s\in \mathbb{C}^{\ast}$ be transcendental over $\mathbb{Q}$ such that $r^{m}s^{n}=1$ implies $m=n=0$. We will obtain some basic properties of the two-parameter quantized enveloping algebra $U_{r,s}^{+}(\mathfrak g)$. In particular, we will verify that the algebra $U_{r,s}^{+}(\mathfrak g)$ satisfies many nice properties such as having normal separation, catenarity and Dixmier-Moeglin equivalence. We shall study a concrete example, the algebra $U_{r,s}^{+}(B_{2})$ in detail. We will first determine the normal ele…
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…
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.
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 $H$ 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 …
We describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra. For primitive ideals we obtain an analogue of Duflo's Theorem, which says that every primitive ideal is the annihilator of a simple highest weight module. This depends on an extension of Lusztig's tensor product theorem.
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.
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.
A class of axial decomposition algebras with Miyamoto group generated by two Miyamoto automorphisms and three eigenvalues $0,1$ and $η$ is introduced and classified in the case with $η\notin\{0,1,\frac{1}{2}\}$. This class includes specializations of 2-generated axial algebras of Majorana type $(ξ,η)$ to the case with $ξ=η$.
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.
Let $A$, $B$ be algebras and $a\in A$, $b\in B$ a fixed pair of elements. We say that a map $φ:A\to B$ preserves products equal to $a$ and $b$ if for all $a_1,a_2\in A$ the equality $a_1a_2=a$ implies $φ(a_1)φ(a_2)=b$. In this paper we study bijective linear maps $φ:I(X,F)\to I(X,F)$ preserving products equal to primitive idempotents of $I(X,F)$, where $I(X,F)$ is the incidence algebra of a finite connected poset $X$ over a field $F$. We fully characterize the situation, when such a map $φ$ exists, and whenever it does, $φ$ is either an automorphism of $I(X,F)$ or the negative of an automorphi…
In this paper, we study rings having the property that every right ideal is automorphism-invariant. Such rings are called right $a$-rings. It is shown that (1) a right $a$-ring is a direct sum of a square-full semisimple artinian ring and a right square-free ring, (2) a ring $R$ is semisimple artinian if and only if the matrix ring $\mathbb{M}_n(R)$ for some $n>1$ is a right $a$-ring, (3) every right $a$-ring is stably-finite, (4) a right $a$-ring is von Neumann regular if and only if it is semiprime, and (5) a prime right $a$-ring is simple artinian. We also describe the structure of an indec…
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…
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.
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.
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.
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.
The left-ideal relation graph on a ring $R$, denoted by $\overrightarrow{Γ_{l-i}}(R)$, is a directed graph whose vertex set is all the elements of $R$ and there is a directed edge from $x$ to a distinct $y$ if and only if the left ideal generated by $x$, written as $[x]$, is properly contained in the left ideal generated by $y$. In this paper, the automorphisms of $\overrightarrow{Γ_{l-i}}(R)$ are characterized, where $R$ is the ring of all $n \times n$ matrices over a finite field $F_q$. The undirected left relation graph, denoted by $Γ_{l-i}(M_n(F_q))$, is the simple graph whose vertices are…
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 …
Let R be a Lie nilpotent algebra of index k over a field K of characteristic zero. If G is an n-element subgroup of Aut(R) of the K-automorphisms, then we prove that R is right integral over Fix(G) of degree n^k. In the presence of a primitive n-th root of unity e in K, for a K-automorphism d in Aut(R) with d^n=id, we prove that the skew polynomial algebra R[w,d] is right integral of degree n^k over Fix(d)[w^n].
We introduce the notions of one-sided dirings, 3-irreducible left modules, 3-primitive left dirings, 3-semi-primitive left dirings, 3-primitive ideals and 3-radicals. The main results consists of two parts. The first part establishes two external characterizations of a 3-semi-primitive left diring. The second part characterizes the 3-radical of a left diring by using 3-primitive ideals.
Suppose that a Lie type algebra L over a field K admits a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H such that the fixed-point subalgebra of F is trivial and the fixed-point subalgebra of H is nilpotent of class c. If the ground field K contains a primitive n-th root of unity, then L is nilpotent and the nilpotency class of L is bounded in terms of |H| and c. The result extends the known theorem of Khukhro, Makarenko and Shumyatsky on Lie algebras with metacyclic Frobenius group of automorphisms.
Let $R$ be a commutative integral domain with unit, $f$ be a nonconstant monic polynomial in $R[t]$, and $I_f \subset R[t]$ be the ideal generated by $f$. In this paper we study the group of $R$-algebra automorphisms of the $R$-algebra without unit $I_f$. We show that, if $f$ has only one root (possibly with multiplicity), then $Aut (I_f) \cong R^\times$. We also show that, under certain mild hypothesis, if $f$ has at least two different roots in the algebraic closure of the quotient field of $R$, then $Aut(I_f)$ is a cyclic group and its order can be completely determined by analyzing the roo…
In this paper, prime as well as primitive Kumjian-Pask algebras $\mathrm{KP}_R(Λ)$ of a row-finite $k$-graph $Λ$ over a unital commutative ring $R$ are completely characterized in graph-theoretic and algebraic terms. By applying quotient $k$-graphs, these results describe prime and primitive graded basic ideals of Kumjian-Pask algebras. In particular, when $Λ$ is strongly aperiodic and $R$ is a field, all prime and primitive ideals of a Kumjian-Pask algebra $\mathrm{KP}_R(Λ)$ are determined.
Let $K$ be a field and $X$ a partially ordered set (poset). Let $FI(X,K)$ and $I(X,K)$ be the finitary incidence algebra and the incidence space of $X$ over $K$, respectively, and let $D(X,K)=FI(X,K)(+)I(X,K)$ be the idealization of the $FI(X,K)$-bimodule $I(X,K)$. In the first part of this paper, we show that $D(X,K)$ has an anti-automorphism (involution) if and only if $X$ has an anti-automorphism (involution). We also present a characterization of the anti-automorphisms and involutions on $D(X,K)$. In the second part, we obtain the classification of involutions on $D(X,K)$ to the case when …
Let $A_{1} := k [t, \partial ]$ be the first algebra over a field $k$ of characteristic zero. One can associate to each right ideal $I$ of $A_1$ its Stafford subgroup, which is a subgroup of $\Aut_k(A_1)$, the automorphism group of the ring $A_1$. In this article we show that each Stafford subgroup is equal to its normalizer. For that, we study when the Stafford subgroup of a right ideal of $A_1$ contains a given Stafford subgroup.
Let $R$ be a 2-torsion free $σ$-prime ring. It is shown here that if $U\not\subset Z(R)$ is a $σ$-Lie ideal of $R$ and $a, b$ in $R$ such that $aUb=σ(a)Ub=0,$ then either $a=0$ or $b=0.$ This result is then applied to study the relationship between the structure of $R$ and certain automorphisms on $R$. To end this paper, we describe additive maps $d: R \longrightarrow R$ such that $d(u^2) = 2ud(u)$ where $u\in U,$ a nonzero $σ$-square closed Lie ideal of $R.$
In this article, we provide an explicit description of a set of generators for any ideal of an ultragraph Leavitt path algebra. We provide several additional consequences of this description, including information about generating sets for graded ideals, the graded uniqueness and Cuntz-Krieger theorems, the semiprimeness, and the semiprimitivity of ultragraph Leavitt path algebras, a complete characterization of the prime and primitive ideals of an ultragraph Leavitt path algebra. We also show that every primitive ideal of an ultragraph Leavitt path algebra is exactly the annihilator of a Chen…
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 …
We give a brief survey of primitivity in ring theory and in particular look at characterizations of primitive ideals in the prime spectrum for various classes of rings.
Let $E$ be an arbitrary (countable) graph and let $R$ be a unital commutative ring. We analyze the ideal structure of the Leavitt path algebra $\lr$ introduced by Mark Tomforde. We first modify the definition of basic ideals and we then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of $\lr$. Then by applying these results we determine prime graded basic ideals and left (or right) primitive graded ideals of $\lr$. In particular, we show that when $E$ satisfies Condition (K) and $R$ is a field, the set …
A generalized pseudo effect algebra (GPEA) is a partially ordered partial algebraic structure with a smallest element 0, but not necessarily with a unit (i.e, a largest element). If a GPEA admits a so-called unitizing automorphism, then it can be embedded as an order ideal in its so-called unitization, which does have a unit. We study unitizations of GPEAs with respect to a unitizing automorphism, paying special attention to the behavior of congruences and ideals in this setting.
Let L be the free m-generated metabelian nilpotent of class c Lie algebra over a field of characteristic 0. An automorphism f of L is called normal if f(I)=I for every ideal I of the algebra L. Such automorphisms form a normal subgroup N(L) of Aut(L) containing the group of inner automorphisms. We describe the group of normal automorphisms of L and the quotient group of Aut(L) modulo N(L).
The group $\rG_n$ of automorphisms of the algebra $\mI_n:=K< x_1, >..., x_n, \frac{\der}{\der x_1}, ... ,\frac{\der}{\der x_n}, \int_1, >..., \int_n>$ of polynomial integro-differential operators is found: $$ \rG_n=S_n\ltimes \mT^n\ltimes \Inn (\mI_n) \supseteq S_n\ltimes \mT^n \ltimes \underbrace{\GL_\infty (K)\ltimes... \ltimes \GL_\infty (K)}_{2^n-1 {\rm times}}, $$ $$ \rG_1\simeq \mT^1 \ltimes \GL_\infty (K),$$ where $S_n$ is the symmetric group, $\mT^n$ is the $n$-dimensional torus, $\Inn (\mI_n)$ is the group of inner automorphisms of $\mI_n$ (which is huge). It is proved that each autom…
Bartholdi and Smoktunowicz constructed finitely generated monomial algebras with prescribed sufficiently fast growth types. We show that their construction need not result in a prime algebra, but it can be modified to provide prime algebras without further limitations on the growth type. Moreover, using a construction of an inverse system of monomial ideals which arise from this construction we are able to further construct finitely generated primitive algebras with many regular enough growth types. Then, inspired by an example by Zelmanov, we show how our prime algebras can be constructed suc…
A classification up to automorphism of the inner ideals of the real finite-dimensional simple Lie algebras is given, jointly with precise descriptions in the case of the exceptional Lie algebras.
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…
