Let $F$ be an algebraically closed field of characteristic zero and let $G$ be a finite group. Consider $G$-graded simple algebras $A$ which are finite dimensional and $e$-central over $F$, i.e. $Z(A)_{e} := Z(A)\cap A_{e} = F$. For any such algebra we construct a \textit{generic} $G$-graded algebra $\mathcal{U}$ which is \textit{Azumaya} in the following sense. $(1)$ \textit{$($Correspondence of ideals$)$}: There is one to one correspondence between the $G$-graded ideals of $\mathcal{U}$ and the ideals of the ring $R$, the $e$-center of $\mathcal{U}$. $(2)$ \textit{Artin-Procesi condition}: $\mathcal{U}$ satisfies the $G$-graded identities of $A$ and no nonzero $G$-graded homomorphic image of $\mathcal{U}$ satisfies properly more identities. $(3)$ \textit{Generic}: If $B$ is a $G$-graded algebra over a field then it is a specialization of $\mathcal{U}$ along an ideal $\mathfrak{a} \in spec(Z(\mathcal{U})_{e})$ if and only if it is a $G$-graded form of $A$ over its $e$-center.
We apply this to characterize finite dimensional $G$-graded simple algebras over $F$ that admit a $G$-graded division algebra form over their $e$-center.
The main purpose of this paper is to investigate epsilon-strongly graded rings that are partial crossed products. Let $G$ be a group, $A=\oplus_{g\in G}\,A_g$ an epsilon-strongly graded ring and ${\bf pic}{R}$ the Picard semigroup of $R:=A_1$. We prove that the isomorphism class $[A_g]$ is an element of ${\bf pic}{R}$, for all $g\in G$. Thus, the association $g\mapsto [A_g]$ determines a partial representation of $G$ on ${\bf pic}{R}$ which induces a partial action $γ$ of $G$ on the center $Z(R)$ of $R$. Sufficient conditions for $A$ to be an Azumaya $R^γ$-algebra are presented in the case tha…
We define a nonassociative generalization of cyclic Azumaya algebras employing skew polynomial rings $D[t;σ]$, where $D$ is an Azumaya algebra of constant rank with center $C$ and $σ$ an automorphism of $D$, such that $σ|_{C}$ has finite order. The automorphisms of these algebras are canonically induced by ring automorphisms of the skew polynomial ring $D[t;σ]$ used in their construction. We achieve a description of their inner automorphisms. Results on the automorphisms of classical Azumaya algebras and central simple algebras of this type are obtained as special cases.
We provide easily-verified necessary and sufficient conditions for a skew group ring, or more generally, a crossed product ring, to be an Azumaya algebra. We use our results to show that (suitable localisations of) skew group rings associated to the quantum Kleinian singularities introduced by Chan, Kirkman, Walton, and Zhang in [CKWZ16a] are Azumaya, and use this to show that these algebras are maximal orders. We also give a new proof of Auslander's Theorem for quantum Kleinian singularities.
We study moduli spaces of maximal orders in a ramified division algebra over the function field of a smooth projective surface. As in the case of moduli of stable commutative surfaces, we show that there is a Kollár-type condition giving a better moduli problem with the same geometric points: the stack of blt Azumaya algebras. One virtue of this refined moduli problem is that it admits a compactification with a virtual fundamental class.
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.
A paper of U. First & Z. Reichstein proves that if $R$ is a commutative ring of dimension $d$, then any Azumaya algebra $A$ over $R$ can be generated as an algebra by $d+2$ elements, by constructing such a generating set, but they do not prove that this number of generators is required, or even that for an arbitrarily large $r$ that there exists an Azumaya algebra requiring $r$ generators. In this paper, for any given fixed $n\ge 2$, we produce examples of a base ring $R$ of dimension $d$ and an Azumaya algebra of degree $n$ over $R$ that requires $r(d,n) = \lfloor \frac{d}{2n-2} \rfloor + 2$ …
Let $ψ: A \to A'$ be a cyclic contraction of dimer algebras, with $A$ non-cancellative and $A'$ cancellative. $A'$ is then prime, noetherian, and a finitely generated module over its center. In contrast, $A$ is often not prime, nonnoetherian, and an infinitely generated module over its center. We present certain Morita equivalences that relate the representation theory of $A$ with that of $A'$. We then characterize the Azumaya locus of $A$ in terms of the Azumaya locus of $A'$, and give an explicit classification of the simple $A$-modules parameterized by the Azumaya locus. Furthermore, we sho…
Generalizing a theorem of Albert, Saltman showed that an Azumaya algebra $A$ over a ring represents a $2$-torsion class in the Brauer group if and only if there is an algebra $A'$ in the Brauer class of $A$ admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose $A'$ such that $\mathrm{deg}\, A'=2\mathrm{deg}\, A$. We show that $2\mathrm{deg}\, A$ is the lowest degree one can expect in general. Specifically, we construct an Azumaya algebra $A$ of degree $4$ and period $2$ such that the degree of any algebra $A'$ in the Brauer class of $A$ admitt…
We describe the derived Picard group of an Azumaya algebra A on an affine scheme X in terms of global sections of the constant sheaf of integers on X, the Picard group of X, and the stabilizer of the Brauer class of A under the action of Aut(X). In particular, we find that the derived Picard group of an Azumaya algebra is generally not isomorphic to that of the underlying scheme. In the case of the trivial Azumaya algebra, our result refines Yekutieli's description of the derived Picard group of a commutative algebra. We also get, as a corollary, an alternate proof of a result of Antieau which…
In arXiv:1606.07885, many different Grothendieck topologies were introduced on the category of Azumaya algebras. Here we give a classification in terms of sets of supernatural numbers. Then we discuss the associated categories of sheaves and their topos-theoretic points, which are related to UHF-algebras. The sheaf toposes that correspond to a single supernatural number have an alternative description, involving actions of the associated projective general linear group.
We develop Azumaya geometry, which is an extension of classical affine geometry to the world of Azumaya algebras, and package the information contained in all quotient stacks $[\mathrm{rep}_n R\,/\,\mathrm{PGL}_n]$ into a presheaf $\mathrm{Rep}_R$ on it. We show that the classical étale and Zariski topologies extend to Grothendieck topologies on Azumaya geometry in uncountably many ways, and prove that $\mathrm{Rep}_R$ is a sheaf for all of them. The restriction to a specific Azumaya algebra $A$ with center $C$ gives us a sheaf in the étale topology which is represented by an affine $C$-scheme…
We prove that, under mild assumptions, for all positive integers $\ell$, the zero set of the discriminant ideal $D_{\ell}(R/Z(R); tr)$ of a prime polynomial identity (PI) algebra $R$ coincides with the zero set of the modified discriminant ideal $MD_{\ell}(R/Z(R); tr)$ of $R$. Furthermore, we prove that, when $\ell$ is the square of the PI-degree of $R$, this zero set is precisely the complement of the Azumaya locus of $R$. This description is used to classify the Azumaya loci of the mutiparameter quantized Weyl algebras at roots of unity. As another application, we prove that the zero set of …
We consider the general circumstance of an Azumaya algebra $A$ of degree $n$ over a locally ringed topos $(\mathbf{X}, {\mathcal{O}}_{\mathbf{ X}})$ where the latter carries a (possibly trivial) involution, denoted $λ$. This generalizes the usual notion of involutions of Azumaya algebras over schemes with involution, which in turn generalizes the notion of involutions of central simple algebras. We provide a criterion to determine whether two Azumaya algebras with involutions extending $λ$ are locally isomorphic, describe the equivalence classes obtained by this relation, and settle the questi…
The differential Brauer monoid of a differential commutative ring R s defined. Its elements are the isomorphism classes of differential Azumaya R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them are differentially isomorphic. The Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the Brauer monoids of R and its ring of constants and the submonoid whose underlying Azumaya algebras are matrix rings.
The Artin-Schreier polynomial $Z^p - Z - a$ is very well known. Polynomials of this type describe all degree $p$ (cyclic) Galois extensions over any commutative ring of characteristic $p$. Equally attractive is the associated Galois action. If $θ$ is a root then $σ(θ) = θ+ 1$ generates the Galois group. Less well known, but equally general, is the so called "differential crossed product" Azumaya algebra generated by $x,y$ subject to the relation $xy - yx = 1$. In characteristic $p$ these algebras are always Azumaya and algebras of this sort generate the $p$ torsion subgroup of the Brauer group…
We study the representation theoretic results of the binary cubic generic Clifford algebra $\mathcal C$, which is an Artin-Schelter regular algebra of global dimension five. In particular, we show that $\mathcal C$ is a PI algebra of PI degree three and compute its point variety and discriminant ideals. As a consequence, we give a necessary and sufficient condition on a binary cubic form $f$ for the associated Clifford algebra $\mathcal C_f$ to be an Azumaya algebra.
Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning identity elements, inverses, existence of limits and colimits and adjointness of certain functors. In the particular instance of noncommutatively graded Lie algebras, we establish the existence of universal graded enveloping algebras and we show a graded version of the Poincaré-Birkhoff-Witt theorem.
In this work, we extend the definition of the graded prime ideals from those in commutative graded rings to the ideals over graded Lie algebras. We prove some facts about graded prime Lie ideals in arbitrary Lie algebras that are similar to those about graded prime ideals over a commutative or non-commutative ring.In addition, the ideas of graded semiprime Lie ideals and graded total prime Lie ideals will be introduced.
For an ample groupoid $\mathcal{G}$ and a unit $x$ of $\mathcal{G}$, Steinberg constructed the induction and restriction functors between the category of modules over the Steinberg algebra $A_R(\mathcal{G})$ and the category of modules over the isotropy group algebra $R\mathcal{G}_x$. In this paper, we prove a graded version of these functors and related results for the graded Steinberg algebra of a graded ample groupoid. As an application, the spectral simple and graded simple modules over the Leavitt path algebra $L_K(E)$ are classified. In particular, we show that many of previously known s…
Let A and B be finite dimensional simple real algebras with division gradings by an abelian group G. In this paper we give necessary and sufficient conditions for the coincidence of the graded identities of A and B. We also prove that every finite dimensional simple real algebra with a G-grading satisfies the same graded identities as a matrix algebra over an algebra D with a division grading that is either a regular grading or a non-regular Pauli grading. Moreover we determine when the graded identities of two such algebras coincide. For graded simple algebras over an algebraically closed fie…
We extend the loop algebra construction for algebras graded by abelian groups to study graded-simple algebras over the field of real numbers (or any real closed field). As an application, we classify up to isomorphism the graded-simple alternative (nonassociative) algebras and graded-simple finite-dimensional Jordan algebras of degree 2. We also classify the graded-division alternative (nonassociative) algebras up to equivalence.
We describe the graded isomorphisms of rings of endomorphisms of graded flags over graded division algebras. As a consequence describe the isomorphism classes of upper block triangular matrix algebras (over an algebraically closed field of characteristic zero) graded by a finite abelian group.
Let $\k$ be a field, and let $A$ and $B$ be connected $\N$-graded $\k$-algebras. The algebra $A$ is said to be a graded right-free extension of $B$ provided there is a surjective graded algebra morphism $π: A \to B$ such that $\kerπ$ is free as a right $A$-module. Suppose that $B$ is graded left coherent, and that $A$ is a graded right-free extension of $B$. We characterize when $A$ is also graded left coherent. We apply our criterion to prove graded coherence of certain non-Noetherian graded twisted tensor products.
We show that if $T$ is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative $T$-graded algebra over a field of characteristic $0$ such that the codimensions of its graded polynomial identities have a non-integer exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic $0$ with a non-integer graded PI-exponent, which is strictly less than the dimension of the algebra. However, if $T$ is a left or right zero band and the $T$-grad…
In this paper, the so-called differential graded (DG for short) Poisson Hopf algebra is introduced, which can be considered as a natural extension of Poisson Hopf algebras in the differential graded setting. The structures on the universal enveloping algebras of differential graded Poisson Hopf algebras are discussed.
We show that every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra. It is known that a graded ideal $I$ of a Leavitt path algebra is isomorphic to the Leavitt path algebra of a graph, known as the generalized hedgehog graph, which is defined based on certain sets of vertices uniquely determined by $I$. However, this isomorphism may not be graded. We show that replacing the short "spines" of the generalized hedgehog graph with possibly fewer, but then necessarily longer spines, we obtain a graph (which we call the porcupine graph) such that its Leavitt path…
We study graded symmetric algebras, which are the symmetric monoids in the monoidal category of vector spaces graded by a group. We show that a finite dimensional graded semisimple algebra is graded symmetric. The center of a symmetric algebra is not necessarily symmetric, but we prove that the center of a finite dimensional graded division algebra is symmetric, provided that the order of the grading group is not divisible by the characteristic of the base field.
We compute point schemes of some regular algebras using (Wolfram) Mathematica. These algebras are Ore extensions of regular graded skew Clifford algebras of global dimension 3.
