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In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) , there is a correspondence between Verma modules and certain line modules that associates a pair ( h , ϕ ) (\mathfrak {h},\,\phi ) , where h \mathfrak {h} is a 2 2 -dimensional Lie subalgebra of s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) and ϕ ∈ h ∗ \phi \in \mathfrak {h}^* satisfies ϕ ( [ h , h ] ) = 0 \phi ([\mathfrak {h}, \, \mathfrak {h}]) = 0 , to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra s l ( 1 | 1 ) \mathfrak {sl}(1|1) and for a color Lie algebra associated to the Lie algebra s l 2 \mathfrak {sl}_2 .

Recently McBreen and Webster constructed a tilting bundle on a smooth hypertoric variety (using reduction to finite characteristic) and showed that its endomorphism ring is Koszul. In this short note we present alternative proofs for these results. We simply observe that the tilting bundle constructed by Halpern-Leistner and Sam on a generic open GIT substack of the ambient linear space restricts to a tilting bundle on the hypertoric variety. The fact that the hypertoric variety is defined by a quadratic regular sequence then also yields an easy proof of Koszulity., Comment: Various fixes

We show that all toric noncommutative crepant resolutions (NCCRs) of affine GIT quotients of "weakly symmetric" unimodular torus representations are derived equivalent. This yields evidence for a non-commutative extension of a well known conjecture by Bondal and Orlov stating that all crepant resolutions of a Gorenstein singularity are derived equivalent. We prove our result by showing that all toric NCCRs of the affine GIT quotient are derived equivalent to a fixed Deligne-Mumford GIT quotient stack associated to a generic character of the torus. This extends a result by Halpern-Leistner and Sam which showed that such GIT quotient stacks are a geometric incarnation of a family of specific toric NCCRs constructed earlier by the authors.

Positivstellensatz is a fundamental result in real algebraic geometry providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this article Positivstellensatze for trace polynomials positive on semialgebraic sets of $n\times n$ matrices are provided. A Krivine-Stengle-type Positivstellensatz is proved characterizing trace polynomials nonnegative on a general semialgebraic set $K$ using weighted sums of hermitian squares with denominators. The weights in these certificates are obtained from generators of $K$ and traces of hermitian squares. For compact semialgebraic sets $K$ Schmudgen- and Putinar-type Positivstellensatze are obtained: every trace polynomial positive on $K$ has a sum of hermitian squares decomposition with weights and without denominators. The methods employed are inspired by invariant theory, classical real algebraic geometry and functional analysis. Procesi and Schacher in 1976 developed a theory of orderings and positivity on central simple algebras with involution and posed a Hilbert's 17th problem for a universal central simple algebra of degree $n$: is every totally positive element a sum of hermitian squares? They gave an affirmative answer for $n=2$. In this paper a negative answer for $n=3$ is presented. Consequently, including traces of hermitian squares as weights in the Positivstellensatze is indispensable.

Let $R$ be the homogeneous coordinate ring of the Grassmannian $\mathbb{G}=\operatorname{Gr}(2,n)$ defined over an algebraically closed field of characteristic $p>0$. In this paper we give a completely characteristic free description of the decomposition of $R$, considered as a graded $R^p$-module, into indecomposables ("Frobenius summands"). As a corollary we obtain a similar decomposition for the Frobenius pushforward of the structure sheaf of $\mathbb{G}$ and we obtain in particular that this pushforward is almost never a tilting bundle. On the other hand we show that $R$ provides a "noncommutative resolution" for $R^p$ when $p\ge n-2$, generalizing a result known to be true for toric varieties. In both the invariant theory and the geometric setting we observe that if the characteristic is not too small the Frobenius summands do not depend on the characteristic in a suitable sense. In the geometric setting this is an explicit version of a general result by Bezrukavnikov and Mirkovi�� on Frobenius decompositions for partial flag varieities. We are hopeful that it is an instance of a more general "$p$-uniformity" principle., We have now been able to prove our conjecture that the Frobenius pushforward yields an NCR

In this paper we generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. We show in particular that quotient singularities for reductive groups always have non-commutative resolutions in an appropriate sense. Moreover we exhibit a large class of such singularities which have (twisted) non-commutative crepant resolutions. We discuss a number of examples, both new and old, that can be treated using our methods. Notably we prove that twisted non-commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew-symmetric matrices. In contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity.

Let $\Bbbk$ be an algebraically closed field of characteristic zero, and let $\Gamma$ be an additive subgroup of $\Bbbk$. Results of Kaplansky-Santharoubane and Su classify intermediate series representations of the generalised Witt algebra $W_\Gamma$ in terms of three families, one parameterised by ${\mathbb A}^2$ and two by ${\mathbb P}^1$. In this note, we use the first family to construct a homomorphism $\Phi$ from the enveloping algebra $U(W_\Gamma)$ to a skew extension of ${\Bbbk}[a,b]$. We show that the image of $\Phi$ is contained in a (double) idealizer subring of this skew extension and that the representation theory of idealizers explains the three families. We further show that the image of $U(W_\Gamma)$ under $\Phi$ is not left or right noetherian, giving a new proof that $U(W_\Gamma)$ is not noetherian. We construct $\Phi$ as an application of a general technique to create ring homomorphisms from shift-invariant families of modules. Let $G$ be an arbitrary group and let $A$ be a $G$-graded ring. A graded $A$-module $M$ is an intermediate series module if $M_g$ is one-dimensional for all $g \in G$. Given a shift-invariant family of intermediate series $A$-modules parametrised by a scheme $X$, we construct a homomorphism $\Phi$ from $A$ to a skew-extension of ${\Bbbk}[X]$. The kernel of $\Phi$ consists of those elements which annihilate all modules in $X$., Comment: 9 pages; to appear in J. Algebra

Using the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein affine toric variety Spec R admits a toric non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of Spec R using standard toric methods. Our proof does not use dimer models., Fix an undefined notation

We give a criterion for the existence of non-commutative crepant resolutions (NCCR's) for certain toric singularities. In particular we recover Broomhead's result that a 3-dimensional toric Gorenstein singularity has a NCCR. Our result also yields the existence of a NCCR for a 4-dimensional toric Gorenstein singularity which is known to have no toric NCCR., 10 pages

In this article we introduce powerful tools and techniques from invariant theory to free analysis. This enables us to study free maps with involution. These maps are free noncommutative analogs of real analytic functions of several variables. With examples we demonstrate that they do not exhibit strong rigidity properties of their involution-free free counterparts. We present a characterization of polynomial free maps via properties of their finite-dimensional slices. This is used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are given by series of generalized polynomials for which we obtain an invariant-theoretic characterization. Finally, we give an inverse and implicit function theorem for free maps with involution., 26 pages

Let Y be the variety of (skew) symmetric nxn-matrices of rank less than or equal to r. In paper we construct a full faithful embedding between the derived category of a non-commutative resolution of Y, constructed earlier by the authors, and the derived category of the classical Springer resolution of Y., Comment: 16 pages Many typos corrected. Acknowledgement expanded

The main result of this note is a tracial Nullstellensatz for free noncommutative polynomials evaluated at tuples of matrices of all sizes: Suppose f_1,...,f_r,f are free polynomials, and tr(f) vanishes whenever all tr(f_j) vanish. Then either 1 or f is a linear combination of the f_j modulo sums of commutators., Comment: 8 pages

If G is a reductive group which acts on a linearized smooth scheme $X$ then we show that under suitable standard conditions the derived category of coherent sheaves of the corresponding GIT quotient stack $X^{ss}/G$ has a semi-orthogonal decomposition consisting of derived categories of coherent sheaves of rings on the categorical quotient $X^{ss}/\!/G$ which are locally of finite global dimension. One of the components of the decomposition is a certain non-commutative resolution of $X^{ss}/\!/G$ constructed earlier by the authors. As a concrete example we obtain in the case of odd Pfaffians a semi-orthogonal decomposition of the corresponding quotient stack in which all the parts are certain specific non-commutative crepant resolutions of Pfaffians of lower or equal rank which had also been constructed earlier by the authors. In particular this semi-orthogonal decomposition cannot be refined further since its parts are Calabi-Yau. The results in this paper also complement a result by Halpern-Leistner (and similar results by Ballard-Favero-Katzarkov and Donovan-Segal) that asserts the existence of a semi-orthogonal decomposition of the derived category of $X/G$ in which one of the components is the derived category of $X^{ss}/G$., We now give in certain cases a semi-orthogonal decomposition consisting of Calabi-Yau parts. So it cannot be refined further

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This paper considers Banach algebras with properties 𝔸 or 𝔹, introduced recently by Alaminos et al. The class of Banach algebras satisfying either of these two properties is quite large; in particular, it includes C*-algebras and group algebras on locally compact groups. Our first main result states that a continuous orthogonally additive n-homogeneous polynomial on a commutative Banach algebra with property 𝔸 and having a bounded approximate identity is of a standard form. The other main results describe Banach algebras A with property 𝔹 and having a bounded approximate identity that admit non-zero continuous symmetric orthosymmetric n-linear maps from An into ℂ.

We describe subalgebras of the Lie algebra $\mf{gl}(n^2)$ that contain all inner derivations of $A=M_n(F)$ (where $n\ge 5$ and $F$ is an algebraically closed field of characteristic 0). In a more general context where $A$ is a prime algebra satisfying certain technical restrictions, we establish a density theorem for the associative algebra generated by all inner derivations of $A$., Comment: 11 pages, accepted for publication in Linear Algebra Appl

The paper begins with short proofs of classical theorems by Frobenius and (resp.) Zorn on associative and (resp.) alternative real division algebras. These theorems characterize the first three (resp. four) Cayley-Dickson algebras. Then we introduce and study the class of real unital nonassociative algebras in which the subalgebra generated by any nonscalar element is isomorphic to C. We call them locally complex algebras. In particular, we describe all such algebras that have dimension at most 4. Our main motivation, however, for introducing locally complex algebras is that this concept makes it possible for us to extend Frobenius' and Zorn's theorems in a way that it also involves the fifth Cayley-Dickson algebra, the sedenions., Comment: Some changes suggested by the referee, accepted for publication in Journal of Algebra

The main result of this short note is a generic version of Paz's conjecture on the lengths of generating sets in matrix algebras. Consider a generic g-tuple A=(A_1,..., A_g) of nxn matrices over a field. We show that whenever $g^{2d}\geq n^2$, the set of all words of degree 2d in A spans the full nxn matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the construction of a special kind of directed multigraphs with few edge-disjoint walks., Comment: 8 pages

Complete solutions of functional identities $\sum_{k\in K}F_k(\bar{x}_m^k)x_k = \sum_{l\in L}x_lG_l(\bar{x}_m^l)$ on the matrix algebra $M_n(\mathbb{F})$ are given. The nonstandard parts of these solutions turn out to follow from the Cayley-Hamilton identity., Comment: 20 pages, comments welcome, version 2- applications have been added

Let A be a centrally closed prime algebra over a characteristic 0 field k, and let q : A → A be the trace of a d-linear map (i.e., q ( x ) = M ( x , … , x ) where M : A d → A is a d-linear map). If [ q ( x ) , x ] = 0 for every x ∈ A , then q is of the form q ( x ) = ∑ i = 0 d μ i ( x ) x i where each μ i is the trace of a ( d − i ) -linear map from A into k. For infinite dimensional algebras and algebras of dimension > d 2 this was proved by Lee, Lin, Wang, and Wong in 1997. In this paper we cover the remaining case where the dimension is ⩽ d 2 . Using this result we are able to handle general functional identities in one variable on A; more specifically, we describe the traces of d-linear maps q i : A → A that satisfy ∑ i = 0 m x i q i ( x ) x m − i ∈ k for every x ∈ A .

We consider a Banach algebra $A$ with the property that, roughly speaking, sufficiently many irreducible representations of $A$ on nontrivial Banach spaces do not vanish on all square zero elements. The class of Banach algebras with this property turns out to be quite large -- it includes $C^*$-algebras, group algebras on arbitrary locally compact groups, commutative algebras, $L(X)$ for any Banach space $X$, and various other examples. Our main result states that every derivation of $A$ that preserves the set of quasinilpotent elements has its range in the radical of $A$., 6 pages

Let $F$ be an algebraically closed field of characteristic zero. We consider the question which subsets of $M_n(F)$ can be images of noncommutative polynomials. We prove that a noncommutative polynomial $f$ has only finitely many similarity orbits modulo nonzero scalar multiplication in its image if and only if $f$ is power-central. The union of the zero matrix and a standard open set closed under conjugation by $GL_n(F)$ and nonzero scalar multiplication is shown to be the image of a noncommutative polynomial. We investigate the density of the images with respect to the Zariski topology. We also answer Lvov's conjecture for multilinear Lie polynomials of degree at most 4 affirmatively., Comment: 13 pages, accepted for publication in J. Algebra

We consider certain functional identities on the matrix algebra $M_n$ that are defined similarly as the trace identities, except that the "coefficients" are arbitrary polynomials, not necessarily those expressible by the traces. The main issue is the question of whether such an identity is a consequence of the Cayley-Hamilton identity. We show that the answer is affirmative in several special cases, and, moreover, for every such an identity $P$ and every central polynomial $c$ with zero constant term there exists $m\in\mathbb{N}$ such that the affirmative answer holds for $c^mP$. In general, however, the answer is negative. We prove that there exist antisymmetric identities that do not follow from the Cayley-Hamilton identity, and give a complete description of a certain family of such identities., Version 2: 24 pages. This paper is a replacement of the paper "Quasi-identities and the Cayley-Hamilton quasi-polynomial" by the first and the third author. The changes are substantial. Version 3: 27 pages. The title has been changed slightly, the exposition improved, references added, and some typos removed

Let $S$ and $T$ be continuous linear operators on an ultraprime Banach algebra $A$. We show that if $S$, $T$, and $ST$ are close to satisfy the derivation identity on $A$, then either $S$ or $T$ approaches to zero. If $T$ is close to satisfy the derivation identity and $[T(a),a]$ is near the centre of $A$ for each $a \in A$, then either $T$ approaches to zero or $A$ is nearly commutative. Further, we give quantitative estimates of these phenomena.

Let $\A$ be an algebra and let $f(x_1,...,x_d)$ be a multilinear polynomial in noncommuting indeterminates $x_i$. We consider the problem of describing linear maps $\phi:\A\to \A$ that preserve zeros of $f$. Under certain technical restrictions we solve the problem for general polynomials $f$ in the case where $\A=M_n(F)$. We also consider quite general algebras $\A$, but only for specific polynomials $f$., Comment: 11 pages, accepted for publication in Linear Algebra Appl

We consider the relationship between derivations $d$ and $g$ of a Banach algebra $B$ that satisfy $\s(g(x)) \subseteq \s(d(x))$ for every $x\in B$, where $\s(\, . \,)$ stands for the spectrum. It turns out that in some basic situations, say if $B=B(X)$, the only possibilities are that $g=d$, $g=0$, and, if $d$ is an inner derivation implemented by an algebraic element of degree 2, also $g=-d$. The conclusions in more complex classes of algebras are not so simple, but are of a similar spirit. A rather definitive result is obtained for von Neumann algebras. In general $C^*$-algebras we have to make some adjustments, in particular we restrict our attention to inner derivations implemented by selfadjoint elements. We also consider a related condition $\|[b,x]\|\leq M\|[a,x]\|$ for all selfadjoint elements $x$ from a $C^*$-algebra $B$, where $a,b\in B$ and $a$ is normal., 12 pages

Let $A$ be a Banach algebra. By $\sigma(x)$ and $r(x)$ we denote the spectrum and the spectral radius of $x\in A$, respectively. We consider the relationship between elements $a,b\in A$ that satisfy one of the following two conditions: (1) $\sigma(ax) = \sigma(bx)$ for all $x\in A$, (2) $r(ax) \le r(bx)$ for all $x\in A$. In particular we show that (1) implies $a=b$ if $A$ is a $C^*$-algebra, and (2) implies $a\in \mathbb C b$ if $A$ is a prime $C^*$-algebra. As an application of the results concerning the conditions (1) and (2) we obtain some spectral characterizations of multiplicative maps., Comment: 10 pages, accepted for publication in J. Math. Anal. Appl

Let A be an ultraprime Banach algebra.We prove that each approximately commuting continuous linear (or quadratic) map on A is near an actual commuting continuous linear (resp. quadratic) map on A. Furthermore, we use this analysis to study how close are approximate Lie isomorphisms and approximate Lie derivations to actual Lie isomorphisms and Lie derivations, respectively.