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This is a general study of twisted Calabi-Yau algebras that are -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi-Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin-Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi-Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi-Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi-Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

The composition operators preserving total non-negativity and totalpositivity for various classes of kernels are classified, following threethemes. Letting a function act by post composition on kernels with arbitrarydomains, it is shown that such a composition operator maps the set of totallynon-negative kernels to itself if and only if the function is constant orlinear, or just linear if it preserves total positivity. Symmetric kernels arealso discussed, with a similar outcome. These classification results are abyproduct of two matrix-completion results and the second theme: an extensionof A.M. Whitney's density theorem from finite domains to subsets of the realline. This extension is derived via a discrete convolution with modulatedGaussian kernels. The third theme consists of analyzing, with tools fromharmonic analysis, the preservers of several families of totally non-negativeand totally positive kernels with additional structure: continuous Hankelkernels on an interval, P\'olya frequency functions, and P\'olya frequencysequences. The rigid structure of post-composition transforms of totallypositive kernels acting on infinite sets is obtained by combining severalspecialized situations settled in our present and earlier works.

We initiate a study of the growth and matrix-valued Hilbert series of N-graded twisted Calabi-Yau algebras that are homomorphic images of path algebras of weighted quivers, generalizing techniques previously used to investigate Artin-Schelter regular algebras and graded Calabi-Yau algebras. Several results are proved without imposing any assumptions on the degrees of generators or relations of the algebras. We give particular attention to twisted Calabi-Yau algebras of dimension d≤3, giving precise descriptions of their matrix-valued Hilbert series and partial results describing which underlying quivers yield algebras of finite GK-dimension. For d=2, we show that these are algebras with mesh relations. For d=3, we show that the resulting algebras are a kind of derivation-quotient algebra arising from an element that is similar to a twisted superpotential.

P.M. Cohn showed in 1971 that given a ring R, to describe, up to isomorphism, a division ring D generated by a homomorphic image of R is equivalent to specifying the set of square matrices over R which map to singular matrices over D, and he determined precisely the conditions that such a set of matrices must satisfy. The present author later developed another version of this data, in terms of closure operators on free R-modules. In this note, we examine the latter concept further, and show how an R-module M satisfying certain conditions can be made to induce such data. In an appendix we make some observations on Cohn's original construction, and note how the data it uses can similarly be induced by an appropriate sort of R-module. Our motivation is the longstanding question of whether, for G a right-orderable group and k a field, the group algebra kG must be embeddable in a division ring. Our hope is that the right kG-module M=k((G)) might induce a closure operator of the required sort. We re-prove a partial result in this direction due to N.I. Dubrovin, note a plausible generalization thereof which would give the desired embedding, and also sketch some thoughts on other ways of approaching the problem.

P.M. Cohn showed in 1971 that given a ring R, to describe, up to isomorphism, a division ring D generated by a homomorphic image of R is equivalent to specifying the set of square matrices over R which map to singular matrices over D, and he determined precisely the conditions that such a set of matrices must satisfy. The present author later developed another version of this data, in terms of closure operators on free R-modules. In this note, we examine the latter concept further, and show how an R-module M satisfying certain conditions can be made to induce such data. In an appendix we make some observations on Cohn's original construction, and note how the data it uses can similarly be induced by an appropriate sort of R-module. Our motivation is the longstanding question of whether, for G a right-orderable group and k a field, the group algebra kG must be embeddable in a division ring. Our hope is that the right kG-module M=k((G)) might induce a closure operator of the required sort. We re-prove a partial result in this direction due to N.I. Dubrovin, note a plausible generalization thereof which would give the desired embedding, and also sketch some thoughts on other ways of approaching the problem.

Commuting integral and differential operators connect the topics of signal processing, random matrix theory, and integrable systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the Korteweg-de Vries (KdV) equation. We prove a general theorem that the integral operator associated to every wave function in the infinite-dimensional adelic Grassmannian [Formula: see text] of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intrinsic property is shown to follow from the symmetries of Grassmannians of Kadomtsev-Petviashvili (KP) wave functions, where the direct commutativity property holds for operators associated to wave functions fixed by Wilson's sign involution but is violated in general. Based on this result, we prove a second main theorem that the integral operators in the computation of the singular values of the truncated generalized Laplace transforms associated to all bispectral wave functions of rank 1 reflect a differential operator. A [Formula: see text] rotation argument is used to prove a third main theorem that the integral operators in the computation of the singular values of the truncated generalized Fourier transforms associated to all such KP wave functions commute with a differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, including as special cases the integral operators associated to all rational solutions of the KdV and KP hierarchies considered by [Airault, McKean, and Moser, Commun. Pure Appl. Math. 30, 95-148 (1977)] and [Krichever, Funkcional. Anal. i Priložen. 12, 76-78 (1978)], respectively, in the late 1970s. Many examples are presented.

In previous work [Adv. Math. 298, pp. 325-368, 2016], the structure of thesimultaneous kernels of Hadamard powers of any positive semidefinite matrixwere described. Key ingredients in the proof included a novel stratification ofthe cone of positive semidefinite matrices and a well-known theorem ofHershkowitz, Neumann, and Schneider, which classifies the Hermitian positivesemidefinite matrices whose entries are $0$ or $1$ in modulus. In this paper,we show that each of these results extends to a larger class of matrices whichwe term $3$-PMP (principal minor positive).

This survey contains a selection of topics unified by the concept of positivesemi-definiteness (of matrices or kernels), reflecting natural constraintsimposed on discrete data (graphs or networks) or continuous objects(probability or mass distributions). We put emphasis on entrywise operationswhich preserve positivity, in a variety of guises. Techniques from harmonicanalysis, function theory, operator theory, statistics, combinatorics, andgroup representations are invoked. Some partially forgotten classical roots inmetric geometry and distance transforms are presented with comments and fullbibliographical references. Modern applications to high-dimensional covarianceestimation and regularization are included.

The Boolean ring B of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, 0) that are closed under the natural metric, but has no prime ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, B is known to be complete in its metric. Together, these facts answer a question posed by J. Gleason. From this example, rings of arbitrary characteristic with the same properties are obtained. The result that B is complete in its metric is generalized to show that if L is a lattice given with a metric satisfying identically either the inequality d(x ∨ y, x ∨ z) ≤ d(y, z) or the inequality d(x ∧ y, x ∧ z) ≤ d(y, z), and if in L every increasing Cauchy sequence converges and every decreasing Cauchy sequence converges, then every Cauchy sequence in L converges; that is, L is complete as a metric space. We show by example that if the above inequalities are replaced by the weaker conditions d(x, x∨y) ≤ d(x, y), respectively d(x, x∧y) ≤ d(x, y), the completeness conclusion can fail. We end with two open questions.

For commutative rings, we introduce the notion of a universal grading, which can be viewed as the “largest possible grading”. While not every commutative ring (or order) has a universal grading, we prove that every reduced order has a universal grading, and this grading is by a finite group. Examples of graded orders are provided by group rings of finite abelian groups over rings of integers in number fields. We also generalize known properties of nilpotents, idempotents, and roots of unity in such group rings to the case of graded orders; this has applications to cryptography. Lattices play an important role in this paper; a novel aspect is that our proofs use that the additive group of any reduced order can in a natural way be equipped with a lattice structure.

Let R be a perfect ring of characteristic p. We show that the group ofcontinuous R-linear automorphisms of the perfect power series ring over R isgenerated by the automorphisms of the ordinary power series ring together withFrobenius; this answers a question of Jared Weinstein.

In this paper we study right S-Noetherian rings and modules, extending notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right S-Noetherian rings are given in terms of completely prime right ideals and point annihilator sets. We also prove an existence result for completely prime point annihilators of certain S-Noetherian modules with the following consequence in commutative algebra: If a module M over a commutative ring is S-Noetherian with respect to a multiplicative set S that contains no zero-divisors for M, then M has an associated prime.

For V a vector space over a field, or more generally, over a division ring, it is well-known that every x ∈ End(V ) has an inner inverse; that is, that there exists y ∈ End(V ) satisfying xyx = x. We show here that a large class of such x have inner inverses y that satisfy with x an infinite family of additional monoid relations, making the monoid generated by x and y what is known as an inverse monoid (definition recalled). We obtain consequences of these relations, and related results. P. Nielsen and J. Šter [16] show that a much larger class of elements x of rings R, including all elements of von Neumann regular rings, have inner inverses satisfying arbitrarily large finite subsets of the abovementioned set of relations. But we show by example that the endomorphism ring of any infinite-dimensional vector space contains elements having no inner inverse that simultaneously satisfies all those relations. A tangential result gives a condition on an endomap x of a set S that is necessary and suffcient for x to have a strong inner inverse in the monoid of all endomaps of S.

We investigate the possibility of constructing Kochen–Specker uncolorable sets of idempotent matrices whose entries lie in various rings, including the rational numbers, the integers, and finite fields. Most notably, we show that there is no Kochen–Specker coloring of the n×n idempotent integer matrices for n≥3, thereby illustrating that Kochen–Specker contextuality is an inherent feature of pure matrix algebra. We apply this to generalize recent no-go results on noncommutative spectrum functors, showing that any contravariant functor from rings to sets (respectively, topological spaces or locales) that restricts to the Zariski prime spectrum functor for commutative rings must assign the empty set (respectively, empty space or locale) to the matrix ring Mn(R) for any integer n≥3 and any ring R. An appendix by Alexandru Chirvasitu shows that Kochen–Specker colorings of idempotents in partial subalgebras of M3(F) for a perfect field F can be extended to partial algebra morphisms into the algebraic closure of F.

If L is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex Δ(L) (definition recalled below). Lattice-theoretically, the resulting object is a subdirect product of copies of L. We note properties of this construction and of some variants, and pose several questions. For M3 the 5–element nondistributive modular lattice, Δ(M3) is modular, but its underlying topological space does not admit a structure of distributive lattice, answering a question of Walter Taylor. We also describe a construction of “stitching together” a family of lattices along a common chain, and note how Δ(M3) can be regarded as an example of this construction.

We investigate conditions that are sufficient to make the Extalgebra of an object in a (triangulated) category into a Frobenius algebra, and compute the corresponding Nakayama automorphism. As an application, we prove the conjecture that hdet(μA) = 1 for any noetherian Artin-Schelter regular (hence skew Calabi-Yau) algebra A.

Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a “Prime Ideal Principle” that gives a uniform method of proving such facts, generalizing the Prime Ideal Principle for commutative rings due to T. Y. Lam and the author. Old and new “maximal implies prime” results are presented, with results touching on annihilator ideals, polynomial identity rings, the Artin–Rees property, Dedekind-finite rings, principal ideals generated by normal elements, strongly noetherian algebras, and just infinite algebras.

If $k$ is a field, $A$ and $B$ $k$-algebras, $M$ a faithful left $A$-module,and $N$ a faithful left $B$-module, we recall the proof that the left$A\otimes_k B$-module $M\otimes_k N$ is again faithful. If $k$ is a generalcommutative ring, we note some conditions on $A,$ $B,$ $M$ and $N$ that do, andothers that do not, imply the same conclusion. Finally, we note a version ofthe main result that does not involve any algebra structures on $A$ and $B.$

We establish an isomorphism between the center of the Heisenberg category defined by Khovanov and the algebra $\Lambda^*$ of shifted symmetric functions defined by Okounkov-Olshanski. We give a graphical description of the shifted power and Schur bases of $\Lambda^*$ as elements of the center, and describe the curl generators of the center in the language of shifted symmetric functions. This latter description makes use of the transition and co-transition measures of Kerov and the noncommutative probability spaces of Biane.

We characterize the diagonalizable subalgebras of End(V), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finite-dimensional vector space. Our characterizations are formulated in terms of a natural topology (the “finite topology”) on End(V), which reduces to the discrete topology in the case where V is finite-dimensional. We further investigate when two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the closure of the set of diagonalizable operators within End(V). Motivated by the classical link between diagonalizability and semisimplicity, we also give an infinite-dimensional generalization of the Wedderburn–Artin theorem, providing a number of equivalent characterizations of left pseudocompact, Jacoboson semisimple rings that parallel various characterizations of artinian semisimple rings. This theorem unifies a number of related results in the literature, including the structure of linearly compact, Jacobson semsimple rings and cosemisimple coalgebras over a field.

Vaughan Pratt has introduced objects consisting of pairs (A,W) where A is a set and W a set of subsets of A, such that (i) W contains \emptyset and A, (ii) if C is a subset of A\times A such that for every a\in A, both (a,b) and b,a are members of W (a 'crossword' with all 'rows' and 'columns' in W then (b,b)\in C (the 'diagonal word') also belongs to W and (iii) for all distinct a,b the set W has an element which contains a but not b. He has asked whether for every A the only such W is the set of all subsets of A We answer that question in the negative. We also obtain several positive results, in particular, a positive answer to the above question if W is closed under complementation. We obtain partial results on whether there can exist counterexamples to Pratt's question with W countable.

We investigate conditions that are sufficient to make the Extalgebra of an object in a (triangulated) category into a Frobenius algebra, and compute the corresponding Nakayama automorphism. As an application, we prove the conjecture that hdet(μA) = 1 for any noetherian Artin-Schelter regular (hence skew Calabi-Yau) algebra A.

Let R be an associative unital algebra over a field k, let p be an element of R, and let R'=R〈q|pqp=p〉. We obtain normal forms for elements of R', and for elements of R'-modules arising by extension of scalars from R-modules. The details depend on where in the chain pR∩Rp⊆pR∪Rp⊆pR+Rp⊆R the unit 1 of R first appears.This investigation is motivated by a hoped-for application to the study of the possible forms of the monoid of isomorphism classes of finitely generated projective modules over a von Neumann regular ring; but that goal remains distant.We end with a normal form result for the algebra obtained by tying together a k-algebra R given with a nonzero element p satisfying 1 ∉ pR+ Rp and a k-algebra S given with a nonzero q satisfying 1 ∉ qS+ Sq, via the pair of relations p= pqp, q= qpq.

Let R be an associative unital algebra over a field k, let p be an element of R, and let R'=R〈q|pqp=p〉. We obtain normal forms for elements of R', and for elements of R'-modules arising by extension of scalars from R-modules. The details depend on where in the chain pR∩Rp⊆pR∪Rp⊆pR+Rp⊆R the unit 1 of R first appears.This investigation is motivated by a hoped-for application to the study of the possible forms of the monoid of isomorphism classes of finitely generated projective modules over a von Neumann regular ring; but that goal remains distant.We end with a normal form result for the algebra obtained by tying together a k-algebra R given with a nonzero element p satisfying 1 ∉ pR+ Rp and a k-algebra S given with a nonzero q satisfying 1 ∉ qS+ Sq, via the pair of relations p= pqp, q= qpq.

We develop a cohomology theory for Jordan triples, including the infinite dimensional ones, by means of the cohomology of TKK Lie algebras. This enables us to apply Lie cohomological results to the setting of Jordan triples. Some preliminary results for von Neumann algebras are obtained.

We develop a cohomology theory for Jordan triples, including the infinite dimensional ones, by means of the cohomology of TKK Lie algebras. This enables us to apply Lie cohomological results to the setting of Jordan triples. Some preliminary results for von Neumann algebras are obtained.

Let R be a left Artinian ring, and M a faithful left R-module such that no proper submodule or homomorphic image of M is faithful. If R is local, and socle(R) is central in R, we show that length(M=J(R)M) + length(socle(M)) ≤ length(socle(R)) + 1. If R is a finite-dimensional algebra over an algebraically closed éld, but not necessarily local or having central socle, we get an inequality similar to the above, with the length of socle(R) interpreted as its length as a bimodule, and the summand +1 replaced by the Euler characteristic of a graph determined by the bimodule structure of socle(R). The statement proved is slightly more general than this summary; we examine the question of whether much stronger generalizations are possible. If a faithful module M over an Artinian ring is only assumed to have one of the above minimality properties-no faithful proper submodules, or no faithful proper homomorphic images-we find that the length of M=J(R)M in the former case, and of socle(M) in the latter, is ≤ length(socle(R)). The proofs involve general lemmas on decompositions of modules.

In this paper we obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen-Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type An-1 for c=mn, and an expression of its quasispherical part (i.e., the isotypic part of "hooks") in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Etingof to give explicit formulas for the characters of the irreducible equivariant D-modules on the nilpotent cone for SLm. Our third result is the construction of the Koszul-BGG complex for the rational Cherednik algebra, which generalizes the construction of the Koszul-BGG resolution from [3] and [21], and the calculation of its homology in type A. We also show in type A that the differentials in the Koszul-BGG complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the Cherednik algebra. Finally, our fourth result is the symmetry theorem, which identifies the quasispherical components in the representations with minimal support over the rational Cherednik algebras Hmn(Sn) and Hnm(Sm). In fact, we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras. This symmetry was essentially established in [8] in the spherical case, and in [24] in the case GCD(m, n)=1, and it has a natural interpretation in terms of invariants of torus knots.

In this paper we obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen-Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type An-1 for c=mn, and an expression of its quasispherical part (i.e., the isotypic part of "hooks") in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Etingof to give explicit formulas for the characters of the irreducible equivariant D-modules on the nilpotent cone for SLm. Our third result is the construction of the Koszul-BGG complex for the rational Cherednik algebra, which generalizes the construction of the Koszul-BGG resolution from [3] and [21], and the calculation of its homology in type A. We also show in type A that the differentials in the Koszul-BGG complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the Cherednik algebra. Finally, our fourth result is the symmetry theorem, which identifies the quasispherical components in the representations with minimal support over the rational Cherednik algebras Hmn(Sn) and Hnm(Sm). In fact, we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras. This symmetry was essentially established in [8] in the spherical case, and in [24] in the case GCD(m, n)=1, and it has a natural interpretation in terms of invariants of torus knots.

We study properties of a group, abelian group, ring, or monoid B which (a) guarantee that every homomorphism from an infinite direct product ΠI Ai of objects of the same sort onto B factors through the direct product of finitely many ultraproducts of the Ai (possibly after composition with the natural map B → B/Z(B) or some variant), and/or (b) guarantee that when a map does so factor (and the index set has reasonable cardinality), the ultrafilters involved must be principal. A number of open questions and topics for further investigation are noted.

Several authors have studied the question of when the monoid ring DM of a monoid M over a ring D is a right and/or left fir (free ideal ring), a semifir, or a 2-fir (definitions recalled in §1). It is known that for M nontrivial, a necessary condition for any of these properties to hold is that D be a division ring. Under that assumption, necessary and sufficient conditions on M are known for DM to be a right or left fir, and various conditions on M have been proved necessary or sufficient for DM to be a 2-fir or semifir. A sufficient condition for DM to be a semifir is that M be a direct limit of monoids which are free products of free monoids and free groups. Warren Dicks has conjectured that this is also necessary. However F. Cedó has given an example of a monoid M which is not such a direct limit, but satisfies all the known necessary conditions for DM to be a semifir. It is an open question whether for this M, the rings DM are semifirs.We note here some reformulations of the known necessary conditions for a monoid ring DM to be a 2-fir or a semifir, motivate Cedó’s construction and a variant thereof, and recoverCedó’s results for both constructions. Any homomorphism from a monoid M into Z induces a Z-grading on DM, and we show that for the two monoids just mentioned, the rings DM are “homogeneous semifirs” with respect to all such nontrivial Z-gradings; i.e., have (roughly) the property that every finitely generated homogeneous one-sided ideal is free of unique rank. If M is a monoid such that DM is an n-fir, and N a “well-behaved” submonoid of M, we prove some properties of the ring DN. Using these, we show that for M a monoid such that DM.

A skew Calabi-Yau algebra is a generalization of a Calabi-Yau algebra which allows for a non-trivial Nakayama automorphism. We prove three homological identities about the Nakayama automorphism and give several applications. The identities we prove show (i) how the Nakayama automorphism of a smash product algebra A#. H is related to the Nakayama automorphisms of a graded skew Calabi-Yau algebra A and a finite-dimensional Hopf algebra H that acts on it; (ii) how the Nakayama automorphism of a graded twist of A is related to the Nakayama automorphism of A; and (iii) that the Nakayama automorphism of a skew Calabi-Yau algebra A has trivial homological determinant in case A is noetherian, connected graded, and Koszul. © 2014 Elsevier Inc.

We prove that operator algebras that have enough projections are completely determined by those projections, their symmetries, and the action of the latter on the former. This includes all von Neumann algebras and all AW*-algebras. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*-algebras is equivalent to a full subcategory of active lattices. Crucial ingredients are an equivalence between the category of piecewise AW*-algebras and that of piecewise complete Boolean algebras, and a refinement of the piecewise algebra structure of an AW*-algebra that enables recovering its total structure. © 2014 Elsevier Inc.

Criteria are obtained for a filter F of subsets of a set I to be an intersection of finitely many ultrafilters, respectively, finitely many κ-complete ultrafilters for a given uncountable cardinal κ. From these, general results are deduced concerning homomorphisms on infinite direct product groups, which yield quick proofs of some results in the literature: the Łoś–Eda theorem (characterizing homomorphisms from a not-necessarily-countable direct product of modules to a slender module), and some results of Nahlus and the author on homomorphisms on infinite direct products of not-necessarily-associative k-algebras. The same tools allow other results of Nahlus and the author to be nontrivially strengthened, and yield an analog to one of their results, with nonabelian groups taking the place of k-algebras. We briefly examine the question of how the common technique used in applying the general results of this note to k-algebras on the one hand, and to nonabelian groups on the other, might be extended to more general varieties of algebras in the sense of universal algebra. In a final section, the Erd˝os–Kaplansky theorem on dimensions of vector spaces DI (D a division ring) is extended to reduced products DI /F, and an application is noted.

We develop a general theory of Markov chains realizable as random walks on $\mathscr R$-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via M\"obius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom-Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop.

Eggert's Conjecture says that if R is a finite-dimensional nilpotentcommutative algebra over a perfect field F of characteristic p, and R^{(p)} isthe image of the p-th power map on R, then dim_F R \geq p dim_F R^{(p)}.Whether this very elementary statement is true is not known. We examine heuristic evidence for this conjecture, versions of the conjecturethat are not limited to positive characteristic and/or to commutative R,consequences the conjecture would have for semigroups, and examples that giveequality in the conjectured inequality. We pose several related questions, andbriefly survey the literature on the subject.

There are two fundamental obstructions to representing noncommutative ringsvia sheaves. First, there is no subcanonical coverage on the opposite of thecategory of rings that includes all covering families in the big Zariski site.Second, there is no contravariant functor F from the category of rings to thecategory of ringed categories whose composite with the global sections functoris naturally isomorphic to the identity, such that F restricts to the Zariskispectrum functor Spec on the category of commutative rings (in a compatible waywith the natural isomorphism). Both of these no-go results are proved byrestricting attention to matrix rings.

Eggert's Conjecture says that if R is a finite-dimensional nilpotentcommutative algebra over a perfect field F of characteristic p, and R^{(p)} isthe image of the p-th power map on R, then dim_F R \geq p dim_F R^{(p)}.Whether this very elementary statement is true is not known. We examine heuristic evidence for this conjecture, versions of the conjecturethat are not limited to positive characteristic and/or to commutative R,consequences the conjecture would have for semigroups, and examples that giveequality in the conjectured inequality. We pose several related questions, andbriefly survey the literature on the subject.

We give a combinatorial interpretation of the determinant of a matrix as a generating function over Brauer diagrams in two different but related ways. The sign of a permutation associated to its number of inversions in the Leibniz formula for the determinant is replaced by the number of crossings in the Brauer diagram. This interpretation naturally explains why the determinant of an even antisymmetric matrix is the square of a Pfaffian.

We study the homology and cohomology groups of super Lie algebra of supersymmetries and of super Poincare Lie algebra in various dimensions. We give complete answers for (non-extended) supersymmetry in all dimensions $\leq 11$. For dimensions $D=10,11$ we describe also the cohomology of reduction of supersymmetry Lie algebra to lower dimensions. Our methods can be applied to extended supersymmetry algebra.

We study the homology and cohomology groups of super Lie algebra of supersymmetries and of super Poincare algebra. We discuss in detail the calculation in dimensions D=10 and D=6. Our methods can be applied to extended supersymmetry algebra and to other dimensions.

We prove that the maximal nilpotent subalgebra of a Kac-Moody Lie algebra has an (essentially unique) Euclidean metric with respect to which the Laplace operator in the chain complex is scalar on each component of a given degree. Moreover, both the Lie algebra structure and the metric are uniquely determined by this property.

For P a poset or lattice, let Id(P) denote the poset, respectively, lattice,of upward directed downsets in P, including the empty set, and letid(P)=Id(P)-\{\emptyset\}. This note obtains various results to the effect thatId(P) is always, and id(P) often, "essentially larger" than P. In the firstvein, we find that a poset P admits no "

We determine the structure of the center and the trace space of the centrally extended preprojective algebra of an ADE quiver, introduced by the first and the third authors in math/0503393. It turns out that this structure has a mysterious relationship to the structure of the maximal nilpotent subalgebra of the corresponding simple Lie algebra.

In our previous paper math.QA/0409261, we defined a deformation of the group algebra of the group of even elements of a Coxeter group W, and showed that it is flat for all values of parameters if and only if all the rank 3 parabolic subgroups of W are infinite. In this paper, we study what happens in the general case. Then the deformation is flat only for some values of parameters, and the set of all such values is called the flatness locus. The main result of the paper is an explicit description of the this flatness locus as a scheme over Z. More specifically, we show that this scheme is the intersection of the flatness loci for the subalgebras corresponding to parabolic subgroups of rank 3. The latter are determined by solving the rigid multiplicative Deligne-Simpson problem. We also define additive analogs of our algebras and study their properties.

We give a condition ensuring that the operators in a nilpotent Lie algebra oflinear operators on a finite dimensional vector space have a commoneigenvector.

For a positive integer n we introduce quadratic Lie algebras tr_n qtr_n and discrete groups Tr_n, QTr_n naturally associated with the classical and quantum Yang-Baxter equation, respectively. We prove that the universal enveloping algebras of the Lie algebras tr_n, qtr_n are Koszul, and find their Hilbert series. We also compute the cohomology rings of these Lie algebras (which by Koszulity are the quadratic duals of the enveloping algebras). We construct cell complexes which are classifying spaces of the groups Tr_n and QTr_n, and show that the boundary maps in them are zero, which allows us to compute the integral cohomology of these groups. We show that the Lie algebras tr_n, qtr_n map onto the associated graded algebras of the Malcev Lie algebras of the groups Tr_n, QTr_n, respectively. We conjecture that this map is actually an isomorphism (this is now a theorem due to P. Lee). At the same time, we show that the groups Tr_n and QTr_n are not formal for n>3.

We define new deformations of group algebras of Coxeter groups W and of subgroups of even elements in them, by deforming the braid relations. We show that these deformations are algebraically flat iff they are formally flat, and that this happens iff the group W has no finite parabolic subgroups of rank 3. The proof uses the theory of constructible sheaves on cell complexes attached to W. We explain the connection of our deformations with the Hecke algebras of orbifolds defined by the first author in math.QA/0406499 and with generalized double affine Hecke algebras defined by the authors and A. Oblomkov in math.QA/0406480. The paper is dedicated to the memory of Walter Feit.

We consider a problem of optimizing convex functionals over matroid bases. It is richly expressive and captures certain quadratic assignment and clustering problems. While generally NP-hard, we show it is polynomial time solvable when a suitable parameter is restricted.

In this dissertation, we investigate the cohomology theory of restricted Lie algebras. The representation theory of restricted Lie algebras is reviewed including a description of the restricted universal enveloping algebra. In the case of an abelian restricted Lie algebra, we construct an augmented complex of free modules over the enveloping algebra that is exact in dimensions less than p and hence define the cohomology theory of these algebras in dimension less than p. In the non-abelian case, we explicitly construct cochain spaces for any coefficient module in dimensions less than 3, and give explicit formulas for the coboundary operators in these dimensions. The corresponding notions of the usual algebraic interpretations of ordinary low dimensional cohomology are defined and we show that our restricted cohomology spaces encode this information as well.