pinecone/core-2020-05-10-deduplication huggingface dataset:
Mod-pReduction for Quantum Groups

AbstractLet Uϵ(G) be the quantized enveloping algebra associated to the Lie algebra G=sl(n+1) at apth-root of unity ε and assume thatpis a prime which does not dividen+1. It is known that the irreducible, finite dimensional representations of Uε(G) are parametrized, up to isomorphisms, by the conjugacy classes of SL(n+1). In the paper we prove that the dimension of any Uε(G)-moduleMparametrized by a conjugacy class O is divided byp1/2dim(O). This result was conjectured by C. De Concini, V. G. Kac, and C. Procesi (J. Amer. Math. Soc.5, 1992, 151–190)
Tame Realisable Classes over Hopf Orders

AbstractLetObe the ring of algebraic integers in a number fieldKand letGbe a finite abelian group. McCulloh has characterised the classes in the locally free classgroup Cl(OG) which are realisable by rings of integers of tame normal extensions ofKwith Galois groupG. We extend this to certain extensions which in general need be neither tame nor normal by replacingOGwith a Hopf order A and introducing the strongly A-tame orders as an analogue of tame rings of integers. We also describe the classes realised by principal homogeneous spaces over the dual of A
Confluence and Koszulity

AbstractA lattice-theoretic approach to Koszulity is well known. This paper gives a lattice-theoretic approach to polynomiality, i.e., existence of a P.B.W. basis in Priddy's sense. This provides a straightforward proof for Priddy's theorem. Polynomiality is interpreted in terms of confluence in noncommutative computational algebra. The basic notion of a reduction operator is investigated in some detail. The reduction algebras are introduced to study the confluence of two reduction operators in terms of representation theory. The meet and the join of two reduction operators are constructed in an algorithmic way. A geometric characterization of the confluence is obtained
Invariants over Curvature Tensor Fields

AbstractIn this paper, we present constructions of higher-order polynomialO(n)-invariants over curvature tensor fields. These invariants are higher-order analogues of the scalar curvature. Our methods are based on certain replicative properties of theO(n)-module structure on a “curvature space” and a realization of a “curvature space” by the action of the symmetric tensor of the Lie algebraso(n) on symmetric matrices. By our methods, we are able to find a complete generator set of functional invariants over the curvature tensor fields of a manifold with lower dimensions
Bipartite Posets of Finite Prinjective Type

AbstractOne of the main results of this paper is Theorem 1.2, which contains a characterization of finite bipartite posetsI=I′∪I″ for which the category prin(kI) of prinjective modules over the incidencek-algebrakIofIis of finite representation type, wherekis a field. In particular, it is shown that for any such posetI, the Auslander–Reiten quiver of the category prin(kI) has no oriented cycle, and there is a bijection between the isomorphism classes of indecomposable objects in prin(kI) and the positive roots of the quadratic formqIassociated withI. An existence of a preprojective component in prin(kI) is proved for faithful posetsIwhich areÃ-free
Formally Integrally Closed Domains and the RingsR((X)) andR{{X}}

LetRbe an integral domain. Forf∈R [ X ] letAfbe the ideal ofRgenerated by the coefficients off. We defineRto be formally integrally closed ⇔ (Afg)t=(AfAg)tfor all nonzerof,g∈R [ X ] . Examples of formally integrally closed domains include locally finite intersections of one-dimensional Prüfer domains (e.g., Krull domains and one-dimensional Prüfer domains). We study the ringsR((X))=R [ X ] NandR{{X}}=R [ X ] NtwhereN={f∈R [ X ] |Af=R} andNt={f∈R [ X ] |(Af)t=R}. We show thatRis a Krull domain (resp., Dedekind domain) ⇔R{{X}} (resp.,R((X))) is a Krull domain (resp., Dedekind domain) ⇔R{{X}} (resp.,R((X))) is a Euclidean domain ⇔ every (principal) ideal ofR{{X}} (resp.,R((X))) is extended fromR⇔Ris formally integrally closed and every prime ideal ofR{{X}} (resp.,R((X))) is extended fromR
Composition Algebras over Rings of Fractions Revisited

AbstractLetkbe a field of characteristic not two, letfh(x0,x1)∈k[x0,x1] be an irreducible homogeneous polynomial and denote the ring of elements of degree zero in the homogeneous localizationk[x0,x1]fhbyk[x0,x1](fh). For degfh=3 it is proved that the composition algebras overk[x0,x1](fh)not containing zero divisors are defined overkand that there is at most one (split) composition algebra not defined overk. For degfh=4 all composition algebras overk[x0,x1](fh)are enumerated and partly classified
Locally Finite Groups with All Subgroups Normal-by-(Finite Rank)

AbstractA group is said to have finite (special) rank ≤sif all of its finitely generated subgroups can be generated byselements. LetGbe a locally finite group and suppose thatH/HGhas finite rank for all subgroupsHofG, whereHGdenotes the normal core ofHinG. We prove that thenGhas an abelian normal subgroup whose quotient is of finite rank (Theorem 5). If, in addition, there is a finite numberrbounding all of the ranks ofH/HG, thenGhas an abelian subgroup whose quotient is of finite rank bounded in terms ofronly (Theorem 4). These results are based on analogous theorems on locally finitep-groups, in which case the groupGis also abelian-by-finite (Theorems 2 and 3)
Polynomial Sequences in Groups

AbstractGiven a groupGwith lower central seriesG=G1⊇G2⊇G3⊇···, we say that a sequenceg: Z→Gispolynomialif for anykthere isdsuch that the sequence obtained fromgby applying the difference operatorDg(n)=g(n)−1g(n+1)dtimes takes its values inGk. We introduce the notion ofthe degree of a polynomial sequenceand we prove that polynomial sequences of degrees not exceeding a given one form a group. As an application we obtain the following extension of the Hall–Petresco theorem:THEOREM.LetG=G1⊇G2⊇G3⊇···be the lower central series of a group G.Let x∈Gk,y∈Gland letp,qbe polynomialsZ→Zof degrees k and l,respectively. Then there is a sequencez0∈G,zi∈Gifori∈N,such thatfor all n∈N
On Almost-Dense Extension Groups of Torsion-Free Groups

AbstractLetAbe a torsion-free group. A mixed groupGis said to be an almost dense extension group (ADE group) ofAifAis almost dense inGandT(G)-high ofG. In the case thatAis of rank 1 andGpis cyclic for every primep, we establish the structure theorem, the representation theorem, the realization theorem, and the classification theorem. Moreover, for such an ADE groupG, we give a necessary and sufficient condition thatGis splitting and establish a characterization that allT(G)-high subgroups ofGare isomorphic
Generalized Sets of Lengths

AbstractLetRbe a Dedekind domain and I(R) the set of irreducible elements ofR. In this paper, we study the sets VR(n)={m|∃α1,…,αn,β1,…,βm∈I(R) such that α1,…,αn=β1,…,βm}, wherenis a positive integer. We show, in constrast to indications in some earlier work, that the sets VR(n) are not completely determined by the Davenport constant of the class group ofR. We offer some specific constructions for Dedekind domains with small class groups, and show how these sets are generalizations of the sets studied earlier by Geroldinger [9,10]
Primitivity in Jordan Systems is Ubiquitous

AbstractThe main result of the paper asserts that if a Jordan pairV=(V+,V−) is (σ)-primitive at some 0≠b∈V−σ, then it is (±)-primitive at any 0≠b′∈V∓. Also, if a Jordan triple systemTis primitive at some 0≠b∈T, then it is primitive at any 0≠b′∈T. As a tool, similar results concerning one-sided primitivity and ∗-primitivity of associative pairs and triple systems are established. Equivalences between our definitions of primitivity and some other appearing in the literature are also obtained as a consequence
Dual Pairs inPin(p,q) and Howe Correspondences for the Spin Representation

AbstractIn this paper we obtain examples of dual pairs in the groupPin(V⊗W) by considering the inverse images of the subgroupsO(V)⊗IdWandIdV⊗O(W) under the double covering map π:Pin(V⊗W)→O(V⊗W). The main technical results are the definition of the group of determinant graded double covers ofO(V) and the fact that the mapW↦π−1(O(V)⊗IdW) defines a homomorphism from the Witt group to this group when the ground fieldksatisfiesk*/(k*)2≅Z2and −1 is not a square ink. We also show that the spin representation ofPin(V⊗W) sets up Howe correspondences for each of these dual pairs whenk=R
Left Cells in the Affine Weyl Group of TypeC4

AbstractWe find a representative set of left cells of the affine Weyl groupWaof typeC4as well as its left cell graphs by applying an algorithm. This algorithm was designed in my previous paper (Tôhoku J. Math.46,1994, 105–124). It is reformulated and improved in a more efficient form here. These representatives of left cells are presented as the vertices of so-called essential graphs so that the generalized τ-invariants of left cells ofWaare actually described explicitly, the latter almost characterize the left cells ofWa. Some comments and conjectures are proposed on cells of affine Weyl groups, mostly for the case of typeCl,l≥2
Ideals of the Enveloping AlgebraU(sl2)

AbstractIn this paper we study the two-sided ideals of the enveloping algebraU=U(sl2(K)) over an arbitrary fieldKof characteristic zero. Starting with two basic ideas, that an irreducible Lie module is generated by its highest weight vector and that the Lie module structure ofUcomes from its ring multiplication, we have found a “good” subset ofUconsisting of highest weight vectors for irreducibleU-submodules ofUso that each two-sided ideal ofUis uniquely generated by at most two elements of that set. Actually, each ideal is generated as a two-sided ideal by just one element. By uniqueness, all the information about the ideal is encoded in the formula for its generator(s). For example, we can list and classify all the prime ideals by height, determine the intersection of an ideal with the center, find the radical ideals and the radical of an ideal, and determine when two ideals are included one in another. An interesting property is that each ideal ofUcan be uniquely written as a product of primes. We also obtain the “least common multiple” and the “greatest common divisor” formulas for the prime ideal factorizations of the intersection and the sum of two ideals. This paper contains many other results of this nature
On Functional Identities in Prime Rings with Involution

AbstractLetAbe a prime ring with involution ∗, letSbe the symmetric elements, letKbe the skew elements, letQmlbe the maximal left ring of quotients,x1,…,xmnoncommuting variables, andEi,Fj,Gk,Hl:Am−1→Qml,i,j,k,l=1,2,…,m. We study functional identities of the form ∑mi=1Eiixi+∑mj=1xjFjj+∑mk=1Gkkx*k+∑ml=1x*lHll=0 for allx1,…,xm∈A(whereEiimeansEi(x1,…,x̂i,…,xm), etc.). In caseS∪Kis not algebraic of bounded degree ≤2mdefinitive results are obtained. As an applicationk-commuting traces of symmetricn-additive maps of eitherSorKintoQmlare characterized
A Family of Commutative Endomorphism Algebras

AbstractLetK(X) be the quotient field of the polynomial ringK[X] over an algebraically closed fieldK. Given an integern≥2, a finite sequencem=(m2,…,mn) ofn−1 positive integers, a sequenceα=(α2,…,αn) ofK-linear functionals onK(X), and a height functionh, there is attached a Kronecker moduleE=E(m,h,α). In this paper we show that whenEhas no finite-dimensional direct summand then EndEis a commutativeK-subalgebra of the algebra ofn×nmatrices overK(X). Consequently, when EndEis an integral domain its transcendence degree overKis at most one. We give sufficient conditions for EndEto be isomorphic toKand also show thatKis the only field that occurs as EndE. Examples whereK[X],K[X,X−1], and some subalgebras ofK(X)noccur as EndEare given. The determination of all the commutativeK-subalgebras ofMn(K(X)) realizable as EndEremains open. Yet surprisingly it can be shown that the coordinate ring of a cubic equation in Weierstrass normal formY2=X3+aX2+bX+cis realizable
Tame Fields and Tame Extensions

AbstractLetVbe a henselian valuation of any rank of a fieldKand letVbe the extension ofVto a fixed algebraic closureKofK. In this paper, it is proved that (K,V) is a tame field, i.e., every finite extension of (K,V) is tamely ramified, if and only if, to each α∈K\K, there correspondsa∈Kfor whichV(α−a)≥ΔK(α), where ΔK(α)=min{V(α′−α)|α′ runs over allK-conjugates of α}. A special case of the previous result, whenKis a perfect field of nonzero characteristic was proved in 1995, with the purpose of completing a result of James Ax [S. K. Khanduja,J. Algebra172(1995), 147–151]
Ratliff–Rush Closures and Coefficient Modules

AbstractLet (R,m) be ad-dimensional Noetherian local domain. SupposeMis a finitely generated torsion-freeR-module and supposeFis a freeR-module containingM. In analogy with a result of Ratliff and Rush [Indiana Univ. Math. J.27(1978), 929–934] concerning ideals, we define and prove existence and uniqueness of theRatliff–RushclosureofMinF. We also discuss properties of Ratliff–Rush closure.In addition to the preceding assumptions, supposeF/Mhas finite length as anR-module. Then we define theBuchsbaum–RimpolynomialofMinF. In analogy with the work of K. Shah [Trans. Amer. Math. Soc.327(1991), 373–384], we definecoefficientmodulesofMinF. Under the assumption thatRis quasi-unmixed, we prove existence and uniqueness of coefficient modules ofMinF
Excellent Rings with Local Generic Formal Fibers

AbstractLet (T,M) be a complete regular local ring of dimension at least 2, containing the rationals, such that the cardinality of the residue fieldT/Mis at least the cardinality of the real numbers. Supposep≠Mis a prime ideal ofT. We construct an excellent regular local ringAsuch that the completion ofAisTand the generic formal fiber ofAis local (this means that the ringT⊗AKis a local ring whereKis the quotient field ofA) withp⊗AKits maximal ideal. In addition, ifqis a prime ideal ofTwithq∩A=P≠(0), thenq=PT
Some Restrictions on the Maps in Minimal Resolutions

AbstractWe show that there are certain restrictions on the entries of the maps in the minimal free resolutions of finitely generated modules of infinite projective dimension over Noetherian local rings. These restrictions provide a way to slightly improve Herzog's characterization of modules of finite projective and injective dimensions in characteristicp>0. We also discuss other homological situations related to these restrictions, including previously known results on the existence of free summands in syzygy modules
Semisimplicity of Parabolic Hecke Algebras

AbstractThe semisimplicity of Iwahori–Hecke algebras has been studied by several authors. A. Gyoja (J. Algebra174, 1995, 553–572) gave a necessary and sufficient condition for Iwahori–Hecke algebras to be semisimple, using the modular representation theory. The author (J. Algebra183, 1996, 514–544) studied the semisimplicity of parabolic Hecke algebras when they have only one parameterq. In this paper we completely determine the cases when parabolic Hecke algebras are semisimple complementing our previous work applying the method of Gyoja
On the Center of Quantized Enveloping Algebras

AbstractLetUbe a quasitriangular Hopf algebra. One may use theR-matrix ofUin order to construct scalar invariants of knots. Analogously, Reshetikhin wrote down tangle invariants which take their values in the center ofU. Reshetikhin's expressions thus define central elements inU. We prove here an identity characterizing some of these elements, whenUis a quantized enveloping algebra. As an application, we give a proof for a statement of Faddeev, Reshetikhin, and Takhtadzhyan concerning the center of a quantized enveloping algebra
On the Nilpotent Length of Polycyclic Groups

AbstractLetGbe a polycyclic group. We prove that if the nilpotent length of each finite quotient ofGis bounded by a fixed integern, then the nilpotent length ofGis at mostn. The casen=1 is a well-known result of Hirsch. As a consequence, we obtain that if the nilpotent length of each 2-generator subgroup is at mostn, then the nilpotent length ofGis at mostn. A more precise result in the casen=2 permits us to prove that if each 3-generator subgroup is abelian-by-nilpotent, thenGis abelian-by-nilpotent. Furthermore, we show that the nilpotent length ofGequals the nilpotent length of the quotient ofGby its Frattini subgroup
Stratifying Endomorphism Algebras Associated to Hecke Algebras

AbstractLetGbe a finite group of Lie type and letkbe a field of characteristicdistinctfrom the defining characteristic ofG. In studying thenon-describingrepresentation theory ofG, the endomorphism algebraS(G,k)=EndkG(⊕JindGPJk) plays an increasingly important role. In typeA, by work of Dipper and James,S(G,k) identifies with aq-Schur algebra and so serves as a link between the representation theories of the finite general linear groups and certain quantum groups. This paper presents the first systematic study of the structure and homological algebra of these algebras forGof arbitrary type. BecauseS(G,k) has a reinterpretation as a Hecke endomorphism algebra, it may be analyzed using the theory of Hecke algebras. Its structure turns out to involve new applications of Kazhdan–Lusztig cell theory. In the course of this work, we prove two stratification conjectures about Coxeter group representations made by E. Cline, B. Parshall, and L. Scott (Mem. Amer. Math. Soc.591, 1996) and we formulate a new conjecture about the structure ofS(G,k). We verify this conjecture here in all rank 2 examples
Generalized Restricted Lie Algebras and Representations of the Zassenhaus Algebra

AbstractLetFbe a field of characteristicp>0,La generalized restricted Lie algebra overF, andP(L) the primitivep-envelope ofL. A close relation betweenL-representations andP(L)-representations is established. In particular, the irreducible κ-reduced modules ofLfor any κ∈L* coincide with the irreducibleκ̄0-reduced modules ofP(L), whereκ̄0∈P(L)* is a trivial extension of κ. From this result, the determination of all irreducible representations of the Zassenhaus algebra is completed, and the dimensions of the corresponding modules are also given
Graded Lie Superalgebras and the Superdimension Formula

AbstractIn this paper, we investigate the structure of graded Lie superalgebras L=⊕(α,a)∈Γ×AL(α,a), where Γ is a countable abelian semigroup and A is a countable abelian group with a coloring map satisfying a certain finiteness condition. Given a denominator identity for the graded Lie superalgebra L, we derive a superdimension formula for the homogeneous subspaces L(α,a)(α∈Γ,a∈A), which enables us to study the structure of graded Lie superalgebras in a unified way. We discuss the applications of our superdimension formula to free Lie superalgebras, generalized Kac–Moody superalgebras, and Monstrous Lie superalgebras. In particular, the product identities for normalized formal power series are interpreted as the denominator identities for free Lie superalgebras. We also give a characterization of replicable functions in terms of product identities and determine the root multiplicities of Monstrous Lie superalgebras
The Center of Some Quantum Projective Planes

AbstractThe homogeneous coordinate ring of a quantum projective plane is a 3-dimensional Artin–Schelter regular algebra with the same Hilbert series as the polynomial ring in three variables; such an algebraAis a graded noncommutative analogue of the polynomial ring in three variables. WhenAis a finite module over its centerZ(A), we define the schemeS=Proj(Z(A)) and the sheaf A of OS-algebras by A(S(f))=A[f−1]0. The center Z of A is defined by Z(S(f))=Z(A[f−1]0) and following Grothendieck we may define the scheme Spec(Z). The algebrasAfall into several families, and for many of these it has been shown that Spec(Z)≅P2whenAis finite over its center. This paper shows that Spec(Z)≅P2for two more families
Fat Points, Inverse Systems, and Piecewise Polynomial Functions

AbstractWe explore the connection between ideals of fat points (which correspond to subschemes of Pnobtained by intersecting (mixed) powers of ideals of points), and piecewise polynomial functions (splines) on ad-dimensional simplicial complex Δ embedded inRd. Using the inverse system approach introduced by Macaulay [11], we give a complete characterization of the free resolutions possible for ideals ink[x,y] generated by powers of homogeneous linear forms (we allow the powers to differ). We show how ideals generated by powers of homogeneous linear forms are related to the question of determining, for some fixed Δ, the dimension of the vector space of splines on Δ of degree less than or equal tok. We use this relationship and the results above to derive a formula which gives the number of planar (mixed) splines in sufficiently high degree
Subgroup Theorems for the Baer-Invariant of Groups

AbstractM. R. Jones and J. Wiegold (J. London Math. Soc. (2)6, 1973, 738) have shown that ifGis a finite group with a subgroupHof finite indexn, then thenth power of the Schur multiplier ofG,M(G)n, is isomorphic to a subgroup ofM(H). In this paper we prove a similar result for the centre by centre bywvariety of groups, wherewis any outer commutator word. Then using a result of M. R. R. Moghaddam (Arch. Math.33,1979, 504–511), we will be able to deduce a result of Schur's type with respect to the variety of nilpotent groups of class at mostc(c≥1), whenc+1 is any prime number or 4
Mixing and Covering in the Symmetric Groups

AbstractWe discuss mixing and covering theorems in the symmetric groups. We present an example of a covering without mixing, and study the conjugacy class [2n/2] of symmetric groupSn, which demonstrates mixing without covering. We derive some new character identities from the computation of [2n/2]2, and also compute [2n/2]3, filling a hole between theorems of J. L. Brenner (J. Austral. Math. Soc. Ser. A25,1978, 210–214) and Y. Dvir (in“Products of Conjugacy Classes in Groups,” Lecture Notes in Math., Vol. 1112, Springer-Verlag, New York/Berlin, 1985). This computation also motivates a certain classification of 3-colored 3-regular graphs
Lots of Hopf Algebras

AbstractThe purpose of this paper is to give a “universal” construction of Hopf algebras over any commutative ring. The mod-pSteenrod algebras are all examples of this construction; other examples of Hopf algebras related to the mod-pSteenrod algebras are also produced. Then we give an argument to produce deformations of these Hopf algebras via “twisting.” The techniques of this paper are, in a way, “scheme-theoretic,” producing many rather complicated Hopf algebras, only a few of which are studied in any detail here; however, we hope to return to the study of some of these Hopf algebras in future work
On Graphs Admitting Arc-Transitive Actions of Almost Simple Groups

AbstractLet Γ be a finite connected regular graph with vertex setVΓ, and letGbe a subgroup of its automorphism group AutΓ. Then Γ is said to beG-locally primitiveif, for each vertex α, the stabilizerGαis primitive on the set of vertices adjacent to α. In this paper we assume thatGis an almost simple group with socle socG=S; that is,Sis a nonabelian simple group andS⊴G≤AutS. We study nonbipartite graphs Γ which areG-locally primitive, such thatShas trivial centralizer in AutΓ andSis not semiregular on vertices. We prove that one of the following holds: (i)S⊴AutΓ≤Aut(S), (ii)G<Y≤AutΓ withYalmost simple and socY≠S, or (iii)Sbelongs to a very restricted family of Lie type simple groups of characteristicp, say, and AutΓ contains the semidirect productZdp:G, whereZdpis a known absolutely irreducibleG-module. Moreover, in certain circumstances we can guarantee thatS⊴AutΓ≤Aut(S). For example, if Γ is a connected (G,2)-arc transitive graph with Sz(q)≤G≤Aut(Sz(q)) (q=22n+1≥8) orG=Ree(q) (q=32n+1≥27), thenG≤AutΓ≤Aut(G)