共查询到20条相似文献,搜索用时 234 毫秒
1.
Tom Braden Anthony Licata Christopher Phan Nicholas Proudfoot Ben Webster 《Selecta Mathematica, New Series》2011,17(3):533-572
We show that the center of a flat graded deformation of a standard Koszul algebra A behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed point set. In particular,
the center of A acts by characters on the deformed standard modules, providing a “localization map”. We construct a universal graded deformation
of A and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the
arrangement coming from the algebra Koszul dual to A. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings
of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal
deformation for the ring governing a block of parabolic category O{\mathcal{O}} for
\mathfrakgln{\mathfrak{gl}_n} is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version
of the “category O{\mathcal{O}}” of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric
variety. 相似文献
2.
The main purpose of this paper is to provide some new criteria for a standard graded algebra A = ⊕
i≥0
A
i
to be a λ-Koszul algebra, which was first introduced in [12] and was another class of “Koszul-type” algebras including Koszul and d-Koszul algebras as special examples. 相似文献
3.
G. Cerulli Irelli 《Journal of Algebraic Combinatorics》2011,33(2):259-276
We provide a technique to compute the Euler–Poincaré characteristic of a class of projective varieties called quiver Grassmannians.
This technique applies to quiver Grassmannians associated with “orientable string modules”. As an application we explicitly
compute the Euler–Poincaré characteristic of quiver Grassmannians associated with indecomposable pre-projective, pre-injective
and regular homogeneous representations of an affine quiver of type [(A)\tilde]p,1\tilde{A}_{p,1}. For p=1, this approach provides another proof of a result due to Caldero and Zelevinsky (in Mosc. Math. J. 6(3):411–429, 2006). 相似文献
4.
We calculate the twisted Hochschild and cyclic homology (in the sense of Kustermans, Murphy and Tuset) of the coordinate algebra
of the quantum SL(2) group relative to twisting automorphisms acting by rescaling the standard generators a,b,c,d. We discover a family of automorphisms for which the “twisted” Hochschild dimension coincides with the classical dimension
of
, thus avoiding the “dimension drop” in Hochschild homology seen for many quantum deformations. Strikingly, the simplest such
automorphism is the canonical modular automorphism arising from the Haar functional. In addition, we identify the twisted
cyclic cohomology classes corresponding to the three covariant differential calculi over quantum SU(2) discovered by Woronowicz. 相似文献
5.
D. V. Millionshchikov 《Proceedings of the Steklov Institute of Mathematics》2008,263(1):99-111
We compute explicitly the adjoint cohomology of two ℕ-graded Lie algebras of maximal class (infinite-dimensional filiform
Lie algebras) m0 and m2. It is known that up to an isomorphism there are only three ℕ-graded Lie algebras of maximal class. The third algebra from
this list is the “positive” part L
1 of the Witt (or Virasoro) algebra, and its adjoint cohomology was computed earlier by Feigin and Fuchs.
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 106–119. 相似文献
6.
7.
P. Baumann 《Transformation Groups》2000,5(1):3-20
Let g be a simple finite-dimensional complex Lie algebra and letG be the corresponding simply-connected algebraic group. A theorem of Kostant states that the universal enveloping algebra of g is a free module over its center. A theorem of Richardson states that the algebra of regular functions ofG is a free module over the subalgebra of regular class functions. Joseph and Letzter extended Kostant's theorem to the case of the quantized enveloping algebra of g. Using the theory of crystal bases as the main tool, we prove a quantum analogue of Richardson's theorem. From it, we recover Joseph and Letzter's result by a kind of quantum duality principle. 相似文献
8.
We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations
of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of
semi-simple Lie algebras. To explain this duality, we introduce an “interpolating quantum group” depending on two parameters
which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending
on two parameters, which interpolate between representations of two Langlands dual quantum groups. 相似文献
9.
In this note we give elementary examples of the naturalness of generalized Clifford algebras appearance, in some particular
quantum mechanical models.
First Weyl’s program [1] for quantum kinematics for the case of simplest Galois fieldsZ
n
is realized in terms of generalized Clifford algebras. Dynamics might then be introduced, following the ideas of Hanney and
Berry [2], as shown in [3].
Second the coherent state picture of the finite dimensional “Z
n
— Quantum Mechanics” is presented.
In the last part the known coherent states ofq-deformed quantum oscillators (q≡ω) are explicitly shown in the generalized Grassman algebras and the generalized Clifford algebras settings.
Presented atThe Polish-Mexican Seminar, Kazimierz Dolny, August 1998 — Poland. 176 相似文献
10.
Abstract. Each choice of a K?hler class on a compact complex manifold defines an action of the Lie algebra sl(2) on its total complex cohomology. If a nonempty set of such K?hler classes is given, then we prove that the corresponding
sl(2)-copies generate a semisimple Lie algebra. We investigate the formal properties of the resulting representation and we
work things out explicitly in the case of complex tori, hyperk?hler manifolds and flag varieties. We pay special attention
to the cases where this leads to a Jordan algebra structure or a graded Frobenius algebra.
Oblatum 21-V-1996 & 15-X-1996 相似文献
11.
Tatiana Gateva-Ivanova 《Algebras and Representation Theory》2011,14(4):779-802
We show the intimate connection between various mathematical notions that are currently under active investigation: a class
of Garside monoids, with a “nice” Garside element, certain monoids S with quadratic relations, whose monoidal algebra A = kSA= \textbf{k}S has a Frobenius Koszul dual
A
! with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were introduced and studied by the author and in 1995 provided a new class of Noetherian Artin-Schelter regular domains, and the square-free set-theoretic solutions of the Yang-Baxter equation. There is a beautiful symmetry in these objects due to their nice combinatorial and algebraic properties. 相似文献
12.
13.
A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple
Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance,
in the case of the orthosymplectic Lie superalgebra osp(1,2), Pinczon gave in [Pi] an example of a Verma module whose annihilator
is not generated by its intersection with the centre of universal enveloping algebra. More generally, Musson produced in [Mu1]
a family of such “singular” Verma modules for osp(1,2l) cases. In this article we give a necessary and sufficient condition
on the highest weight of a osp(1,2l)-Verma module for its annihilator to be generated by its intersection with the centre.
This answers a question of Musson. The classical proof of the Duflo theorem is based on a deep result of Kostant which uses
some delicate algebraic geometry reasonings. Unfortunately these arguments can not be reproduced in the quantum and super
cases. This obstruction forced Joseph and Letzter, in their work on the quantum case (see [JL]), to find an alternativeapproach
to the Duflo theorem. Following their ideas, we compute the factorization of the Parthasarathy–Ranga-Rao–Varadarajan (PRV)
determinants. Comparing it with the factorization of Shapovalov determinants we find, unlike to the classical and quantum
cases, that the PRV determinant contains some extrafactors. The set of zeroes of these extrafactors is precisely the set of
highest weights of Verma modules whose annihilators are not generated by their intersection with the centre. We also find
an analogue of Hesselink formula (see [He]) giving the multiplicity of every simple finite dimensional module in the graded
component of the harmonic space in the symmetric algebra.
Oblatum 1-IX-1998 & 4-XII-1998 / Published online: 10 May 1999 相似文献
14.
Edward Richmond 《Journal of Algebraic Combinatorics》2009,30(1):1-17
Horn recursion is a term used to describe when non-vanishing products of Schubert classes in the cohomology of complex flag
varieties are characterized by inequalities parameterized by similar non-vanishing products in the cohomology of “smaller”
flag varieties. We consider the type A partial flag variety and find that its cohomology exhibits a Horn recursion on a certain
deformation of the cup product defined by Belkale and Kumar (Invent. Math. 166:185–228, 2006). We also show that if a product of Schubert classes is non-vanishing on this deformation, then the associated structure
constant can be written in terms of structure constants coming from induced Grassmannians. 相似文献
15.
A. Van Daele 《Advances in Mathematics》1998,140(2):11
A Hopf algebra is a pair (A, Δ) whereAis an associative algebra with identity andΔa homomorphism formAtoA⊗Asatisfying certain conditions. If we drop the assumption thatAhas an identity and if we allowΔto have values in the so-called multiplier algebraM(A⊗A), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A, Δ) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (Â, Δ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (Â, Δ) is canonically isomorphic with the original multiplier Hopf algebra (A, Δ). It is possible to generalize many aspects of abstract harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting. 相似文献
16.
This paper presents solutions or partial solutions for several problems in the theory of relation algebras. In a simple relation
algebra an element x satisfying the condition (a) must be an atom of . It follows that x must also be an atom in every simple extension of . Andréka, Jónsson and Németi [1, Problem 4] (see [12, Problem P5]) asked whether the converse holds: if x is an atom in every simple extension of a simple relation algebra, must it satisfy (a)? We show that the answer is “no”.?
The only known examples of simple relation algebras without simple proper extensions are the algebras of all binary relations
on a finite set. Jónsson proposed finding all finite simple relation algebras without simple proper extensions [12, Problem
P6]. We show how to construct many new examples of finite simple relation algebras that have no simple proper extensions, thus
providing a partial answer for this second problem. These algebras are also integral and nonrepresentable.? Andréka, Jónsson,
Németi [1, Problem 2] (see [12, Problem P7]) asked whether there is a countable simple relation algebra that cannot be embedded in a one-generated relation algebra.
The answer is “yes”. Givant [3, Problem 9] asked whether there is some k such that every finitely generated simple relation algebra can be embedded in a k-generated simple relation algebra. The answer is “no”.
Received November 27, 1996; accepted in final form July 3, 1997. 相似文献
17.
The relationships between piecewise-Koszul algebras and other “Koszul-type” algebras are discussed. The Yoneda-Ext algebra
and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A
! to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers
in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary “period” and piecewise-Koszul
algebras with arbitrary “jump-degree”. 相似文献
18.
19.
Gábor Czédli 《Algebra Universalis》2007,57(1):63-73
A ternary term m(x, y, z) of an algebra is called a majority term if the algebra satisfies the identities m(x, x, y) = x, m(x, y, x) = x and m(y, x, x) = x. A congruence α of a finite algebra is called uniform if all of its blocks (i.e., classes) have the same number of elements. In particular, if all the α-blocks are two-element then α is said to be a 2-uniform congruence. If all congruences of A are uniform then A is said to be a uniform algebra. Answering a problem raised by Gr?tzer, Quackenbush and Schmidt [2], Kaarli [3] has recently proved that uniform finite lattices
are congruence permutable.
In connection with Kaarli’s result, our main theorem states that for every finite algebra A with a majority term any two 2-uniform congruences of A permute. Examples show that we can say neither “algebra” instead of “algebra with a majority term”, nor “3-uniform” instead
of “2-uniform”.
Given two nonempty sets A and B, each relation
gives rise to a pair of closure operators, which are called the Galois closures on A and B induced by ρ. Galois closures play an important role in many parts of algebra, and they play the main role in formal concept analysis
founded by Wille [4]. In order to prove our main theorem, we introduce a pair of smaller closure operators induced by ρ. These closure operators will hopefully find further applications in the future.
Dedicated to the memory of Kazimierz Głazek
Presented by E. T. Schmidt.
Received November 29, 2005; accepted in final form May 23, 2006.
This research was partially supported by the NFSR of Hungary (OTKA), grant no. T049433 and T037877. 相似文献