scholarly journals KOSZUL CALCULUS

2017 ◽  
Vol 60 (2) ◽  
pp. 361-399 ◽  
Author(s):  
ROLAND BERGER ◽  
THIERRY LAMBRE ◽  
ANDREA SOLOTAR
Keyword(s):  
Duality Theorem ◽  
Derived Categories ◽  
Quadratic Algebra ◽  
Koszul Duality ◽  
True Nature ◽  
Quadratic Algebras ◽  
Koszul Homology ◽  
Cup Products ◽  
Koszul Cohomology

AbstractWe present a calculus that is well-adapted to homogeneous quadratic algebras. We define this calculus on Koszul cohomology – resp. homology – by cup products – resp. cap products. The Koszul homology and cohomology are interpreted in terms of derived categories. If the algebra is not Koszul, then Koszul (co)homology provides different information than Hochschild (co)homology. As an application of our calculus, the Koszul duality for Koszul cohomology algebras is proved foranyquadratic algebra, and this duality is extended in some sense to Koszul homology. So, the true nature of the Koszul duality theorem is independent of any assumption on the quadratic algebra. We compute explicitly this calculus on a non-Koszul example.

Solution of the Dirac equation with some superintegrable potentials by the quadratic algebra approach

2014 ◽  
Vol 29 (06) ◽  
pp. 1450028 ◽  
Author(s):  
S. Aghaei ◽  
A. Chenaghlou
Keyword(s):  
Dirac Equation ◽  
Quadratic Algebra ◽  
Two Dimensional ◽  
Oscillator Algebra ◽  
Quadratic Algebras ◽  
Energy Eigenvalues ◽  
Vector Potentials ◽  
Algebra Approach ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

The Dirac equation with scalar and vector potentials of equal magnitude is considered. For the two-dimensional harmonic oscillator superintegrable potential, the superintegrable potentials of E8 (case (3b)), S4 and S2, the Schrödinger-like equations are studied. The quadratic algebras of these quasi-Hamiltonians are derived. By using the realization of the quadratic algebras in a deformed oscillator algebra, the structure function and the energy eigenvalues are obtained.

scholarly journals Linear Koszul duality

2009 ◽  
Vol 146 (1) ◽  
pp. 233-258 ◽  
Author(s):  
Ivan Mirković ◽  
Simon Riche
Keyword(s):  
Vector Bundle ◽  
Hecke Algebras ◽  
Derived Categories ◽  
Koszul Duality

AbstractIn this paper we construct, for F1 and F2 subbundles of a vector bundle E, a ‘Koszul duality’ equivalence between derived categories of 𝔾m-equivariant coherent(dg-)sheaves on the derived intersection $F_1 \rcap _E F_2$, and the corresponding derived intersection $F_1^{\perp } \rcap _{E^*} F_2^{\perp }$. We also propose applications to Hecke algebras.

scholarly journals Quadratic algebra contractions and second-order superintegrable systems

2014 ◽  
Vol 12 (05) ◽  
pp. 583-612 ◽  
Author(s):  
Ernest G. Kalnins ◽  
W. Miller
Keyword(s):  
Constant Curvature ◽  
Two Dimensions ◽  
Second Order ◽  
Quadratic Algebra ◽  
Quadratic Algebras ◽  
Physical Systems ◽  
Special Cases ◽  
Superintegrable Systems ◽  
Wilson Polynomials ◽  
Symmetry Algebras

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of second-order superintegrable systems in two dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. For constant curvature spaces, we show that the free quadratic algebras generated by the first- and second-order elements in the enveloping algebras of their Euclidean and orthogonal symmetry algebras correspond one-to-one with the possible superintegrable systems with potential defined on these spaces. We describe a contraction theory for quadratic algebras and show that for constant curvature superintegrable systems, ordinary Lie algebra contractions induce contractions of the quadratic algebras of the superintegrable systems that correspond to geometrical pointwise limits of the physical systems. One consequence is that by contracting function space realizations of representations of the generic superintegrable quantum system on the 2-sphere (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.

On composition and absolute-valued algebras

2006 ◽  
Vol 136 (4) ◽  
pp. 717-731 ◽  
Author(s):  
José Antonio Cuenca Mira
Keyword(s):  
Finite Dimension ◽  
Quadratic Algebra ◽  
Central Idempotent ◽  
Quadratic Algebras ◽  
Large Classes ◽  
Anisotropic Norm ◽  
Composition Algebras ◽  
Characteristic 2 ◽  
The Absolute

In this note we give a complete description of the composition algebras A over fields of characteristic ≠ 2, 3 in the following cases: if A has an anisotropic norm and x2x = xxx2 for every element; when A has a unitary central idempotent, it satisfies the identity (x2x2)x2 = x2(x2x2), and A is of finite dimension or has anisotropic norm. As a consequence, we obtain the existence, up to an isomorphism, of only seven absolute-valued algebras with a non-zero central idempotent where the last identity holds. This result completes the study of the absolute-valued algebras of this kind that was initiated by El-Mallah and Agawany.We also introduce the class of e-quadratic algebra, which contains the quadratic algebras, but also includes large classes of composition and absolute-valued algebras. Many results on composition, absolute-valued and e-quadratic algebras are shown, and new proofs of some well-known theorems are given.

scholarly journals SUBALGEBRAS OF GOLOD–SHAFAREVICH ALGEBRAS

2009 ◽  
Vol 19 (03) ◽  
pp. 423-442 ◽  
Author(s):  
THOMAS VODEN
Keyword(s):  
Associative Algebra ◽  
Quadratic Algebra ◽  
Quadratic Algebras

A graded associative algebra generated by m elements of degree one is called Golod–Shafarevich (GS) if it is presented with less than m2/4 relators of degree at least two. We explore conditions under which subalgebras of graded GS algebras are themselves GS. We prove that infinitely many Veronese powers of an algebra presented by m generators and r relators are GS if [Formula: see text]. For quadratic algebras, the bound is improved to [Formula: see text]. We also show that if A is a generic quadratic algebra presented by m generators and r relators, then all Veronese powers of A are GS if [Formula: see text], and all but finitely many Veronese powers of A are not GS if [Formula: see text].

scholarly journals The symplectic geometry of higher Auslander algebras: Symmetric products of disks

2021 ◽  
Vol 9 ◽  
Author(s):  
Tobias Dyckerhoff ◽  
Gustavo Jasso ◽  
Yankι Lekili
Keyword(s):  
Unit Disk ◽  
Symplectic Geometry ◽  
Morita Equivalence ◽  
Derived Categories ◽  
Symmetric Product ◽  
Koszul Duality ◽  
Fukaya Categories ◽  
Simplicial Space ◽  
Dimensional Unit ◽  
Geometric Realisation

Abstract We show that the perfect derived categories of Iyama’s d-dimensional Auslander algebras of type ${\mathbb {A}}$ are equivalent to the partially wrapped Fukaya categories of the d-fold symmetric product of the $2$ -dimensional unit disk with finitely many stops on its boundary. Furthermore, we observe that Koszul duality provides an equivalence between the partially wrapped Fukaya categories associated to the d-fold symmetric product of the disk and those of its $(n-d)$ -fold symmetric product; this observation leads to a symplectic proof of a theorem of Beckert concerning the derived Morita equivalence between the corresponding higher Auslander algebras of type ${\mathbb {A}}$ . As a by-product of our results, we deduce that the partially wrapped Fukaya categories associated to the d-fold symmetric product of the disk organise into a paracyclic object equivalent to the d-dimensional Waldhausen $\text {S}_{\bullet }$ -construction, a simplicial space whose geometric realisation provides the d-fold delooping of the connective algebraic K-theory space of the ring of coefficients.

scholarly journals A Classification of Some Quadratic Algebras

2006 ◽  
Vol 13 (01) ◽  
pp. 133-148
Author(s):  
Heather C. McGilvray
Keyword(s):  
Quadratic Algebra ◽  
Quadratic Algebras ◽  
Linear Resolution

We investigate a group of quadratic algebras and the associated resolutions of the field. When the quadratic algebra is Koszul, we provide the associated linear resolution; when not Koszul, we describe the maps of the resolution up to the point of non-linearity. In all cases, the resolutions are shown to be surprisingly regular and quickly stabilizing.

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