Representations of Super Yang-Mills Algebras

Herscovich, Estanislao

doi:10.1007/s00220-012-1648-z

Cited by 4 publications

(8 citation statements)

References 28 publications

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“…A finitely generated local algebra is coherent if, for every finitely generated ideal J , the space Tor A 1 (k, J ) is finite dimensional over the residue field k. In particular, it allows to prove that if a graded algebra has a double-sided ideal J , such that the quotient algebra is right noetherian, and J is free as a left A-module, then A is right coherent [13,14]. This criterion allows to prove coherence for some classes of algebras, see [9,10,15].…”

Section: Examples and Related Resultsmentioning

confidence: 99%

Coherence of relatively quasi-free algebras

Bondal

Zhdanovskiy

2015

European Journal of Mathematics

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Section: Examples and Related Resultsmentioning

confidence: 99%

Coherence of relatively quasi-free algebras

Bondal

Zhdanovskiy

2015

European Journal of Mathematics

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“…, s, We suppose further that the matrices (Γ i a,b ) satisfy the nondegeneracy assumption explained in the third paragraph before Rmk. 1 of [18].…”

Section: If S ∈ S We Will Denotementioning

confidence: 98%

“…Using the explicit description of the minimal projective resolution of the trivial module k over the super Yang-Mills algebra YM(n, s) Γ (for (n, s) = (1, 0), (1, 1)) given in [18], Prop. 2, and Corollary 3.26 given below, we see that these graded algebras are multi-Koszul.…”

Section: If S ∈ S We Will Denotementioning

confidence: 99%

“…We would like to point out that even tough this new definition could still seem to be rather restrictive, it allows several interesting examples, for it includes the super Yang-Mills algebras introduced by M. Movshev and A. Schwarz in [28] and further studied by the author in [18], which were one of the main motivations for the present work (see Example 3.21). These algebras are not generated in degree 1, so they cannot be multi-Koszul in the sense defined in [19], and in particular they cannot be generalized Koszul either.…”

Section: Introductionmentioning

confidence: 99%

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On the multi-Koszul property for connected algebras

Herscovich¹

2013

Preprint

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In this article we introduce the notion of multi-Koszul algebra for the case of a locally finite dimensional nonnegatively graded connected algebra, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras, which were in turn an extension of Koszul algebras introduced by S. Priddy. This notion also extends and generalizes the one recently introduced by the author and A. Rey, which was for the particular case of algebras further assumed to be finitely generated in degree 1 and with a finite dimensional space of relations. The idea of this new notion in the realm of arbitrary locally finite dimensional nonnegatively graded connected algebras, which should be perhaps considered as a probably interesting common property for several of these algebras, was to find a grading independent description of some of the interesting properties shared by all generalized Koszul algebras. In order to simplify the exposition and unify the left and right possible sides of the multi-Koszul property, the very definition of multi-Koszul algebras starts by considering the minimal graded projective resolution of the algebra A as an A-bimodule, which may be used to compute the corresponding Hochschild (co)homology groups. This new definition includes several new interesting examples, e.g. the super Yang-Mills algebras introduced by M. Movshev and A. Schwarz, which are not generalized Koszul or even multi-Koszul for the previous definition given by the author and Rey in any natural manner. On the other hand, we provide an equivalent description of the new definition in terms of the Tor (or Ext) groups, similar to the existing one for homogeneous algebras, and we show that several of the typical homological computations performed for the generalized Koszul algebras are also possible in this more general setting. In particular, we give a very explicit description of the A∞algebra structure of the Yoneda algebra of a multi-Koszul algebra, which has a similar pattern as for the case of generalized Koszul algebras. We also show that the Yoneda algebra of a finitely generated multi-Koszul algebra with a finite dimensional space of relations is generated in degrees 1 and 2, so a K2 algebra in the sense of T. Cassidy and B. Shelton.

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“…2. Theoretical physics: Yang-Mills algebras are Koszul cubic algebras [14,15], their PBW deformations were determined [6], and their representation theory was studied in [27,29]. Other cubic algebras linked to parastatistics were studied in [19].…”

Section: Introductionmentioning

confidence: 99%

Koszul calculus for N-homogeneous algebras

Berger

2019

Journal of Algebra

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We extend Koszul calculus defined on quadratic algebras by Berger, Lambre, Solotar (Koszul calculus, Glasg. Math. J.) to N -homogeneous algebras for any N ≥ 2, quadratic algebras corresponding to N = 2. We emphasize that N -homogeneous algebras are considered in full generality, with no Koszulity assumption. Koszul cup and cap products are introduced and are reduced to usual cup and cap products if N = 2, but if N > 2, they are defined by very specific expressions. These specific expressions are compatible with the Koszul differentials and provide associative products on classes.There is no associativity in general on chains-cochains, suggesting that Koszul cochains should constitute an A∞-algebra, acting as an A∞-bimodule on Koszul chains. 2010 MSC: 16S37, 16S38, 16E40, 16E45. Keywords: N -homogeneous algebras, N -Koszul algebras, Koszul (co)homology, Hochschild (co)homology, cup and cap products.by Priddy when N = 2 [36]. Since then, N -homogeneous algebras and N -Koszul algebras have been connected to the following domains.

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Representations of Super Yang-Mills Algebras

Cited by 4 publications

References 28 publications

Coherence of relatively quasi-free algebras

Coherence of relatively quasi-free algebras

On the multi-Koszul property for connected algebras

Koszul calculus for N-homogeneous algebras

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