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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.48 n.3 Bahía Blanca 2007
Non-homogeneous N-Koszul Algebras
Roland Berger
Abstract. This is a joint work with Victor Ginzburg [4] in which we study a class of associative algebras associated to finite groups acting on a vector space. These algebras are non-homogeneous N-Koszul algebra generalizations of symplectic reflection algebras. We realize the extension of the N-Koszul property to non-homogeneous algebras through a Poincaré-Birkhoff-Witt property.
PART I - HOMOGENEOUS N-KOSZUL ALGEBRAS
I introduced these algebras in [2]. These algebras extend classic Koszul algebras (Priddy, 1970) corresponding to . A natural question is: why higher
's? I list below four answers.
1. There are some relevant examples coming from
- noncommutative projective algebraic geometry: cubic Artin-Schelter regular algebras [1] of global dimension 3, as
![x](/img/revistas/ruma/v48n3/3a037x.png)
![y](/img/revistas/ruma/v48n3/3a038x.png)
![x](/img/revistas/ruma/v48n3/3a039x.png)
![y](/img/revistas/ruma/v48n3/3a0310x.png)
- representation theory: skew-symmetrizer killing algebras (introduced in [2]):
![2 ≤ p ≤ n](/img/revistas/ruma/v48n3/3a0312x.png)
![1,2,...,p](/img/revistas/ruma/v48n3/3a0313x.png)
![n](/img/revistas/ruma/v48n3/3a0314x.png)
![p](/img/revistas/ruma/v48n3/3a0315x.png)
![( ) n p](/img/revistas/ruma/v48n3/3a0316x.png)
- theoretical physics: Yang-Mills algebras (A. Connes and M. Dubois-Violette [7]):
![gλμ](/img/revistas/ruma/v48n3/3a0318x.png)
![(s + 1) × (s + 1)](/img/revistas/ruma/v48n3/3a0319x.png)
![s + 1](/img/revistas/ruma/v48n3/3a0320x.png)
![s + 1](/img/revistas/ruma/v48n3/3a0321x.png)
2. Poincaré duality in Hochschild (co)homology (R.B. and N. Marconnet [5]): if is
-Koszul and AS-Gorenstein, then
![d](/img/revistas/ruma/v48n3/3a0325x.png)
![A](/img/revistas/ruma/v48n3/3a0326x.png)
![ɛd+1φ](/img/revistas/ruma/v48n3/3a0327x.png)
![A](/img/revistas/ruma/v48n3/3a0328x.png)
![M](/img/revistas/ruma/v48n3/3a0329x.png)
3. Extension of -Koszulity to quiver algebras with relations by E. Green, E. Marcos, R. Martínez-Villa, P. Zhang [10].
4. Extension of Koszul duality in terms of -algebras by J.-W. He and D.-M. Lu [11].
PART II - SYMPLECTIC REFLECTION ALGEBRAS
These algebras were introduced by P. Etingof and V. Ginzburg [8], and play an important role in representation theory and algebraic geometry (desingularization). Let be a finite dimensional complex vector space which is endowed with a symplectic 2-form
. Let
be a finite subgroup of Sp
and
be the smash product of the tensor algebra
of
with the group algebra
of
. From these data, a
-invariant linear map
![ℂ Γ](/img/revistas/ruma/v48n3/3a0343x.png)
![H ψ](/img/revistas/ruma/v48n3/3a0345x.png)
![H0 = S (V )# Γ → gr(H ψ)](/img/revistas/ruma/v48n3/3a0346x.png)
Theorem (P.E.-V.G. [8]) This morphism is an isomorphism, i.e., the Poincaré-Birkhoff-Witt (PBW) property holds for .
Ginzburg and I are able to provide an -version of this theorem [4]. First we define an
-version of
with
(the notation
is more convenient as far as symplectic reflection algebras are concerned). These generalized
's are called higher symplectic reflection algebras [4].
PART III - HIGHER SYMPLECTIC REFLECTION ALGEBRAS
Fix ,
. We have generalizations
where
![Alt(v ,...,v ) = ∑ sgn(σ) v ...v 1 p σ σ(1) σ(p)](/img/revistas/ruma/v48n3/3a0358x.png)
Theorem (R.B.-V.G. [4]) The PBW property holds for generalized .
The undeformed algebra is the skew-symmetrizer killing algebra of Part I (up to the change of rings
) which is still
-Koszul for the new ground ring. In order to prove the previous theorem, we state and prove the following.
-PBW Theorem (R.B.-V.G. [4]) Assume that
is a von Neumann regular ring,
is a
-
-bimodule,
, and
is a sub-
-
-bimodule of
, where
for any
. Set
and
, where
and
is the projection of
onto
modulo
.
Assume that is
-Koszul (this assumption can be weakened). Then the combination of the two conditions
![]() | (0.1) |
![]() | (0.2) |
is equivalent to the PBW property for .
Next, we check conditions (0.1) and (0.2) for generalized . Condition (0.1) is easily drawn from the
-invariance of
, while condition (0.2) (which can be viewed as an
-version of the Jacobi identity) is obtained by a close analysis of a standard Koszul complex.
Comments on the -PBW Theorem
- For and
field, this theorem is due to A. Braverman-D. Gaitsgory [6], and A. Polishchuk-L. Positselski [12] (during the 1990's).
- The -PBW theorem for
field and
finite-dimensional is independently stated and proved by G. Fløystad and J. Vatne [9].
Definitions Let us keep notations and assumptions of the -PBW theorem. If the PBW property holds for
, one says that
is Koszul (R.B.-V.G.), or that
is a PBW-deformation of
(G.F.-J.V.).
The first definition extends nicely the definition of homogeneous -Koszul algebras. A historical argument in favour of this terminology is given by the first Lie theory use by J. L. Koszul of his complex (mentioned in Cartan-Eilenberg's book, p. 281): working in the filtered context of the enveloping algebra of a Lie algebra, J. L. Koszul used the classical PBW property as a tool to carry over the exactness of his complex to the standard complex. The second definition is useful when one wants to find all the
's corresponding to a given
.
There are already some various applications of the -PBW theorem:
1. G. Fløystad and J. Vatne have found [9] all the PBW-deformations of
- any cubic Artin-Schelter regular algebra of global dimension 3,
- any skew-symmetrizer killing algebra for between (note that, for this second point, the intersection of their result and our result is very small since it corresponds to a trivial group
).
2. The PBW-deformations of Yang-Mills algebras have been determined by M. Dubois-Violette and R.B. [3].
In these applications, the -PBW theorem of G.F.-J.V. suffices. However, our general setting for the
-PBW theorem allows us to include significant examples (as higher symplectic reflection algebras) for which the ground field
is enlarged to group algebras
with non trivial
.
[1] M. Artin, W. F. Schelter, Graded algebras of global dimension 3, Adv. Math. 66 (1987), 171-216. [ Links ]
[2] R. Berger, Koszulity for nonquadratic algebras, J. Algebra 239 (2001), 705-734. [ Links ]
[3] R. Berger, M. Dubois-Violette, Inhomogeneous Yang-Mills Algebras, Lett. Math. Phys. 76 (2006), 65-75. [ Links ]
[4] R. Berger, V. Ginzburg, Symplectic reflection algebras and non-homogeneous N-Koszul property, J. Algebra 304 (2006), 577-601. [ Links ]
[5] R. Berger, N. Marconnet, Koszul and Gorenstein properties for homogeneous algebras, Alg. and Rep. Theory 9 (2006), 67-97. [ Links ]
[6] A. Braverman, D. Gaitsgory, Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type, J. Algebra 181 (1996), 315-328. [ Links ]
[7] A. Connes, M. Dubois-Violette, Yang-Mills algebra, Lett. Math. Phys. 61 (2002), 149-158. [ Links ]
[8] P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. math. 147 (2002), 243-348. [ Links ]
[9] G. Fløystad, J.E. Vatne, PBW-deformations of N-Koszul algebras, J. Algebra 302 (2006), 116 -155. [ Links ]
[10] E.L. Green, E.N. Marcos, R. Martínez-Villa, P. Zhang, D-Koszul algebras, J. Pure and Applied Algebra 193 (2004), 141-162. [ Links ]
[11] J.-W. He, D.-M. Lu, Higher Koszul algebras and A-infinity algebras. J. Algebra 293 (2005), no. 2, 335-362. [ Links ]
[12] A. Polishchuk, L. Positselski, Quadratic algebras. University Lecture Series 37, American Mathematical Society, Providence, RI, 2005. [ Links ]
Roland Berger
LaMUSE,
Faculté des Sciences et Techniques,
23, Rue P. Michelon,
42023 Saint-Etienne Cedex 2, France
Roland.Berger@univ-st-etienne.fr
Recibido: 6 de abril de 2006
Aceptado: 10 de octubre de 2006