G-algebras, Lie Algebras, Hopf Algebras II

International Journal of Algebra, Vol. 10, 2016, no. 8, 351 - 372
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ija.2016.6748


G-algebras, Lie Algebras, Hopf Algebras II
R. Martı́nez-Villa

Centro de Ciencias Matemáticas UNAM, Morelia, Mexico
http://www.matmor.unam.mx

Dedicated to Edward L. Green who introduced me to the world
of Koszul algebras and non commutative Groebner basis.

Copyright c 2016 R. Martı́nez-Villa. This article is distributed under the Creative Com-
mons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.

Abstract
In a series of papers the author jointly with J. Mondragón stud-
ied in a systematic way the structure of homogeneous Groebner basis
algebras, or G-algebras, and in two recent papers the author applied
these results to the homogenized enveloping algebra of the Lie algebra
S`(2,C). The results obtained in the study of this particular algebra
inspired our previous paper on the Hopf algebra structure of the ho-
mogenized enveloping algebra of finite dimensional Lie algebras.
In this first section of this paper we continue further the study of
the homogenization des homogenization process, this time at the level
of Ext and Tor. In the second part we study quantized homogeneous
G-algebras Bn and give the structure of thieir Yoneda algebra. Using
this construction we find the structure of arbitrary quadratic G-algebras
in terms of their structure of constants.

Mathematics Subject Classification: Primary 16S30, 17B35; Secondary
16T05

Keywords: G-algebras, Lie algebra, Hopf.


1 Introduction

Let k be a field and k <X1 ,X2 ,...Xn > the free algebra in n generators. A
quadratic Groebner basis algebra or a G-algebra is an algebra defined by gen-

,352 R. Martı́nez-Villa

n
bkij Xk -aij ,
P
erators and relations as An =k <X1 ,X2 ,...Xn >/ < Xj Xi -cij Xi Xj -
k=1
i<j> and such that X1 ,X2 ,...Xn form a Poincare Birkoff Witt basis. In case
aij =0, cij =1 for i<j a G-algebra An is isomorphic to the enveloping algebra
of a finite dimensional Lie algebra, and in the case aij =1, cij =1 for i<j, and
bkij =0 for all k, An is the Weyl algebra [5], [6].
A homogeneous G-algebra is an algebra given P k by generators and relations
2
as Bn =k <X1 ,X2 ,...Xn ,Z >/ < Xj Xi -cij Xi Xj - bij Xk Z -aij Z , Xi Z-ZXi > such
that X1 ,X2 ,...Xn ,Z form a Poincare Birkoff Witt (or PBW) basis. We proved
in [15], [18] that an homogeneous algebra Bn has a PBW basis if and only if
Bn /(Z-1)Bn has a PBW basis, moreover An is isomorphic to Bn /(Z-1)Bn .We
called to the relation between An and Bn , the homogenization deshomoge-
nization process. The homogeneous version Bn of the G-algebra An has the
advantage of being Koszul so we can use Koszul theory to relate the Koszul
Bn -modules with the Koszul modules over the Yoneda algebra B!n , this has
been the approach we followed in [15],[18],[19].
The paper consists of two parts. The first one is dedicated to the study
of Ext and Tor for a homogenized G-algebra Bn and their relations with the
corresponding Ext and Tor of the graded localization BnZ , and those of the
deshomogenized G-algebra Bn /(Z-a)Bn .
In the second part of the paper we look to the Yoneda algebra Bn! of a
homogenized G-algebra Bn with quantized relations. The algebra Bn! has
as a subalgebra the quantized exterior algebra Cn! , we prove that there is
graded derivation ∂:Cn! → Cn! [1] with ∂ 2 =0, and that the tensor product
k[Z]/(Z)2 ⊗ Cn! , twisted with ∂ is isomorphic to Bn! . Conversely, given a graded
derivation ∂ of Cn! with ∂ 2 =0, the skew tensor product k[Z]/(Z)2 ⊗ Cn! is iso-
morphic to the Yoneda algebra Bn! of a homogeneous G-algebra Bn . Using
this characterization we describe the structure of constants of an arbitrary
quadratic G-algebra Bn . We end the paper with the remark that the quan-
tized polynomial algebra Cn and the quantized exterior algebra Cn! have a
Hopf structure under a skew tensor product that generalizes the graded tensor
product.


2 Ext and Tor for homogeneous G-algebras
and the des homogenization process
2.1 The homogenization des homgenization of
k[Z]-modules
Through the paper k will denote a field of characteristic zero. In this section
we study in further detail the properties of the graded localization k[Z]Z of the

, G-algebras, Lie, Hopf 353


polynomial ring in one variable, considered in our previous papers. We start
recalling results from [15], [16], [18],[19] stating them in the form that we need
for this paper.
We consider next the graded functors Ext∗Bn (-,?) and TorB ∗ (-,?) and their
n

relations with the functors Ext∗BnZ (-,?) and TorB ∗
nZ
(-,?) over the graded local-
∗ B /(z−a)Bn
ization and the functors ExtBn /(z−a)Bn (-,?) and Tor∗ n (-,?) over the des
homogenized algebra Bn /(z-a)Bn .
The ring k[Z] has the usual graduation. We denote by k[Z]Z the localiza-
tion k[Z]S with S the multiplicative set S={1,Z,Z2 ,,,Zn ,,,}. The ring k[Z]Z is
Z-graded with homogeneous elements of the degree m, (k[Z]Z )m ={aZ m | a ∈
k}=kZ m . Hence the homogeneous elements of degree m form a one dimen-
sional k-vector space.
For a∈ k-{0} (Z-a) is a maximal ideal of k[Z], and k[Z]/(Z-a) is a one
dimensional k-vector space.
j π
We have a composition of algebra maps: k[Z]→ k[Z]Z → k[Z]Z /(Z-a)k[Z]Z
with ϕ=πj given by ϕ(f)=f+(Z-a)k[Z]Z . The map ϕ is onto and has kernel
(Z-a)k[Z].
We have:

Proposition 2.1. There are ring isomorphisms: k[Z]/(Z-a)k[Z]∼
= k[Z]Z /
= k, and for any m∈ Z (k[Z]Z )m =kZm ∼
(Z-a)k[Z]Z ∼ = k.

The proposition can be generalized for Z-graded k[Z]-module M, to do this
we need first the following:

Definition 2.2. For a Z-graded k[Z]-module M we define the Z-torsion part
as:
tZ (M)={m∈M|there is a non negative integer k such that Zk m=0}
and the usual torsion part
t(M)={m∈M|there is f∈ k[Z] f6= 0 and fm=0}

Clearly tZ (M)⊆t(M).

Lemma 2.3. Let M be a Z-graded k[Z]-module. Then tZ (M)=t(M). In par-
ticular t(M) is Z-graded.

Proof. Let m=mk +mk−1 +...m1 be the decomposition in homogeneous compo-
nents of m∈t(M) and assume deg(mi )≥deg(mi−1 ) and let f(Z)=c0 +c1 Z+...c` Z`
be a non zero polynomial of degree ` such
P that f(Z)m=0. Then f(Z)m has a ho-
mogeneous components (f(Z)m)n = ci Z mj =0. In particular, c` Z` mk =0
i
deg(mj )+i=n
and mk ∈ tZ (M)⊆t(M). Then m-mk ∈t(M) and m-mk = mk−1 +...m1 .It fol-
lows by induction on the number of homogeneous summands that each mi ∈
tZ (M).