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On Graded Global Dimension of Color Hopf Algebras | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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On Graded Global Dimension of Color Hopf Algebras

Yan-HuaWang

Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China. Email: [email protected]

Received Date: 21 January 2011; Accepted Date: 3 April 2011

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Abstract

In this paper, we prove the fundamental theorem of color Hopf module similar to the fundamental theorem of Hopf module. As an application, we prove that the graded global dimension of a color Hopf algebra coincides with the projective dimension of the trivial module K.

Introduction

Let G be a group. The notion of color Hopf algebras first appeared in the book of Montgomery [6, 10.5.11]. The most important examples are Li-Zhang’s twisted Hopf algebras in [4], universal enveloping algebras of Lie superalgebras and universal enveloping algebras of color Lie algebras in [1] (or [2,6,9,11]). Roughly speaking, a color Hopf algebra means a G-graded algebra and G-graded coalgebra satisfying some compatibility conditions. Its unique difference from a Hopf algebra is that the comultiplication Δ : AAA is an algebra homomorphism, not for the componentwise multiplication on AA, but for the twisted multiplication on AA by Lusztig’s rule.

Lorenz-Lorenz proved that the global dimension of a Hopf algebra is exactly the projective dimension of the trivial module Equation; see [5, Section 2.4]. One may ask a similar question for color Hopf algebras. Following Schauenburg [10] and Doi [3], we prove the fundamental theorem of color Hopf module. As an application, we show that the graded global dimension of a color Hopf algebra coincides with the graded projective dimension of the trivial module Equation, which also is equal to the projective dimension of Equation.

The paper is organized as follows: in Section 2, we provide some background material for color Hopf algebras. In Section 3, we prove the fundamental theorem of color Hopf module; and we prove the main theorem: let A be a color Hopf algebra, then the graded global dimensional of A is equal to the (graded) projective dimensional of right A-module Equation, where Equation is viewed as the trivial graded right A-module via the counit of A; see Theorem 9.

Throughout, Equation will be a field. All algebras and coalgebras are over Equation. All unspecified spaces (algebras, coalgebras, etc.) are graded by the group G, all unadorned Hom and ⊗ are taken over Equation. Equation× denotes Equation \ {0}.

Preliminaries

Let G be a group with identity element e. We will write G as a multiplication group. An associative algebra A with unit 1A is said to be G-graded if there is a family {Ag | gG} of subspaces of A such that A = ⊕gGAg with 1AAe and AgAhAgh, for all g, hG. Any element aAg is called a homogenous element of degree g, and we write |a| = g. In this paper, all unspecified elements are homogenous.

A graded right A-module M is a right A-module with a decomposition M = ⊕gGMg such that MgAhMgh. We denote the module as MAM, mama for any mM, aA. Let M and N be graded right A-modules. Define

Equation

We obtain the category of graded right A-modules, denoted by A-gr; for details see [8]. A module M is said to be a gr-free module if M is isomorphic to a direct sum of graded modules of the form A(g); see [7, page 5]. In the following, we will refer to projective objects of A-gr as gr-projective modules.

Recall from [12] that a graded coalgebra C is a graded Equation-space C = ⊕gGCg with counit ε : CEquation and comultiplication Δ : CCC satisfying the following conditions: Equation and ε(Cg) = 0 for ge, gG.

A graded right A-comodule M is a right A-comodule with a decomposition M = ⊕gGMg such that such that ρ : MMA, where Equation for any mxM.

A bicharacter χ : G × GEquation× means

Equation

where g, h, lG and Equation× is the multiplication group of the unit in Equation.

Definition 1. A color Hopf algebra A is a 6-tuple (A, m, u, Δ, ε, S) such that

(G1) A = ⊕gGAg is a graded algebra with multiplication m : AAA and the unit map u : EquationA. In the meantime, (A, Δ, ε) is a graded coalgebra with respect to the same grading;

(G2) the counit ε : AEquation and comultiplication Δ : AAA are algebra maps in the sense that

Equation     (2.1)

(G3) the antipode S : AA is a graded map such that

Equation

for all homogenous elements aA, where Equation.

Remark 2. The antipode preserves the degree, that is, |S(a)| = |a| for all homogenous aA.

The antipode of color Hopf algebras has similar results with Hopf algebras; see [1] (compare with [12, page 74], and [4, Theorem 2.10]).

Lemma 3. Let A be a color Hopf algebra, then the antipode S satisfies

Equation     (2.2)

Graded global dimension of color Hopf algebras

Let M be a graded right A-comodule. The coinvariants of M form the set

Equation.

Note that McoA is a graded subspace of M.

Definition 4. Let A be a color Hopf algebra. A graded right color Hopf module is a graded Equation-space M such that

(1) M is a graded right A-module;

(2) M is a graded right A-comodule with comodule map ρ : MMA defined by Equation;

(3) ρ is a right A-module map, that is

Equation     (3.1)

Example 5. Let M be a graded Equation-space. Then we define on MA a graded right A-module structure by (ma)b = mab for any mM, a, bA, and a graded right A-comodule structure given by the map ρ : MAMAA, Equation for any mM, aA. Thus MA becomes a graded right color Hopf module with these two structures. Indeed

Equation

Lemma 6. Let A be a color Hopf algebra. If a, b ∈ A are homogenous, then

Equation.    (3.2)

Proof. If |a| ≠ e, then ε(a) = 0 and hence the equation holds. If |a| = e, then χ(|a|, |b|) = 1, thus ε(a)χ(|a|, |b|) = ε(a).

The following theorem can be viewed as the fundamental theorem of color Hopf module (compare with [12, page 84]).

Theorem 7. Let A be a color Hopf algebra and M be a graded right color Hopf module. Then Equation is a graded right color Hopf module, where McoA ⊗ A is a trivial right color Hopf module. In particular, M is a graded free right color Hopf module.

Proof. Consider the map α : MM defined by Equation for any mM. If mM, then

Equation

Thus α(m) ∈ McoA.

It makes then sense to define the map Equation, for all mM. Define map G : McoAAM by G(ma) = ma, for all mMcoA, aA. We will show that F is the inverse of G. Indeed, if mMcoA and aA, then Equation. Thus,

Equation

Hence, GF = id M and FG = Equation.

It remains to show that G is a morphism of a graded color Hopf module, that is, it is a morphism of a graded right A-module and a morphism of a graded right A-comodule.

The first assertion is clear since

Equation.

In order to show that G is a morphism of a graded right A-comodule, we have to prove that

(ρG)(ma) = (G ⊗ id)ρ(ma).

This is immediate since for maMcoAA we have

Equation

This ends the proof.

Proposition 8. Let A be a color Hopf algebra and M be a graded right A-module. Then M ⊗ A is a graded right color Hopf module using comodule map ρ = idMΔ.

Proof. Define the graded right A-module structure of MA as

Equation

Indeed, MA is a graded right A-module and for any a, b, cA, mM, we have

Equation

Since

Equation

we have ((ma)b)c = (ma)(bc). Thus MA is a graded right A-module.

Define the graded right A-comodule of MA as

Equation.

Then MA is a graded right A-comodule since

Equation

Thus MA is a right A-comodule.

Moreover, MA is a graded right color Hopf module. Since

Equation

This completes the proof.

We will refer to projective objects of graded A-module as gr-projective modules. Taking the notations of [7], we denote the graded global dimensional of A as gr. gl. dim A.

Theorem 9. Let A be a color Hopf algebra. Then one has

Equation

where gr. gl.dim A and gr. p. dim A denote the graded global dimension and graded projective dimension of A, respectively; p.dim A denotes the projective dimension of A.

Proof. Consider the projective resolution of Equation in the category of graded right A-modules:

Equation

Assume that M is a graded right A-module. Then for any graded right A-module P, we have a graded right A-module structure on MP with the action given by

Equation.

In this way, we obtain an exact sequence of graded right A-modules

Equation

We claim that this is a projective resolution of M and this will complete the proof.

Now we recall the degree-shift functor on A-gr. Let gG and M = ⊕gGMg be a graded right A-module. We can define a new graded right A-module M(g) which has the same module structure with M, and has the gradation given by M(g)h = Mgh for all hG (see [7,8]). Indeed, if P is a projective graded right A-module, then P is a direct summand in a free graded right A-module, thus Equation as a graded right A-module for some graded right A-module X and some set I. Then

Equation

where it is enough to show that each MA(g) is projective. Note MA(g) = (MA)(g), so we only prove that MA is projective. But this is true since MA has a graded right color Hopf module structure if we take the graded right A-module structure and graded right A-comodule structure as Proposition 8.

The last equality gr. p.dimA Equation = p.dimA Equation is derived from [7, I.2.7].

Acknowledgments

The author would like to thank Professor Quanshui Wu and Professor Xiaowu Chen for their discussions and help. This project was supported by the Chinese NSF (Grant no. 10901098).

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