On the Lie transformation algebra of monoids in symmetric monoidal categories

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On the Lie transformation algebra of monoids in symmetric monoidal categories

ABHISHEKBANERJEE

ABSTRACT- We define the Lie transformation algebra of a (not necessarily associative) monoid objectAin aK-linear symmetric monoidal category (C;;1), whereK is a field. WhenAis associative and satisfies certain conditions, we describe explicity the Lie transformation algebra and inner derivations ofA.

Additionally, we also show that derivations preserve the nucleus of the monoidA

MATHEMATICSSUBJECTCLASSIFICATION(2010). 17A36, 18D10.

KEYWORDS. Inner derivations, Lie transformation algebra.

1. Introduction

Given an associativeZ-algebraAand an elementa2A, the morphism D_a:Aÿ!Adefined byD_a(x):axÿxadefines an inner derivation onA.

However, if A is not associative, the morphism D_a is not necessarily a derivation. For nonassociative algebras, a theory of inner derivations has been developed by Schafer [4]. The purpose of this paper is to extend this theory to monoids over aK-linear symmetric monoidal category (C;;1), whereKis a field.

More precisely, letAbe a (not necessarily associative) unital monoid object in a K-linear symmetric monoidal category (C;;1). A morphism f :1ÿ!Ainduces a morphismLA(f):Aÿ!A(resp. RA(f):Aÿ!A) by left multiplication (resp. right multiplication) on the monoidA(in the sense of (2.9)). We consider the subspaceL(A) (resp.R(A)) ofHom(A;A) generated by morphisms of the formL_A(f) (resp.R_A(f)). Then, we start by defining the Lie transformation algebraL(A) ofAto be the smallest Lie algebra con-

(*) Indirizzo dell'A.: ColleÁge de France, 3, rue d'Ulm, 75231, Paris cedex 05, France.

E-mail: [email protected]

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taining the subspaceL(A) R(A) ofHom(A;A). IfDis a derivation onA(see Definition 2.10), we will say thatDis an inner derivation ifD2L(A).

When Ais an associative monoid object, we show that the Lie transformation algebra L(A) ofAis actually equal toL(A) R(A). In particular, ifAis associative and has no left (or right) ``absolute divisors of zero'', we show that a derivationDonAis inner if and only ifDis of the form DLA(f)ÿRA(f) for some morphismf :1ÿ!A. Moreover, we verify that for any (not necessarily associative) monoidA, the collection of inner derivations is always an ideal in the Lie algebraDer(A) of derivations onA.

Finally, we also show that iff :1ÿ!Ais a morphism in the nucleus ofA (see (2.13)), for any derivationD2Der(A),Dfis also in the nucleus ofA.

We mention here that the notion of derivations on monoid objects ap- pears elsewhere in the literature (see, for instance, Baues, Jibladze and Tonks [1]). For more on derivations and nonassociative algebras, we refer the reader to Jacobson [2] and Schafer [4], [5].

2. Derivations on monoids

Let (C;;1) be a K-linear symmetric monoidal category. Since Cis symmetric, for every pairX,Y of objects inC, we have an isomorphism:

tX;Y :XY ÿ! YXsuch thattX;YtY;X 1YXandtY;XtX;Y1XY. When there is no danger of confusion, we shall omit the subscripts and simply writet:XYÿ! YX. Further, for any objectXinC, we have two isomorphisms lX:Xÿ! 1X and rX :Xÿ! X1 satisfying rX tlX.

Given (C;;1), we shall letMon(C) denote the category of unital, not necessarily associative, monoids object inC. For any monoidAinC, we will denote bym_A:AAÿ!Aande_A:1ÿ!Aresp. the ``multiplication map'' and the ``unit map'' on the monoidA. We start by defining the notion of a derivation onA.

DEFINITION2.1. Let A be an object of Mon(C). A morphism D:Aÿ!A is referred to as a derivation on A if it satisfies the following condition:

mA(D11D)DmA:AAÿ!A (2:1)

Given derivationsD,D⁰of a monoidA, it may be easily verified that the commutator [D;D⁰]:DD⁰ÿD⁰D is also a derivation on A. Then, since the categoryCisK-linear, the spaceDer(A) of derivations onAis a Lie algebra.

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Further, for any monoidA, given a morphismf :1ÿ!A, we define:

L_A(f):ÿ Aÿ!^l^A

1Aÿ!^f¹AAÿ!^m^A A RA(f):ÿ

Aÿ!^r^A

A1ÿ!¹^f AAÿ!^m^A A :ÿ

Aÿ!^l^A

1Aÿ!^f¹ AAÿ!^m^A^t A (2:2)

We denote by L(A) (resp. R(A)) the subspace of Hom(A;A) generated by morphisms of the formLA(f) (resp.RA(f)) wheref 2Hom(1;A). We will say that f :1ÿ!A is a left (resp. right) absolute divisor of zero if L_A(f)0 (resp.R_A(f)0).

Suppose that M : L(A) R(A)Hom(A;A). If we define the se- quence of spacesfMig_i2N as follows:

M₁M M_i:[M₁;M_iÿ1]; i2;3; ::::

(2:3)

then, as in [4, § 2], the spaceL(A):M1M2. . .is the smallest Lie algebra containing M1M. Then L(A) is referred to as the Lie transformation algebra ofA. Following [4], we will say that a derivation D:Aÿ!Ais inner ifD2L(A).

In particular, suppose that Ais an associative monoid. Then, for any morphism f :1ÿ!A, it is known (see [1, § 4]) that LA(f)ÿRA(f)2 Hom(A;A) is a derivation onA. Further, for anyf;g:1ÿ!A, it follows from associativity ofAthat either of the compositionsL_A(f)R_A(g) and R_A(g)L_A(f) is equal to the composition

(2:4) L_A(f)R_A(g)

RA(g)LA(f):Aÿ! 1(A1)!^f⁽¹^g) A(AA) !^mÂ⁽¹^mÂ⁾ A i.e., we have [L_A(f);R_A(g)]0. We will denote bym(f;g) the morphism m(f;g):1ÿ! 11ÿ!^f^g AAÿ!^mÂ A. We now have the following result.

PROPOSITION2.2. Let A be an associative monoid object in(C;;1).

Then, the Lie transformation algebraL(A)of A is given by L(A) L(A) R(A)

(2:5)

Further, if A has no left (resp. right) absolute divisor of zero, then D2Hom(A;A)is an inner derivation if and only if DL_A(f)ÿR_A(f) for some f :1ÿ!A.

PROOF. Let us choose elementsL_A(f)R_A(g),L_A(f⁰)R_A(g⁰)2 L(A) R(A) for morphisms f;f⁰;g;g⁰:1ÿ!A. Then, since [L_A(f);R_A(g⁰)]

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[LA(f⁰);RA(g)]0, it follows that

[LA(f)RA(g);LA(f⁰)RA(g⁰)][LA(f);LA(f⁰)][RA(g);RA(g⁰)]

(2:6)

We now notice that:

(2:7)

LA(f)LA(f⁰)

Aÿ! 1Aÿ!^f⁰¹AAÿ!^m^A Aÿ! 1Aÿ!^f¹ AAÿ!^m^A A

Aÿ! 1 1A!^f^f

01

A AA^m^A¹^m!^a A

Aÿ! 11A!^f^f

0 1

AAA^m^A^m^a!¹ A

LAmf;f⁰

Hence, it follows that [L_A(f);L_A(f⁰)]L_A(m(f;f⁰)ÿm(f⁰;f)). Similarly, we may show thatRA(g)RA(g⁰)RA(m(g⁰;g)) and hence [RA(g);RA(g⁰)] R_A(m(g⁰;g)ÿm(g;g⁰)). From (2.6), it follows that

(2:8) [L_A(f)R_A(g);L_A(f⁰)R_A(g⁰)]

L_A(m(f;f⁰)ÿm(f⁰;f))ÿR_A(m(g⁰;g)ÿm(g;g⁰)) and henceL(A) R(A) is a Lie algebra. Since L(A) is the smallest Lie algebra containingL(A) R(A), we haveL(A) L(A) R(A).

We now suppose that L_A(f)R_A(g)2L(A) is a derivation. Since L_A(g)ÿR_A(g) is a derivation as mentioned before, so is L_A(fg) (LA(f)RA(g))(LA(g)ÿRA(g)). Since A is associative, the following commutative diagram

AA^l^A!¹ 1AA!^f^g¹¹AA A! AAA

#

# ^m^A¹

# AA!^l^AA 1 AA!^f^g11A AA AA

m_A

# ^1m^A

# ^m^A

# A ^l^A! 1A ^f^g1! AA !^m^A A (2:9)

shows that m_A(L_A(fg)1)L_A(f g)m_A:AAÿ!A. Since L_A(fg) is a derivation, it follows from (2.1) thatm_A(1L_A(fg))0.

On the other hand, we note that

(2:10) 0m_A(1L_A(fg))(e_A1)l_AL_A(m(e_A;f g)) L_A(e_A)L_A(f g)L_A(fg)

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and henceLA(fg)0. Now, ifAhas no left absolute divisors of zero, it follows thatLA(f g)0 and the inner derivationLA(f)RA(g)2L(A) is actually of the form L_A(f)R_A(g)R_A(g)ÿL_A(g)L_A(ÿg)ÿ R_A(ÿg). The result follows similarly for the case of no right absolute di-

visors of zero. p

PROPOSITION 2.3. Let A be an object of Mon(C). Then, the space L(A)\Der(A)of inner derivations on A is an ideal in the Lie algebra Der(A).

PROOF. We choose a morphism f :1ÿ!A and some D2Der(A).

Our first objective is to show that [D;L_A(f)]DL_A(f)ÿL_A(f)D L_A(Df). For this, we note that:

DLA(f) DmA(f1)lA

m_A(D11D)(f1)l_A

m_A((Df)1)l_Am_A(1D)(f 1)l_A

LA(Df)mA(f1)(1D)lA

L_A(Df)m_A(f1)l_AD

L_A(Df)L_A(f)D (2:11)

It follows from (2.11) that [D;L(A)] L(A). Similarly, we may show that [D;R(A)] R(A). It follows, therefore, that forM₁M L(A) R(A), [D;M₁]M₁.

We now suppose that [D;Mj]Mjfor alljifor some giveni. Then, given any elementD⁰2M_i1, by definition ofM_i1 in (2.3),D⁰ may be written as a sumD⁰P^k

l1D⁰_l with eachD⁰_l2[M1;Mi]. We now note that for each 1lk, we have

(2:12) [D;D⁰_l]2[D;[M1;Mi]][M1;[D;Mi]][Mi;[D;M1]]

[M1;Mi][Mi;M1]Mi1

From (2.12), it follows that [D;M_i1]M_i1 and hence [D;M_i]M_i for all i1 by induction. It follows that [D;L(A)]L(A). Since [D;Der(A)]Der(A), it follows that [D;L(A)\Der(A)]L(A)\Der(A).

p REMARK2.4. It follows from Proposition2.3that if A is a monoid such that the derivation algebra Der(A)is simple (as a Lie algebra) and there exist non zero inner derivations of A, then every derivation of A is inner.

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Given a monoid objectA, we will say that a morphism f:1ÿ!Ais in the nucleus ofAif each of the following three morphisms is identically 0:

(2:13) A0(f):

AA!^lÂ¹ (1A)A^f¹¹!(AA)A!^mÂ^(mÂ^1ÿ(1mÂ^)a) A A1(f):

AA!^1lÂ A(1A)¹⁽^f¹⁾!A(AA)!^mÂ^((mÂ^1)a

ÿ1ÿ1m_A)

A A2(f):

AA!^1rÂ A(A1)!^1(1f) A(AA)!^mÂ^((mÂ^1)a

ÿ1ÿ1m_A)

A whereais the natural isomorphisma:(AA)Aÿ! A(AA). The set of all morphisms in the nucleus ofAwill be denoted byNuc(A) (compare [3, § 1.13]). Clearly, ifAis associative,Nuc(A)Hom(1;A).

PROPOSITION2.5. Let A be an object of Mon(C). Let D be a derivation on A and let f :1ÿ!A be an element of the nucleus of A. Then, Df 2Nuc(A).

PROOF. Since f2Nuc(A), we know that A₀(f)A₁(f)A₂(f)0 as defined in (2.13). We proceed as follows:

A0(Df)

mA(mA1ÿ(1mA)a)((Df 1)1)(lA1)

m_A(m_A1ÿ(1m_A)a)((D1)1)((f 1)1)(l_A1)

m_A((Dm_Aÿm_A(1D))1)((f1)1)(l_A1) ÿmA(D1)(1mA)a((f1)1)(lA1)

(m_A(D1)(m_A1)ÿm_A(m_A1)((1D)1)

ÿDm_A(1m_A)am_A(1Dm_A)a)((f1)1)(l_A1)

(DmA(mA1)ÿmA(1D)(mA1)ÿmA(mA1)((1D)1) ÿDm_A(1m_A)am_A(1Dm_A)a)((f 1)1)(l_A1)

DA₀(f)ÿm_A((m_A1)((11)D)(m_A1)((1D)1) ÿ(1DmA)a)((f1)1)(lA1)

ÿmA(mA1)((1D)1(11)D)((f1)1)(lA1)

m_A(1m_A)(1(D1)1(1D))a((f 1)1)(l_A1)

m_A((1m_A)aÿm_A1)((f 1)1)((1D)1)(l_A1)

m_A((1m_A)aÿm_A1)((f1)1)((11)D)(l_A1)

ÿA₀(f)(D1)ÿA₀(f)(1D)0

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Similarly, one may check that bothA1(Df) andA2(Df) are 0. Hence,

Df 2Nuc(A). p

REFERENCES

[1] H.-J. BAUES, M. JIBLADZE, A. TONKS, Cohomology of monoids in monoidal categories. Operads: Proceedings of Renaissance Conferences (Hartford, CT/

Luminy, 1995), 137-165, Contemp. Math.,202, Amer. Math. Soc., Providence, RI, 1997.

[2] N. JACOBSON,Derivation algebras and multiplication algebras of semi-simple Jordan algebras.Ann. of Math. (2)50(1949), 866-874.

[3] O. LOOS, P. H. PETERSSON, M. L. RACINE, Inner derivations of alternative algebras over commutative rings. Algebra Number Theory2 (2008), no. 8, 927-968.

[4] R. D. SCHAFER, Inner derivations of non-associative algebras. Bull. Amer.

Math. Soc.55(1949), 769-776.

[5] R. D. SCHAFER,An introduction to nonassociative algebras.Pure and Applied Mathematics,22, Academic Press, New York-London 1966.

Manoscritto pervenuto in redazione il 7 Ottobre 2012.

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