Abstract The purpose of this note is to add to the evidence that the algebra structure of a p-block of a finite group is closely related to the Lie algebra structure of its first Hochschild cohomology group. We show that if B is a block of a finite group algebra kG over an algebraically closed field k of prime characteristic p such that HH1(B) is a simple Lie algebra and such that B has a unique isomorphism class of simple modules, then B is nilpotent with an elementary abelian defect group P of order at least 3, and HH1(B) is in that case isomorphic to the Witt algebra HH1(kP) . In particular, no other simple modular Lie algebras arise as HH1(B) of a block B with a single isomorphism class of simple modules. 1. Introduction Let p be a prime and k an algebraically closed field of characteristic p . For G a finite group, a block of kG is an indecomposable direct factor of the group algebra kG . There is an abundance of stable equivalences in block theory, but it is notoriously difficult to pin down even the most basic numerical invariants—such as the number of isomorphism classes of simple modules—through stable equivalences. The main motivation for the present note is that on the one hand, the Lie algebra structure of HH1(B) of a block B of a finite group algebra kG is invariant under stable equivalences of Morita type (cf. [4, Theorem 10.7]), and on the other hand, there is evidence for close structural connections between the algebra structure of B and the Lie algebra structure of HH1(B) (cf. [1]). Understanding those connections might therefore ultimately contribute towards determining block invariants in some cases. We describe some of the structural connections between B and HH1(B) in two extreme cases for blocks with a single isomorphism class of simple modules. Theorem 1.1 Let Gbe a finite group and let Bbe a block algebra of kGhaving a unique isomorphism class of simple modules. Then HH1(B)is a simple Lie algebra if and only if Bis nilpotent with an elementary abelian defect group Pof order at least 3. In that case, we have a Lie algebra isomorphism HH1(B)≅HH1(kP) . Theorem 1.1 implies in particular that none of the other simple modular Lie algebras occur as HH1(B) of some block algebra of a finite group with the property that B has a single isomorphism class of simple modules. See [7, 8] for details and further references on the classification of simple Lie algebras in positive characteristic. We do not know whether the hypothesis on B to have a single isomorphism class of simple modules is necessary in Theorem 1.1. Theorem 1.2 Let Gbe a finite group and let Bbe a block algebra of kGhaving a non-trivial defect group and a unique isomorphism class of simple modules. Then dimk(HH1(B))≥ 2. The hypothesis that B has a single isomorphism class of simple modules is necessary in Theorem 1.2. For instance, if P is cyclic of order p≥ 3 and if E is the cyclic automorphism group of order p−1 of P , then HH1(k(P⋊E)) has dimension one. This follows immediately from the centralizer decomposition of Hochschild cohomology; see [3, Theorem 1.4] for a more general result. 2. Quoted results We collect in this section results needed for the proof of Theorem 1.1. Theorem 2.1 (Okuyama and Tsushima [5]) Let Gbe a finite group and Ba block algebra of kG . Then Bis a nilpotent block with an abelian defect group if and only if J(B)=J(Z(B))B . Let A be a finite-dimensional (associative and unital) k-algebra. A derivation on A is a k-linear map f:A→A satisfying f(ab)=f(a)b+af(b) for all a , b∈A . The set Der(A) of derivations on A is a Lie subalgebra of Endk(A) , with respect to the Lie bracket [f,g]=f◦g−g◦f , for any f , g∈ Endk(A) . For c∈A , the map sending a∈A to the additive commutator [c,a]=ca−ac is a derivation on A ; any derivation arising this way is called an inner derivation on A . The set IDer(A) of inner derivations is a Lie ideal in Der(A) , and we have a canonical identification HH1(A)≅Der(A)/IDer(A) . See [9, Chapter 9] for more details on Hochschild cohomology. If A is commutative, then HH1(A)≅Der(A) . A k-algebra A is symmetric if A is isomorphic to its k-dual A* as an A– A-bimodule; this implies that A is finite-dimensional. Theorem 2.2 ([1, Theorem 3.1]) Let Abe a symmetric k-algebra and let Ebe a maximal semisimple subalgebra. Let f:A→Abe an E– E-bimodule homomorphism satisfying E+J(A)2⊆ker(f)and Im(f)⊆soc(A) . Then fis a derivation on Ain socZ(A)(Der(A)) , and if f≠ 0, then fis an outer derivation of A . In particular, we have ∑Sdimk(ExtA1(S,S))≤dimk(socZ(A)(HH1(A))),where in the sum Sruns over a set of representatives of the isomorphism classes of simple A-modules. Corollary 2.3 ([1, Corollary 3.2]) Let Abe a local symmetric k-algebra. Let f:A→Abe a k-linear map satisfying k·1+J(A)2⊆ker(f)and Im(f)⊆soc(A) . Then fis a derivation on Ain socZ(A)(Der(A)) , and if f≠0 , then fis an outer derivation of A . In particular, we have dimk(J(A)/J(A)2)≤dimk(socZ(A)(HH1(A))). Theorem 2.4 (Jacobson [2, Theorem 1]) Let Pbe a finite elementary abelian p-group of order at least 3. Then HH1(kP)is a simple Lie algebra. The converse to this theorem holds as well. Proposition 2.5 Let Pbe a finite abelian p-group. If HH1(kP)is a simple Lie algebra, then Pis elementary abelian of order at least 3. Proof Suppose that P is not elementary abelian; that is, its Frattini subgroup Q=Φ(P) is nontrivial. Since P is abelian, we have HH1(kP)=Der(kP) . We will show that the set of derivations with image contained in I(kQ)kP=ker(kP→kP/Q) is a non-zero Lie ideal in Der(kP) , where I(kQ) is the augmentation ideal of kQ . Indeed, every element in Q is equal to xp for some x∈P , and hence every element in I(kQ) is a linear combination of elements of the form (x−1)p , where x∈P . Every derivation on kP annihilates all elements of this form (using the fact that k has characteristic p ), and hence every derivation on kP preserves I(kQ)kP . Thus there is a canonical Lie algebra homomorphism Der(kP)→Der(kP/Q) . This homomorphism is non-zero; indeed, it is an isomorphism on the components of Hochschild cohomology corresponding to H1(P;k)≅H1(P/Q;k) under the centralizer decomposition. The kernel of this canonical Lie algebra homomorphism contains all derivations with image in soc(kP) , so this kernel is non-zero by Corollary 2.3. Thus HH1(kP) is not simple as a Lie algebra, whence the result.□ Remark 2.6 Theorem 1.1 implies that the hypothesis on P being abelian is not necessary in the statement of Proposition 2.5. 3. Auxiliary results In order to exploit the hypothesis on HH1 being simple in the statement of Theorem 1.1, we consider Lie algebra homomorphisms into the HH1 of subalgebras and quotients. Lemma 3.1 Let Abe a finite-dimensional k-algebra and fa derivation on A . Then fsends Z(A)to Z(A) , and the map sending fto the induced derivation on Z(A)induces a Lie algebra homomorphism HH1(A)→HH1(Z(A)) . Proof Let z∈Z(A) . For any a∈A we have az=za , hence f(az)=f(a)z+af(z)=f(z)a+zf(a)=f(za). Comparing the two expressions, using zf(a)=f(a)z , yields af(z)=f(z)a , and hence f(z)∈Z(A) . The result follows.□ Lemma 3.2 Let Abe a local symmetric k-algebra such that J(Z(A))A≠J(A) . Then the canonical Lie algebra homomorphism HH1(A)→HH1(Z(A))is not injective. Proof Since J(Z(A))A<J(A) , it follows from Nakayama’s lemma that J(Z(A))A+J(A)2<J(A) . Thus there is a non-zero linear endomorphism f of A which vanishes on J(Z(A))A+J(A)2 and on k·1A , with image contained in soc(A) . In particular, f vanishes on Z(A)= k·1A+J(Z(A)) . By Corollary 2.3, the map f is an outer derivation on A . Thus the class of f in HH1(A) is non-zero, and its image in HH1(Z(A)) is zero, whence the result.□ Lemma 3.3 Let Abe a local symmetric k-algebra and let fbe a derivation on Asuch that Z(A)⊆ker(f) . Then f(J(A))⊆J(A) . Proof Since A is local and symmetric, we have soc(A)⊆Z(A) , and J(A) is the annihilator of soc(A) . Let x∈J(A) and y∈soc(A) . Then xy=0 , hence 0=f(xy)=f(x)y+xf(y) . Since y∈soc(A)⊆Z(A) , it follows that f(y)=0 , hence f(x)y=0 . This shows that f(x) annihilates soc(A) , and hence that f(x)∈J(A) .□ Lemma 3.4 Let Abe a finite-dimensional k-algebra and Jan ideal in A . Let fbe a derivation on Asuch that f(J)⊆J . Then f(Jn)⊆Jnfor any positive integer n . Let f , gbe derivations on Aand let m , nbe positive integers such that f(J)⊆Jmand g(J)⊆Jn . Then [f,g](J)⊆Jm+n−1 . Proof In order to prove (i), we argue by induction over n . For n=1 there is nothing to prove. If n>1 , then f(Jn)⊆f(J)Jn−1+Jf(Jn−1) . Both terms are in Jn , the first by the assumptions, and the second by the induction hypothesis f(Jn−1)⊆Jn−1 . Let y∈J . Then [f,g](y)=f(g(y))−g(f(y)) . We have g(y)∈Jn ; that is, g(y) is a sum of products of n elements in J . Applying f to any such product shows that the image is in Jm+n−1 . A similar argument applied to g(f(y)) implies (ii).□ Proposition 3.5 Let Abe a finite-dimensional k-algebra. For any positive integer mdenote by Der(m)(A)the k-subspace of derivations fon Asatisfying f(J(A))⊆J(A)m . For any two positive integers mand nwe have [Der(m)(A),Der(n)(A)]⊆Der(m+n−1)(A) . The space Der(1)(A)is a Lie subalgebra of Der(A) . For any positive integer m , the space Der(m)(A)is an ideal in Der(1)(A) . Suppose that Ais local. The space Der(2)(A)is a nilpotent Lie subalgebra of Der(A) . Proof Statement (i) follows from Lemma 3.4 (ii). The statements (ii) and (iii) are immediate consequences of (i). Since A is local and since 1 is annihilated by any derivation on A , statement (iii) follows from (i) and the fact that J(A) is nilpotent.□ 4. Proof of Theorems 1.1 and 1.2 Let G be a finite group and B a block of kG . Suppose that B has a single isomorphism class of simple modules. If B is nilpotent and P a defect group of B , then by [6], B is Morita equivalent to kP , and hence there is a Lie algebra isomorphism HH1(B)≅HH1(kP) . Thus if B is nilpotent with an elementary abelian defect group P of order at least 3, then HH1(B) is a simple Lie algebra by Theorem 2.4. Suppose conversely that HH1(B) is a simple Lie algebra. If J(B)= J(Z(B))B , then B is nilpotent with an abelian defect group P by Theorem 2.1. As before, we have HH1(B)≅ HH1(kP) , and hence Proposition 2.5 implies that P is elementary abelian of order at least 3. Suppose that J(Z(B))B≠J(B) . Let A be a basic algebra of B . Then J(Z(A))A≠J(A) . Moreover, A is local symmetric, since B has a single isomorphism class of simple modules. Thus soc(A) is the unique minimal ideal of A . We have J(A)2≠ {0} . Indeed, if J(A)2={0} , then soc(A) contains J(A) , and hence J(A) has dimension 1, implying that A has dimension 2. In that case B is a block with defect group of order 2. But then HH1(A)≅HH1(kC2) is not simple, a contradiction. Thus J(A)2≠{0} , and hence soc(A)⊆J(A)2 . By Lemma 3.2, the canonical Lie algebra homomorphism HH1(A)→ HH1(Z(A)) is not injective. Since HH1(A) is a simple Lie algebra, it follows that this homomorphism is zero. In other words, every derivation on A has Z(A) in its kernel. It follows from Lemma 3.3 that every derivation on A sends J(A) to J(A) . Thus, by Lemma 3.4, every derivation on A sends J(A)2 to J(A)2 . This implies that the canonical surjection A→ A/J(A)2 induces a Lie algebra homomorphism HH1(A)→ HH1(A/J(A)2) . Note that the algebra A/J(A)2 is commutative, and hence HH1(A/J(A)2)=Der(A/J(A)2) . Since J(A)2 contains soc(A) , it follows that the kernel of the canonical map HH1(A)→HH1(A/J(A)2) contains the classes of all derivations with image in soc(A) . Since there are outer derivations with this property (cf. Corollary 2.3), it follows from the simplicity of HH1(A) that the canonical map HH1(A)→HH1(A/J(A)2)=Der(A/J(A)2) is zero. Thus every derivation on A has image in J(A)2 . But then Proposition 3.5 implies that Der(A)=Der(2)(A) is a nilpotent Lie algebra. Thus HH1(A) is nilpotent, contradicting the simplicity of HH1(A) . The proof of Theorem 1.1 is complete. Proof of Theorem 1.2 Denote by A a basic algebra of B . Since B has a unique isomorphism class of simple modules and a non-trivial defect group, it follows that A is a local symmetric algebra of dimension at least 2. By Corollary 2.3, we have dimk(HH1(A))≥ dimk(J(A)/J(A)2) . Thus dimk(HH1(A))≥1 . Moreover, if dimk(HH1(A))=1 , then dimk(J(A)/J(A)2)=1 , and hence A is a uniserial algebra. In that case B is a block with a cyclic defect group P and a unique isomorphism class of simple modules, and hence B is a nilpotent block. Thus A≅kP . We have dimk(HH1(kP))=∣P∣ , a contradiction. The result follows.□ Remark 4.1 All finite-dimensional algebras in this paper are split thanks to the assumption that k is algebraically closed. It is not hard to see that one could replace this by an assumption requiring k to be a splitting field for the relevant algebras. The statements 3.1 and 3.4 do not require any hypothesis on k . Funding The present paper was partially funded by EPSRC grant EP/M02525X/1 of the first author. Acknowledgements The authors would like to thank the referee for his or her helpful comments and suggestions. References 1 D. Benson , R. Kessar and M. Linckelmann , On blocks of defect two and one simple module, and Lie algebra structure of HH1 , J. Pure Appl. Algebra 221 ( 2017 ), 2953 – 2973 . Google Scholar Crossref Search ADS 2 N. Jacobson , Classes of restricted Lie algebras of characteristic p, II , Duke Math. J. 10 ( 1943 ), 107 – 121 . Google Scholar Crossref Search ADS 3 R. Kessar and M. Linckelmann , On Blocks with Frobenius Inertial Quotient , J. Algebra 249 ( 2002 ), 127 – 146 . Google Scholar Crossref Search ADS 4 S. König , Y. Liu and G. Zhou , Transfer maps in Hochschild cohomology and applications to stable and derived invariants and to the Auslander–Reiten conjecture , Trans. Amer. Math. Soc. 364 ( 2012 ), 195 – 232 . Google Scholar Crossref Search ADS 5 T. Okuyama and Y. Tsushima , Local properties of p-block algebras of finite groups , Osaka J. Math. 20 ( 1983 ), 33 – 41 . 6 L. Puig , Nilpotent blocks and their source algebras , Invent. Math. 93 ( 1988 ), 77 – 116 . Google Scholar Crossref Search ADS 7 H. Strade , Simple Lie Algebras over Fields of Positive Characteristic, I. de Gruyter Expositions in Mathematics 38, de Gruyter, Berlin, 2004 , viii+540pp. 8 H. Strade , Simple Lie Algebras over Fields of Positive Characteristic, II. de Gruyter Expositions in Mathematics 42, de Gruyter, Berlin, 2009 , vi+385pp. 9 C. A. Weibel , An introduction to homological algebra, Cambridge Studies Adv. Math. Vol. 38, Cambridge University Press , Cambridge , 1994 . © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
The Quarterly Journal of Mathematics – Oxford University Press
Published: Dec 1, 2018
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