JMATH Journal of Mathematics 2314-4785 2314-4629 Hindawi Publishing Corporation 10.1155/2016/3983895 3983895 Research Article Automorphisms and Inner Automorphisms Jaber Ameer 1 Yasein Moh’D 1 Hong Shaofang Department of Mathematics The Hashemite University Zarqa 13115 Jordan hu.edu.jo 2016 722016 2016 25 08 2015 31 12 2015 2016 Copyright © 2016 Ameer Jaber and Moh’D Yasein. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let K be a field of characteristic not 2 and let A=A0+A1 be central simple superalgebra over K, and let be superinvolution on A. Our main purpose is to classify the group of automorphisms and inner automorphisms of (A,) (i.e., commuting with ) by using the classical theorem of Skolem-Noether. Also we study two examples of groups of automorphisms and inner automorphisms on even central simple superalgebras with superinvolutions.

1. Introduction

An associative super ring R=R0¯+R1¯ is nothing but a (Z/2Z)-graded associative ring. A (Z/2Z)-graded ideal I=I0¯+I1¯ of an associative super ring R is called a superideal of R. An associative super ring R is simple if it has no nontrivial superideals. Let R be an associative super ring with 1R0¯; then R is said to be a division super ring if all nonzero homogeneous elements are invertible; that is, every 0rαRα has an inverse rα-1 necessarily in Rα.

Let K be a field of characteristic not 2; an associative (Z/2Z)-graded K-algebra A=A0¯+A1¯ is a finite dimensional central simple superalgebra over a field K, if Z^(A)=K is the center of A, where (1)Z^Aα=aαAαaαbβ=-1αβbβaα  bβAβ,and the only superideals of A are (0) and A itself. We say that A0¯A1¯ is the set of all homogeneous elements of A.

Finite dimensional central simple associative superalgebras over a field K are isomorphic to EndVMn(D), where D=D0+D1 is a finite dimensional associative division superalgebra over K; that is, all nonzero elements of Dα, α=0,1, are invertible, and V=V0+V1 is an n-dimensional D-superspace. If D1=0, the grading of Mn(D) is induced by that of V=V0+V1, A=Mp+q(D), p=dimDV0, and q=dimDV1, so p+q is a nontrivial decomposition of n. Meanwhile, if D10, then the grading of Mn(D) is given by (Mn(D))α=Mn(Dα), α=0,1.

For completeness, we recall the structure theorem for central simple associative division superalgebras.

Theorem 1 (Division Superalgebra Theorem [<xref ref-type="bibr" rid="B10">1</xref>, <xref ref-type="bibr" rid="B11">2</xref>]).

If D=D0+D1 is a finite dimensional associative division superalgebra over a field K, then exactly one of the following holds, where throughout E denotes a finite dimensional associative division algebra over K:

D=D0=E, and D1=0.

D=EKK[u], u2=λK×, D0=EK1, and D1=EKu.

D=E or M2(E); uD such that u2=λK/K2  (λK/α+α2αK if Char(K)=2)D0=CD(u), D1=SD(u), where CDu=dDdu=ud and SDu=dDdu=uσd, for some quadratic Galois extension K[u]D with Galois automorphism σ. Moreover, in the second case, u=01λ0(u=01λ1 if Char(K)=2) and K[u] does not embed in E.

Following  we say that a division superalgebra D is even if Z(D)D1=0; that is, D is even if its form is (i) or (iii), and we say that D is odd otherwise; that is, D is odd if its form is (ii). Also, if A=Mn(D) is a finite dimensional central simple superalgebra over a field K, then we say that A is an even K-superalgebra if D is even division superalgebra and A is odd K-superalgebra if D is odd division superalgebra.

In  Racine described all types of superinvolutions on A=Mp+q(D). It appears that if is a superinvolution on A such that (A0,) is simple algebra, then p=q and is conjugate to the transpose involution. Otherwise, is conjugate to the orthosymplectic involution.

In  we proved that if A is a finite dimensional central simple associative superalgebra over a field K of characteristic not 2 such that A has a superinvolution of the first kind, then A=Mp+q(D), where D is a division algebra over K.

In  we proved that A=Mn(D), where D10 has a pseudosuperinvolution of the first kind if and only if A is of order 2 in the Brauer-Wall group BW(K), where K is a field of characteristic not 2. But if K is a field of characteristic 2, and A is a central simple associative superalgebra over K, then a superinvolution (which is a pseudosuperinvolution) on A is just an involution on A respecting the grading. Moreover, if A is of order 2 in the Brauer-Wall group BW(K), then the supercenter of A equals the center of A and ^K=K, which means that A is of order 2 in the Brauer group Br(K). Thus, by theorem of Albert, A has an involution of the first kind, but since A is of order 2 in the Brauer-Wall group BW(K), A has superantiautomorphism of the first kind respecting the grading; therefore by [6, Chapter 8, Theorem  8.2] A has an involution of the first kind respecting the grading, which means that A has a superinvolution (which is a pseudosuperinvolution) of the first kind if and only if A is of order 2 in the Brauer-Wall group BW(K).

Let K/k be a separable quadratic field extension over k with Galois group Gal(K/k), where Gal(K/k)=1,σ and θ2k-k2 and σ(θ)=-θ (θ2k/α+α2αk, σ(θ)=θ+1 if Char(k)=2). We recall a theorem of Albert-Reihm on the existence of K/k-involution (involution of the second kind) which states that finite dimensional central simple algebra A over K has a K/k-involution if and only if the corestriction of A splits over k. In  Elduque and Villa gave much better exposition and motivation for the whole theory of the existence of superinvolutions.

Throughout this work we say that if u is homogeneous element in A, then ψu:AA is an inner automorphism on A, with conjugation by u. Also Aut(A,), where A is a central simple superalgebra of any type, means the set of all automorphisms on A commuting with a given superinvolution defined on A, and inAut(A,) means the set of all inner automorphisms on A commuting with .

In this paper we examine the characterization of automorphisms and inner automorphisms on (A,), where A is a central simple superalgebra of any type, which commute with a given superinvolution defined on A. Then we produce two examples of even superalgebras with superinvolution to investigate in detail groups of automorphisms and inner automorphisms in these examples.

2. Group of Automorphisms of Superalgebras Definition 2.

Let A be any K-superalgebra; we define the map φ:AA by (2)aαφ=-1αaαaαAα,  α=0¯,1¯.

This map, φ, is a superalgebra automorphism, called the sign automorphism, since (3)aαbβφ=-1α+βaαbβ=aαφbβφfor all aαAα and bβAβ. The automorphism φ has order 2, if Char(K)2 (unless A1¯=0), and φ=idA if Char(K)=2.

Lemma 3.

Let A=A0-+A0-v be an odd superalgebra over a field K with a superinvolution , and let φ be the sign automorphism. Then (4)AutA,=inAutA,φ·inAutA,.

Proof.

If ξAut(A,), then by [4, Theorem  3.2] Skolem-Noether theorem forces that ξ or ξφ is inner but not both of them, so, ξinAut(A,) or ξφinAut(A,).

Now, let be a superinvolution on an odd central simple superalgebra A=A0¯+A0¯v of any kind and let ξinAut(A,). Then ξ=ψcγ for some invertible cγAγ. But ξA0¯ is inner, so ξA0¯=ψb0¯. Therefore, cγ(b0¯-1ab0¯)cγ-1=a, aA0¯, and cγ(b0¯-1vb0¯)cγ-1=v which implies that ψcγb0¯-1=idAψcγ=ψb0¯. So, we can choose γ to be 0¯, and hence ψc0¯=ψb0¯c0¯=αb0¯ for some αK.

Now, aαψb0¯=aαψb0¯b0¯b0¯=λKb0¯=λb0¯-1. Let (5)G=b0¯A0¯:b0¯=αb0¯-1  for  some  αK×.Let ~ be a relation on G defined as follows: a0¯~b0¯b0¯a0¯-1=αK×. Then one can easily show that ~ is an equivalence relation on G, and G/~={[b0¯]b0¯G}, where b0¯={αb0¯αK×}, is a group with the well-defined operation [b0¯][c0¯]=[c0¯b0¯].

Theorem 4.

Let A be an odd central simple superalgebra over a field K with a superinvolution ; then inAut(A,)G/~.

Proof.

Let δ:inAut(A,)G/~ such that δ(ψb0¯)=[b0¯]; then (6)δψb0¯ψc0¯=δψc0¯b0¯=c0¯b0¯=b0¯c0¯=δψb0¯δψc0¯.So, δ is an onto homomorphism such that ker(δ)=ψ1. Therefore, inAut(A,)G/~.

Let A=Mn(D) be an even central simple superalgebra over K, where D is a nontrivial grading division superalgebra. Then by [1, Division Superalgebra Theorem] D=D0¯+D1¯=CD(u)+CD(u)v, where uv=-vu and Z(A0¯)=K[u] and CD(u) is the centralizer of u in D.

Let be a superinvolution on A, and let ξAut(A,). Then by the Skolem-Noether theorem there exists an invertible element cγAγ such that aξ=cγacγ-1, aA. Since aξ=aξ, aA, we have cγcγ=λ for some λK. If ξA0¯=ψcγA0¯ is inner, then ξA0¯=ψb0¯ for some invertible element b0¯A0¯ which implies that aξ=b0¯ab0¯-1, aA0¯, and therefore ψb0¯-1cγ=idA0¯. Let xγ=b0¯-1cγ; if γ=1¯, then xγaxγ-1=a  aA0¯. But A1¯=A0¯xγ, so xγ centralizes A1¯; thus xγZ(A)=K. This is a contradiction and therefore γ=0¯. Therefore, for any ξAut(A,) such that ξA0¯ is inner, ξ=ψb0¯, for some invertible element b0¯A0¯, and hence (7)inAutA,=ξAutA,:ξ=ψb0¯,  b0¯A0¯.

Corollary 5.

Let A=Mn(D) be an even central simple superalgebra over a field K, where (8)D=CDu+CDuvand uv=-vu is a nontrivial grading division superalgebra, and let be a superinvolution on A. Then(9)AutA,=inAutA,ψv·inAutA,:if  ψv=ψvinAutA,:if  ψvψv.

Proof.

If ψv=ψv, then for any ξAut(A,) such that uξ=-u we get uξψv=-uψv=u which implies that ξψvA0¯ is inner and so ξψv=ψb0¯ for some invertible b0¯ in A0¯. Thus ξ=ψvψb0¯. But if ψvψv, then it is easy to check that Aut(A,)=inAut(A,). Therefore (10)AutA,=inAutA,ψv·inAutA,:if  ψv=ψvinAutA,:if  ψvψv.

Theorem 6.

Let A=Mn(D), where D=D0-+D1-=CD(u)+CD(u)v, and let uv=-vu be a superalgebra with a superinvolution . Then inAut(A,)G/~.

Proof.

Let δ:inAut(A,)G/~ such that δ(ψb0¯)=[b0¯]. Then (11)δψb0¯ψc0¯=δψc0¯b0¯=c0¯b0¯=b0¯c0¯=δψb0¯δψc0¯.So, δ is an onto homomorphism, where kerδ=idA=ψ1¯. Therefore, inAut(A,)G/~.

Next we introduce this example to show that uu may equal λ with λαα for any αK.

Example 7.

Let k=Q(e), let K=k(2), and let u=ei, v=24j, where i2=-1, j2=-1, and ij=-ji. Then A=(K+Ku)(Kv+Kuv) is a quaternion division superalgebra over field K. Let be a superinvolution on A defined by

(a+b2)=a-b2, a,bk; u=u; v=v; then (uv)=vu=-uv.

Now, ψuAut(A,), since uu=-eK. But uuαα, where αK, because if (12)-e=a+b2a+b2,where a,bk, then (13)-e=a+b2a-b2=a2-2b2=p1eq1e2-2p2eq2e2,where pi(e),qi(e)Q[e] and qi(e)0. Hence, (14)-eq1e2q2e2=p1eq2e2-2p2eq1e2.Since 2Q, the highest power of e in the right-hand side is even, but the highest power of e in the left-hand side is odd, a contradiction.

Now we try to classify the group of automorphisms of central simple superalgebras A=M2p(D0¯), and A=Mp+q(D0¯), where D0¯ is a division algebra over K and (pq).

First let A=M2p(D0¯), where D0¯ is a division algebra over K, and let be any superinvolution on A. If ξAut(A,), then, by [1, Isomorhpism Theorem], ξ=ψMα for some invertible element MαAα and (15)MαPγMα-1=MαPγMα-1,PγAγ;thus,(16)MαPγMα-1=-1αα+γ-1αγMα-1PγMα=-1αMα-1PγMα=Mα-1PγMα,which implies that MαMαPγMα-1(Mα)-1=Pγ, Pγ, and therefore MαMα=λI2p, where λK×. If is a superinvolution of the first kind, then(17)MαMα=-1α2MαMα=λI2p=λI2p=MαMα.Hence α2=0¯, which implies that α=0¯ and therefore (18)AutA,=ψMM  is  an  invertible  element  in  A0¯,  with  MM=λI2p.Moreover, if is a superinvolution of the second kind on A, then(19)AutA,=ψMM  is  an  invertible  element  in  A0¯,  with  MM=λI2pψMM  is  an  invertible  element  in  A1¯,  with  MM=γI2p,  γ=-γ.

Theorem 8.

Let A=M2p(D0-), where D0- is a division algebra over K, and let be any superinvolution on A. If is a superinvolution of the first kind, then Aut(A,)G/~, where G/~ is the group defined in Theorem 4. If is a superinvolution of the second kind, then Aut(A,)G/~, where G={b0-A0-b0-b0-=αK×}b1-A1-b1-b1-=λK×,  where  λ=-λ and the equivalence relation ~ on G is defined by aα~bαbα=γaα for some γK×.

Proof.

If is a superinvolution of the first kind, then (20)δ:AutA,G~,ψa0¯a0¯is a group isomorphism.

Similarly, if is a superinvolution of the second kind, then it is easy to check that G/~ is a group under the well-defined operation aα[bβ]=[bβaα], and so (21)δ:AutA,G~,ψaαaαis a group isomorphism. Thus in all cases we get Aut(A,)G/~.

Second let A=Mp+q(D0¯), where D0¯ is a division algebra over K and (pq). Let be any superinvolution on A of any kind. If ξAut(A,), then by [1, Isomorhpism Theorem] ξ=ψMα, for some invertible element Mα in Aα, since pq, the homogeneous invertible elements in A are the invertible elements in A0¯. Thus α=0¯, and, therefore, if ξAut(A,), then ξ=ψM, where M is an invertible element in A0¯. Now, if PγAγ, then(22)MPγM-1=MPγM-1=M-1PγM=M-1PγM.Therefore, MMPγM-1(M)-1=Pγ, PγAγ, and hence MM=λIp+q, where λK×. Thus it is easy to check that AutA,={ψMM  is  an  invertible  element  in  A0¯,  where  MM=λIp+q}G/~, where G={b0¯A0¯b0¯b0¯=αK×}.

3. Examples of Group of Automorphisms of Even Superalgebras

In this section we classify in detail the groups of automorphisms of superalgebras A=M2p(D), and A=Mp+q(D), where D=H or R, where H is the Hamiltonian quaternion algebra and R is the field of real numbers.

Let A=M2p(D), where D=H or R, and let be a superinvolution of the first kind on A such that (A0¯,A0¯) is simple; then by [1, Proposition  13] is defined by (23)abcdd~-b~c~a~,where ~ is the involution on Mp(D) induced by . Then H=AutA,={ψMM  is  an  invertible  element  in  A0¯,  where  MM=αI2p,  and  αR}.

If α>0, then by replacing M by 1/αM we get MM=I2p.

If α<0, then by replacing M by 1/-αM we get MM=-I2p.

So H={ψMMM=I2p  or  -I2p}.

Theorem 9.

In the situation described above, let (24)G=AutA0-,A0-.Then H/R×G0, where G0 is the connected component of the identity in G.

Proof.

By [8, page 593], G=G0ψNG0, where N=0IpIp0 and G0 is the connected component of the identity. Let (25)π:HG0ψMψMA0¯.Then π is onto group homomorphism because if g0G0, then g0=ψMA0¯ for some invertible element MA0¯ since G0 is the connected component of the identity. Then it is easy to check that MM=αI2p (αR) since ψM=ψM.

If α>0, then by replacing M by 1/αM we get MM=I2p.

If α<0, then by replacing M by 1/-αM we get MM=-I2p. Therefore, ψMH and π(ψM)=g0. Also it is easy to check that kerπ={ψMM=λIp00γIp,  where  λ,γR}. But (26)λIp00γIpγIp00λIp=λγIp00γλIp=I2por  λIp00γIpγIp00λIp=λγIp00γλIp=-I2p.So, γ=1/λ or -1/λ. Therefore, kerπ={ψMM=λIp001/λIp, or M=λIp00-1/λIp,whereλR×}R×, since ψM=ψ-M. Therefore G0H/kerπH/R×.

In the second case, let A=Mp+q(D), where D=H or R, and let be a superinvolution of the first kind on A such that (A0¯,A0¯) is not simple; then by [1, Proposition  14] is defined by (27)xyzwx~-Nz¯tM-1My¯tN-1w~,where x~=Nx¯tN-1 (N¯t=-NMp(D)) and w~=Mw¯tM-1 (M¯t=MMq(D)) and is the standard involution on D.

Since is a superinvolution of the first kind we have H=AutA,={ψMM  is  an  invertible  element  in  A0¯,  where  MM=αIp+q,  and  αR}. Similarly, H={ψMMM=Ip+q  or  -Ip+q}.

Theorem 10.

In the situation described above, let (28)G=AutA0-,A0-.Then H/kerππ(H)G, where (29)π:HGψNψNA0-.

Proof.

Since A0¯ is semisimple and pq, we get G=AutA0¯,A0¯={ψMM=A00B,  where  A,B  are  invertible  matrices}. If ψMG, then (30)ψMA0¯=A0¯ψMMM=λIp00γIq,λ,γR;thereforeψMA0¯=A0¯ψMMMV=Ip+q,-Ip+q,Ip00-Iq,-Ip00Iq.So G=ψMMMV. Therefore, (31)π:HGψNψNA0¯is a group homomorphism and kerπ={ψNψNA0¯=idA0¯}={idA,ψM,M=Ip00-Iq}. To see this let N=A00B such that ψNkerπ; then NN=Ip+q or -Ip+q (since ψNH), and (32)A00BX00YA-100B-1=X00Yif and only if A=λIp and B=γIq (λ,γR); since (33)NN=λIp00γIqλIp00γIq=λ2Ip00γ2IqIp+q,-Ip+q,we get NN=Ip+q. Therefore, λ2=γ2=1λ, γ1,-1, which implies that kerπ={ψIp+q,ψ-Ip+q,ψN,ψM}, where N=-Ip00Iq and M=Ip00-Iq and hence kerπ=idA,ψM since ψIp+q=ψ-Ip+q and ψN=ψM. Therefore, H/kerππ(H)G.

Theorem 11.

As in the theorem above, if pq, then G=π(H)ψMA0-π(H) for a fixed invertible matrix M=A00B such that (34)MMIp00-Iq,-Ip00Iq.

Proof.

Since MMIp+q,-Ip+q, then ψMH, which implies that ψMA0¯π(H). Let N=X00Y such that (35)NNIp00-Iq,-Ip00Iq;then(36)NM-1NM-1=NM-1M-1N=NIp00-IqN:MM=Ip00-IqN-Ip00IqN:MM=-Ip00Iq=XX~00-YY~:MM=Ip00-Iq-XX~00YY~:MM=-Ip00Iq.Therefore, (NM-1)(NM-1){Ip+q,-Ip+q}; thus, by Theorem 10, (37)ψNM-1A0¯=ψM-1A0¯ψNA0¯πHHkerπ.Now, ψNA0¯=ψNM-1MA0¯=ψMA0¯ψM-1A0¯ψNA0¯=ψMA0¯ψNM-1A0¯ψMA0¯π(H). Therefore, (38)G=πHψMA0¯πH.

For the case p=q, G=Aut(A0¯,A0¯)=G0ψNA0¯G0, where N=0IpIp0 (it is easy to check that ψNA0¯A0¯=A0¯ψNA0¯) and G0={ψMA0¯MMV} is the connected component of the identity in G. Now, let π:HG0 (ψLψLA0¯), where H={ψMMM=Ip+q  or  -Ip+q}; then π is a group homomorphism and kerπ=idA,ψN where N=Ip00-Ip. Therefore, H/kerππ(H)G0.

Theorem 12.

In the situation described above, if p=q, then (39)G0=πHψMA0-πH,for a fixed invertible matrix M=A00B, such that (40)MMIp00-Iq,-Ip00Iq,and G0 is the connected component of the identity in G.

Proof.

Consider ψMA0¯π(H) since MM{Ip00-Ip,-Ip00Ip}. Let N=X00Y such that NN{Ip00-Ip,-Ip00Ip}; then(41)NM-1NM-1=NM-1M-1N=NIp00-IpN:MM=Ip00-IpN-Ip00IpN:MM=-Ip00Ip=XX~00-YY~:MM=Ip00-Ip-XX~00YY~:MM=-Ip00Ip.Therefore, (NM-1)(NM-1){I2p,-I2p} and so (42)ψNM-1A0¯=ψM-1A0¯ψNA0¯πHHkerπ.Now, ψNA0¯=ψNM-1MA0¯=ψMA0¯ψM-1A0¯ψNA0¯=ψMA0¯ψNM-1A0¯ψMπ(H). Therefore, (43)G0=πHψMA0¯πH.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Racine M. L. Primitive superalgebras with superinvolution Journal of Algebra 1998 206 2 588 614 10.1006/jabr.1997.7412 MR1637088 ZBL0915.16036 2-s2.0-0000684372 Racine M. L. Zelmanov E. I. Simple Jordan superalgebras with semisimple even part Journal of Algebra 2003 270 2 374 444 10.1016/j.jalgebra.2003.06.012 MR2019625 2-s2.0-0348159873 Lam T. Y. The Algebraic Theory of Quadratic Forms 1973 The Benjamin/Cummings Publishing Company Jaber A. Central simple superalgebras with anti-automorphisms of order two of the first kind Journal of Algebra 2010 323 7 1849 1859 10.1016/j.jalgebra.2010.01.008 MR2594651 2-s2.0-76749097253 Jaber A. Existence of pseudo-superinvolutions of the first kind International Journal of Mathematics and Mathematical Sciences 2008 2008 13 386468 10.1155/2008/386468 MR2377358 2-s2.0-41949117451 Scharlau W. Quadratic and Hermitian Forms 1985 Heidelberg, Germany Springer Elduque A. Villa O. The existence of superinvolutions Journal of Algebra 2008 319 10 4338 4359 10.1016/j.jalgebra.2007.10.044 MR2407903 2-s2.0-41549158661 Weil A. Algebras with involutions and the classical groups Indian Mathematical Society 1960-1961 24 589 623