Our main purpose is to develop the theory of existence of pseudo-superinvolutions of the first kind on finite dimensional central simple associative superalgebras over K, where K
is a field of characteristic not 2. We try to
show which kind of finite dimensional central simple associative superalgebras
have a pseudo-superinvolution of the first kind.
We will show that a division superalgebra 𝒟
over a field K
of characteristic not 2 of even type has pseudo-superinvolution (i.e., K-antiautomorphism J such that (dδ)J2=(−1)δdδ)
of the first kind if and only if 𝒟
is of order 2 in the Brauer-Wall group BW(K). We will also show that a division superalgebra 𝒟
of odd type over a field K
of characteristic not 2 has a pseudo-superinvolution of the first kind if and only if −1∈K,
and 𝒟
is of order 2 in the Brauer-Wall group BW(K).
Finally, we study the existence of pseudo-superinvolutions on central simple superalgebras
𝒜=Mp+q(𝒟0).

1. Introduction

Let K be a field of
characteristic not 2. An associative superalgebra is a ℤ2-graded
associative K-algebra 𝒜=𝒜0+𝒜1. A superalgebra 𝒜 is central simple over K, if Z^(𝒜)=K, where (Z^(𝒜))α={aα∈𝒜α|aαbβ=(−1)αβbβaα for allbβ∈𝒜β}, and the only
superideals of 𝒜 are (0) and 𝒜.

Finite dimensional central simple associative
superalgebras over a field K are isomorphic
to End V≅Mn(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,q=dim𝒟V1, so p+q is a nontrivial
decomposition of n. While if 𝒟1≠{0}, then the grading of Mn(𝒟) is given by (Mn(𝒟))α=Mn(𝒟α), α=0,1, as we recall.

Let 𝒜=𝒜0+𝒜1 be any
associative superalgebra over a field K of
characteristic not 2, and let ∗:𝒜→𝒜 be an
antiautomorphism on 𝒜, then ∗ is called a pseudo-superinvolution on 𝒜 if (a0+b1)∗∗=a0−b1.

In recent work on the representations of Jordan
superalgebras which has yet to appear, Martinez and Zelmanov make use of
pseudo-superinvolutions.

We recall a theorem of Albert which shows that a
finite dimensional central simple algebra over a field k has an
involution of the first kind if and only if it is of order 2 in the Brauer
group Br(k). The proof of this classical theorem is in many books
of algebra, for example, see [1, Chapter 8, Section 8].

Throughout my work on the existence of
superinvolutions of the first kind which has yet to appear, we prove that
finite dimensional central simple division superalgebras of odd or even type
with nontrivial grading over a field K of
characteristic not 2 have no superinvolutions of the first kind, also these
results were introduced in [2, Proposition 9], [3]. Moreover, we introduce an
example of a central simple superalgebra 𝒜=Mn(𝒟) over a field K of
characteristic not 2, where 𝒟1≠{0}, such that 𝒜 has no
superinvolution of the first kind, but it is of order 2 in the
Brauer-Wall group BW(K), which means that Albert's theorem does not hold for
superinvolutions and this is one of the reasons why one introduces a
generalization for which it does.

In [2, Theorem 7], Racine proved that 𝒜=Mn(𝒟) has a
superinvolution if and only if 𝒟 has. Therefore, 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.

Let 𝒟 be a division
superalgebra with nontrivial grading over a field K of
characteristic not 2. Since if 𝒜 is a central
simple associative superalgebra over K, then by [2, Theorem 3] 𝒜=Mn(𝒟), where 𝒟1≠{0} or 𝒜=Mp+q(𝒟), where 𝒟1={0}. In Section 2, we give
some basic definitions for the
supercase.

In Section 3, we classify the existence of
pseudo-superinvolution of the first kind on 𝒟 and we prove
the following results.

If 𝒜=Mn(𝒟), where 𝒟1≠{0}, then 𝒜 has a
pseudo-superinvolution of the first kind if and only if 𝒟 has. Therefore,
it is enough to classify the existence of a pseudo-superinvolution of the first
kind on 𝒟.

A division superalgebra 𝒟 of even type
over a field K of
characteristic not 2 has a pseudo-superinvolution
of the first kind if and only if 𝒟 is of order 2
in the Brauer-Wall group BW(K).

A division superalgebra 𝒟 of odd type
over a field K of
characteristic not 2 has a
pseudo-superinvolution of the first kind if and only if −1∈K and 𝒟 is of order 2
in the Brauer-Wall group BW(K).

In Section 4, we classify the existence of a
pseudo-superinvolution of the first kind on 𝒜=Mp+q(𝒟), where 𝒟 is a division
algebra over K.

Finally, if K is a field of
characteristic 2, and 𝒜 is a central
simple associative superalgebra over K, then a superinvolution (which is a
pseudo-superinvolution) 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 an antiautomorphism
of the first kind respecting the grading, therefore by [1, 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 pseudo-superinvolution) of the first kind if and
only if 𝒜 is of order 2
in the Brauer-Wall group BW(K).

2. Basic DefinitionsDefinition 2.1.

If R=R0+R1 is an associative
super-ring, a (right) R-supermodule M is a right R-module with a
grading M=M0+M1 as R0-modules such
that mαrβ∈Mα+β for any mα∈Mα,rβ∈Rβ,α,β∈Z2. An R-supermodule M is simple if MR≠{0} and M has no proper
subsupermodule.

Following [2],
we have the following definition of R-supermodule
homomorphism.

Definition 2.2.

Suppose that M and N are R-supermodules.
An R-supermodule homomorphism from M into N is an R0-module
homomorphism hγ:M→N, γ∈Z2, such that Mαhγ⊆Nα+γ and (mαrβ)hγ=(mαhγ)rβ,∀mα∈Mα,rβ∈Rβ,α,β∈Z2.

Definition 2.3.

The opposite
super-ring R° of the
super-ring R
is defined to
be R°=R
as an additive
group, with the multiplication given by bβ°cγ:=(−1)βγcγbβ,bβ∈Rβ,cγ∈Rγ.

So if 𝒜 is a
superalgebra, then 𝒜° is just the
opposite super-ring of 𝒜; one can easily show that if 𝒜 is a central
simple associative superalgebra over a field K, then 𝒜° is also a
central simple associative superalgebra over K.

Definition 2.4.

Let 𝒜=𝒜0+𝒜1, ℬ=ℬ0+ℬ1 be associative
superalgebras. Then the graded tensor product 𝒜⊗^Kℬ=[(𝒜0⊗ℬ0)⊕(𝒜1⊗ℬ1)]⊕[(𝒜0⊗ℬ1)⊕(𝒜1⊗ℬ0)], where the multiplication on 𝒜⊗^Kℬ is induced by(aα⊗bβ)(cγ⊗dδ)=(−1)βγaαcγ⊗bβdδ,aα∈𝒜α,cγ∈𝒜γ,bβ∈ℬβ,dδ∈ℬδ. If 𝒜 and ℬ are associative
superalgebras, then 𝒜⊗^Kℬ is an
associative superalgebra.

The commuting
super-ring of R on M is defined to
be 𝒞=𝒞0+𝒞1, where 𝒞γ:={cγ∈EndγM∣cγrα=(−1)αγrαcγ∀rα∈Rα,α∈ℤ2}.

Definition 2.5.

Two finite dimensional central simple
superalgebras 𝒜 and ℬ over a field K are called similar (𝒜~ℬ)
if there exist
graded K-vector spaces V=V0⊕V1, W=W0⊕W1, such that 𝒜⊗^KEndKV≅ℬ⊗^KEndKW
as K-superalgebras.

Similarity is
obviously an equivalence relation. The set of similarity classes will be
denoted by BW(K) (the
Brauer-Wall group of K). If [𝒜] denotes the
class of A in BW(K) by using [4,
Chapter 4,Theorem 2.3(3)], the operation [𝒜][ℬ]=[𝒜⊗^Kℬ] is
well-defined, and makes the set of similarity classes of finite dimensional
central simple superalgebras over K into a
commutative group, BW(K), where the class of the matrix algebras Mp+q(K) is a neutral
element for this product. Moreover, it was proved in [4, 5] that a central
simple associative superalgebra A is of order 2 in BW(K) if and only if A≈A°, the opposite
superalgebra.

3. Existence of Pseudo-Superinvolution on <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M193"><mml:mrow><mml:mi>𝒟</mml:mi></mml:mrow></mml:math></inline-formula>Theorem 3.1 (Division Superalgebra Theorem [<xref ref-type="bibr" rid="B5">3</xref>]).

If 𝒟=𝒟0+𝒟1is a finite
dimensional associative division superalgebra over a fieldK, then exactly one of the following holds where
throughoutℰdenotes a
finite dimensional associative division algebra overK.

𝒟=𝒟0=ℰ, and 𝒟1={0}.

𝒟=ℰ⊗KK[u],u2=λ∈K×,𝒟0=ℰ⊗K1,𝒟1=ℰ⊗Ku.

𝒟=ℰ or M2(ℰ), u∈𝒟 such that u2=λ∈K/K2, 𝒟0=C𝒟(u),𝒟1=S𝒟(u), where C𝒟(u)={d∈𝒟∣du=ud},S𝒟(u)={d∈𝒟|du=−ud}, moreover, in
the second case, u=(01λ0) and K[u] does not embed
in ℰ.

Following [4],
we say that a division superalgebra 𝒟 is even if Z(𝒟)∩𝒟1={0}, where Z(𝒟) is the center
of 𝒟, that is, 𝒟 is even if its
form is (i) or (iii), and that 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 an even
division superalgebra and 𝒜 is an odd K-superalgebra
if 𝒟 is an odd
division superalgebra.

Let V=V0+V1 be a (left)
superspace over a division superalgebra 𝒞 and W=W0+W1 a right
superspace over 𝒞. A bilinear pairing (,)ν is a biadditive
map (,)ν:V×W→𝒞 satisfying(vα,wβ)ν∈𝒞α+β+ν,(cγvα,wβ)ν=cγ(vα,wβ)ν,(vα,wβcγ)ν=(vα,wβ)νcγ for all vα∈Vα, wβ∈Wβ, and cγ∈𝒞γ. The bilinear pairing (,)ν is nondegenerate if(vα,W)ν={0}⇒vα=0,(V,wβ)ν={0}⇒wβ=0. If (,)ν is
nondegenerate, we say that the superspaces V and W are dual.

The right 𝒞-superspace W may be viewed
as a (left) 𝒞°-superspace viacγwβ:=(−1)βγwβcγ. An element aα∈End𝒞(V)α is said to have
an adjointaα∗∈End𝒞°(W)α if(vβaα,wδ)ν=(−1)αδ(vβ,wδaα∗)ν,∀vβ∈Vβ,wδ∈Wδ. Therefore if 𝒟 is a division
superalgebra and σ is an
antiautomorphism of 𝒟, then it is an isomorphism of 𝒟 onto 𝒟° and a right 𝒟°-superspace W is a left 𝒟 superspace
under the actiondδwβ:=(−1)δβwβdδσ,dδ∈𝒟δ,wβ∈Wβ. Thus, (,)ν:V×W→𝒟 is a pseudo-sesquilinear
pairing of (left) 𝒟 superspaces,
that is,(dδvα,wβ)ν=dδ(vα,wβ)ν,(vα,dδwβ)ν=(−1)βδ(vα,wβ)νdδσ,(vαdδ,wβ)ν=(−1)δ(β+ν+1)(vα,wβ)νdδ for all vα∈Vα,wβ∈Wβ,dδ∈𝒟δ. If ¯ is a
pseudo-superinvolution of 𝒟, then 𝒟 is isomorphic
to 𝒟° and we may
consider pseudo-sesquilinear pairings of V×V. If ϵ∈Z(𝒟) with ϵϵ¯=1, andδν={−1ν=1,1ν=0, an ϵ-Hermitian
pseudo-superform is a pseudo-sesquilinear pairing satisfying(vα,wβ)ν=(−1)α(β+1)ϵδν(wβ,vα)ν¯,∀vα∈Vα,wβ∈Vβ.

The pseudo-superform (,)ν is said to be even or odd according
to either ν=0 or 1. If ϵ=1 (resp., −1), (,)ν is said to be Hermitian (resp., skew-Hermitian).

We say that a super-ring R is prime if for any nonzero superideals I,J, the product IJ≠{0}. If R=Mn(𝒟), where 𝒟 is a division
superalgebra over a field K, then R is a prime. We
also have the usual characterization for homogeneous elements:Risprime⟺aαRbβ≠{0}∀0≠aα∈Rα,0≠bβ∈Rβ.

Theorem 3.2.

If a
central simple superalgebra 𝒜=Mn(𝒟)≅End𝒟(I) over a field K such that −1∈K, where I is a minimal
right superideal of 𝒜 and 𝒟° is the
commuting super-ring of 𝒜 on I, has a pseudo-superinvolution∗, then𝒟has and∗is the adjoint
with respect to a nondegenerate Hermitian or skew-Hermitian pseudo-superform onI.

𝒟=e0𝒜e0, and I=e0𝒜 is a left 𝒟 superspace for
some symmetric primitive even idempotent e0.

If ∗ is a
pseudo-superinvolution on 𝒜 and e0∗=e0, then ∗|𝒟=− is a
pseudo- superinvolution on 𝒟, and for vα=e0aα∈Iα, wβ=e0bβ∈Iβ, define(vα,wβ)0:=e0aα(e0bβ)∗=e0aαbβ∗e0∈𝒟α+β. One checks that for all dδ∈𝒟δ, vα∈Iα, wβ∈Iβ, (dδvα,wβ)0=dδ(vα,wβ)0,(vα,dδwβ)0=(−1)βδ(vα,wβ)0dδ¯,(vα,wβ)0=(−1)α(β+1)(wβ,vα)0¯, that I is self dual
with respect to (,)0, and that ∗ is the adjoint
with respect to the Hermitian pseudo-superform (,)0.

If the minimal right superideal I contains a
homogeneous ϵ-symmetric
element aα∗=ϵaα, ϵ=±1 such that aαI≠{0}, then aαI=I, so by [2, Lemma 5], there exists an idempotent f0∈I0 such that aαf0=aα and I=f0𝒜. Thus, f0aα=aα andaα=ϵaα∗=ϵ(f0aα)∗=ϵaα∗f0∗=aαf0∗=(aαf0)f0∗. Again the proof of [2, Lemma 5] shows that e0=f0f0∗∈I0 is a nonzero
even symmetric idempotent and I=e0𝒜 and since for 𝒞=e0𝒜e0, 𝒞° is the
commuting super-ring of 𝒜 on I, 𝒟=𝒞=e0𝒜e0.

Assume from now on that if aα∗=ϵaα∈Iα, ϵ=±1, then aαI={0}.

We will show that if bβbβ∗≠0 for some bβ∈Iβ, then I∗I={0}. Indeed, by [2, Lemma 2], bβbβ∗≠0 implies that {0}≠bβbβ∗𝒜⊆I. Therefore, bβbβ∗𝒜=I and 𝒜bβbβ∗=I∗. Since bβbβ∗∈I is ϵ-symmetric, I∗I=𝒜bβbβ∗I={0}.

We claim that aα∗aα=0 for all aα∈Iα. Let 0≠aα∈Iα, by [2, Lemma 5] I=aα𝒜=e0𝒜 and 𝒜e0=𝒜aα is a minimal
left superideal. If bβbβ∗=0 for all bβ∈aα𝒜α+β, then we are done. Otherwise, by the preceding
argument,{0}=I∗I=𝒜aα∗aα𝒜∀aα∈Iα. Thus, aα∗aα=0, since aα=aαr0 for some r0∈𝒜0 which implies
that aα∗aα=r0∗aα∗aαr0∈𝒜aα∗aα𝒜={0}.

From now on, we let I be a minimal
right superideal of 𝒜 such that aα∗aα=0 for all aα∈Iα. As in [2, Lemma 5], I=e0𝒜=e0𝒜0+e0𝒜1 and hence we
have e0𝒜e0∗≠{0} by primeness.
Therefore e0𝒜νe0∗≠{0} for at least
one ν∈ℤ2. We choose ν to be 0, if possible.
This will always be the case if 𝒟1=e0𝒜1e0≠{0}, for if e0𝒜1e0∗≠{0}, since e0∗𝒜e0∗=(e0𝒜e0)∗ is a division
superalgebra, e0𝒜0e0∗⊇e0𝒜1e0∗𝒜1e0∗≠{0}. We may therefore
assume that if ν=1, then 𝒟1={0}.

Assume e0𝒜νe0∗≠{0}. If for some rν∈𝒜ν, rν∗=δνrν, then (e0rνe0∗)∗=δνe0rνe0∗. If for all rν∈𝒜ν, rν∗−δνrν≠0, then

if ν=1, then we have(e0(rν∗−δνrν)e0∗)∗=e0((−1)νrν−δνrν∗)e0∗=e0(−rν−δνrν∗)e0∗=−δνe0(rν∗−δνrν)e0∗, if ν=0, then we have(e0(rν∗−δνrν)e0∗)∗=e0(rν−δνrν∗)e0∗=−e0(rν∗−δνrν)e0∗=−δνe0(rν∗−δνrν)e0∗. Thus in all
cases, we can choose tν≠0∈𝒜ν such that(e0tνe0∗)∗=ϵδνe0tνe0∗,ϵ=±1. Since e0∗𝒜e0tνe0∗≠{0}, by primeness, and since e0∗𝒜0e0∗ is a division
algebra, one can choose sν∈𝒜ν such thate0∗sνe0tνe0∗=e0∗. Applying ∗,e0=(−1)ν2e0tν∗e0∗sν∗e0=(−1)ν2e0tν∗e0∗sν∗e0. Therefore,e0∗sνe0=e0∗sν((−1)νϵδνe0tνe0∗sν∗e0)=(−1)νϵδν(e0∗sνe0tνe0∗)sν∗e0=(−1)νϵδνe0∗sν∗e0. If ν=1, then e0∗sν∗e0=ϵδν(e0∗sνe0). Thus(e0∗sνe0)∗=ϵδν(e0∗sνe0). If ν=0, then e0∗sν∗e0=ϵδν(e0∗sνe0). Thus(e0∗sνe0)∗=ϵδν(e0∗sνe0). So in all cases, we have(e0∗sνe0)∗=ϵδν(e0∗sνe0). We therefore havee0∗sνe0tνe0∗=e0∗,e0tνe0∗sνe0=e0,(e0tνe0∗)∗=ϵδνe0tνe0∗,(e0∗sνe0)∗=ϵδνe0∗sνe0. For vα=e0aα∈Iα, wβ=e0bβ∈Iβ,vαwβ∗=e0aαbβ∗e0∗=e0aαbβ∗e0∗sνe0tνe0∗. Define(vα,wβ)ν:=e0aαbβ∗e0∗sνe0∈e0𝒜α+β+νe0=𝒟α+β+ν. By the last claim, (vα,vα)ν:=e0aαaα∗e0∗sνe0=0, for all vα∈Iα. If (vα,I)ν={0},e0aα𝒜e0∗sνe0={0}, and since e0∗sνe0≠0,e0aα=0,byprimeness. Similarly, (I,wβ)ν={0} implies wβ=0 and (,)ν is
nondegenerate. If dδ∈𝒟δ, (dδvα,wβ)ν=dδ(vα,wβ)ν. Moreover(vα,dδwβ)ν=(e0aα,dδe0bβ)ν=(−1)δβe0aαbβ∗e0∗dδ∗e0∗sνe0=(−1)δβe0aαbβ∗e0∗sνe0tνe0∗dδ∗e0∗sνe0=(−1)δβ(vα,wβ)νe0tνe0∗dδ∗e0∗sνe0=(−1)δβ(vα,wβ)νdδ¯, wheredδ¯:=e0tνe0∗dδ∗e0∗sνe0. For dδ∈𝒟δ,dδ¯¯=e0tνe0∗(e0tνe0∗dδ∗e0∗sνe0)∗e0∗sνe0=(−1)ν2+δe0tνe0∗sν∗e0dδe0tν∗e0∗sνe0=(−1)ν2+δϵδνe0dδϵδνe0=(−1)ν2+δ(δν)2dδ=(−1)δdδ. For cγ∈𝒟γ and dδ∈𝒟δ,cγdδ¯=e0tνe0∗(cγdδ)∗e0∗sνe0=(−1)γδe0tνe0∗dδ∗cγ∗e0∗sνe0=(−1)γδe0tνe0∗dδ∗e0∗sνe0tνe0∗cγ∗e0∗sνe0=(−1)γδdδ¯cγ¯. Thus “−” is a
pseudo-superinvolution of 𝒟 and (,)ν is a
nondegenerate pseudo-sesquilinear superform on I whose adjoint
is ∗. Finally,(vα,wβ)ν¯=e0tνe0∗(e0aαbβ∗e0∗sνe0)∗e0∗sνe0=(−1)αβ+β(−1)ν(α+β)e0tνe0∗sν∗e0bβaα∗e0∗sνe0=(−1)αβ+β(−1)ν(α+β)ϵδνe0bβaα∗e0∗sνe0=(−1)αβ+β(−1)ν(α+β)ϵδν(wβ,vα)ν. If ν=0, then (vα,wβ)0¯=(−1)αβ+βϵδ0(wβ,vα)0, and hence(wβ,vα)0=(−1)αβ+βϵδ0(vα,wβ)0¯. Thus (,)0 is ϵ-Hermitian
pseudo-superform. If ν=1, then we have assumed that 𝒟1={0} and therefore (vα,wα)1=0, for all vα,wα∈Iα. Hence the right-hand side is 0 unless α+β=1. Thus for all vα∈Iα, wβ∈Iβ,(vα,wβ)ν¯=(−1)ν+αβ+βϵδν(wβ,vα)ν=(−1)αβ+βϵ(−δν)(wβ,vα)ν. Thus(wβ,vα)1=(−1)αβ+βϵδ1(vα,wβ)1¯ and (,)1 is an ϵ-Hermitian
pseudo-superform.

If 𝒜=Mn(𝒟) is a finite
dimensional central simple super algebra over a field K, where 𝒟 is a finite
dimensional division superalgebra with nontrivial grading over K then, by
Theorem 3.2, it is enough to study the existence of pseudo-superinvolutions on 𝒟 to ascertain
the existence of pseudo-superinvolutions on 𝒜.

Theorem 3.3.

Let𝒟=𝒟0+𝒟0vbe an even
division superalgebra over a fieldKof
characteristic not2, then𝒟has aK-pseudo-superinvolution
if and only if𝒟≈𝒟°,the opposite
superalgebra.

Proof.

Suppose that 𝒟 has a K-pseudo-superinvolution ∗, then ∗ is a K- antiautomorphism
on 𝒟 which implies
that 𝒟≈𝒟°.

Conversely, suppose that 𝒟≈𝒟°, then there exists a K-antiautomorphism J on 𝒟. Since J2 is a K-automorphism
on 𝒟, there exists aα∈𝒟α such thatxJ2=aαxaα−1∀x∈𝒟. Now, uJ∈Z(𝒟0)=K(u) implies that uJ=c+du for some c,d∈K, and uJvJ=(vu)J=(−uv)J=−vJuJ implies that (c+du)vJ=−vJ(c+du)=−(c−du)vJ, thus c+du=−c+du implies c=0, and hence uJ=du, d∈K. Moreover, (u2)J=(uJ)2 implies that u2=d2u2, so d=1 or d=−1, which means that uJ=u or uJ=−u. So, in all cases uJ2=u, thus uJ2=aαuaα−1=u implies that α=0, and hence aα=a0∈𝒟0.

Case(1): if uJ=u, then 𝒟0≈𝒟0° implies that 𝒟0 has an
involution of the first kind, so by [1, Chapter 8, Theorem 8.2], a0a0J=α2 for some α∈K(u), thus (a0/α)(a0/α)J=(a0/α)J(a0/α)=1. If a0/α=−1, then a0=−α∈K(u). If not, then let I:𝒟0→𝒟0 be a map
defined by xI=(1+a0/α)−1xJ(1+a0/α), an easy computation shows that I is an
involution of the first kind on 𝒟0, since uI=u, and hence xI=(1+a0/α)−1xJ(1+a0/α) for all x∈𝒟 defines a K-antiautomorphism
of the first kind on 𝒟, such that xI2=αxα−1 for all x∈𝒟, where α∈Z(𝒟0)=K(u).

So, we find that for the case(1) we can define a K-antiautomorphism
(say h) such that for
some α∈K(u), xh2=αxα−1 for all x∈𝒟, and uh=u, and moreover, ααh=αhα∈K(u). Suppose that α=c+du, where c,d∈K, then vh3=(vh2)h=(αvα−1)h=(α−1)hvhαh, and vh3=(vh)h2=αvhα−1, implies that αvhα−1=(α−1)hvhαh, thus αhαvh(αhα)−1=vh, so, αhα∈Z^(𝒟)=K. Therefore,(c+du)h(c+du)=(c+du)2=c2+2cdu+d2u2∈K, which implies that 2cd=0, so c=0 or d=0, but by [3], 𝒟 does not have a
superinvolution of the first kind, implies that d≠0, hence c=0, therefore α=du. Now, vh2=(du)v(du)−1=−v(du)(du)−1=−v, thus h is a K-pseudo-superinvolution
on 𝒟.

Case(2): if uJ=−u, then ∗:𝒟→𝒟 defined by x∗=vxJv−1 for all x∈𝒟 is a K- antiautomorphism
on 𝒟, and u∗=u, also for any x∈𝒟, x∗∗=bxb−1, where b=v(vJ)−1a0∈𝒟0. Therefore, by case(1), 𝒟 has a K-pseudo-superinvolution.

Theorem 3.4.

Let
𝒟=𝒟0+𝒟0u, where u∈Z(𝒟), be a division
superalgebra of odd type over K, then 𝒟has a
pseudo-superinvolution of the first kind if and only if −1∈K, and 𝒟≈𝒟°, the opposite
superalgebra.

Proof.

Let ∗ be any
pseudo-superinvolution of the first kind on 𝒟, then u∗=αu for some α in K, so u∗∗=−u=(αu)∗=α2u, thus α2=−1 implies that −1∈K.

Conversely, suppose that α=−1∈K and 𝒟≈𝒟°, then 𝒟0≈𝒟0°, so 𝒟0 has an involution
of the first kind (say J). Therefore,
if ∗:𝒟→𝒟 is defined by (a+bu)∗=aJ+αbJu, where a,b∈𝒟0, then ∗ is a
pseudo-superinvolution on 𝒟, since(a+bu)∗∗=(aJ+αbJu)∗=a+α2bu=a−bu,(aubu)∗=(abu2)∗=(ab)∗(u2)∗=−(ab)∗(u∗)2=−(b∗u∗)(a∗u∗)=−(bu)∗(au)∗.

Corollary 3.5.

Let𝒟=𝒟0+𝒟0u, whereu∈Z(𝒟),be a division
superalgebra of odd type over a fieldK, such thatα=−1∈K. Then the following hold.

If ∗ is a
pseudo-superinvolution on 𝒟, then we can not choose u∈𝒟1 such that u∗=u or u∗=−u.

If − is an
involution of 𝒟0, then the superalgebra 𝒟 has a
pseudo-superinvolution ∗ extending − given by(a+bu)∗=a¯+αb¯u.

Proof.

(1) If u∗=u, then u∗∗=−u=u∗=u, a contradiction. Also, if u∗=−u, then u∗∗=−u=−u∗=u, a contradiction.

(2) Given an involution “−” of 𝒟0, one checks that(a+bu)∗=a¯+αb¯u
defines a pseudo-superinvolution on the superalgebra 𝒟=𝒟0⊗K[u], extending “−,” such that (u2)∗=u2.

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We say the
central simple superalgebra (𝒜,∗) with
pseudo-superinvolution is simple if the only ∗-stable
superideals of 𝒜 are (0) and 𝒜. The first lemma is a version of a standard result
for super-rings with superinvolution, and the proof of this lemma is the same
as the proof of [2, Lemma 11].

Lemma 4.1.

If
𝒜 is an
associative super-ring with pseudo-superinvolution ∗ such that (𝒜,∗) is simple, then
either 𝒜 is simple (as a
super-ring) or 𝒜=ℬ⊕ℬ∗, with ℬ a simple
super-ring.

In the second
case, ℬ∗ is isomorphic
to the opposite super-ring ℬ° of ℬ. We will consider a super-ring 𝒜 with nonzero odd
part. To avoid double indices, we will write 𝒜=A+B, where A=𝒜0 is the even
part and B=𝒜1 the odd part.
The proof of the next theorem is the same as the proof of [2, Theorem 12].

Theorem 4.2.

Let
𝒜=A+B be an
associative super-ring with B≠{0}, and “∗” a
pseudo-superinvolution of 𝒜. If (𝒜,∗) is simple, then
either (A,∗|A) is simple, orA=A1⊕A2,B=B1⊕B2, where (Ai,∗|Ai) are simple and Bi are irreducible A-bimodules withB1∗=B2,B2∗=B1, such thatA1B1=B1=B1A2,A2B2=B2=B2A1,B1B2=A1,B2B1=A2,{0}=A2B1=A1B2=B1A1=B2A2=B1B1=B2B2.

We will need
more information on the pseudo-superinvolutions of 𝒜 when the
grading is not inherited from that of 𝒟, that is, 𝒟=𝒟0, and 𝒜 is finite
dimensional. If 𝒜=Mp+q(𝒟), 𝒜0=Mp(𝒟)⊕Mq(𝒟), p,q>0, then we are either in that situation
or in the other, described in Theorem 4.2. We consider each case
in turn using the notation of Theorem 4.2.

Theorem 4.3.

If
𝒜=Mp+q(𝒟0), where 𝒜0=Mp(𝒟0)+Mq(𝒟0),p,q>0 is a finite
dimensional central simple superalgebra over a field K such that −1∈K, and ∗ is a
pseudo-superinvolution on 𝒜
and
(𝒜0,∗|𝒜0) is simple then p=q,Mp(𝒟0) has an
involution ~ and (𝒜,∗) is isomorphic
to M2p(𝒟0) with the
pseudo-superinvolution ∗ given by (abcd)∗=(d˜αb˜α˜c˜a˜), for a,b,c,d∈Mp(𝒟0), and α∈K such that αα˜=−1.

Conversely if Mp(𝒟0) has an
involution ~ then (4.4) defines a
pseudo-superinvolution on the simple superalgebra 𝒜=Mp+p(𝒟0) over K such that −1∈K.

Proof.

Since 𝒜 has a
pseudo-superinvolution then, by Theorem 3.2,
so has 𝒟. In this case since 𝒟=𝒟0, 𝒟 has an
involution “−” and Mp(𝒟) has an
involution a˜=a¯t, t the transpose.
Since (𝒜0,∗|𝒜0) is simple, Mq(𝒟) is
anti-isomorphic to Mp(𝒟) and p=q. Up to isomorphism, (𝒜0,∗|𝒜0) is given by (Mp(𝒟)⊕Mp(𝒟),∗) with (a,b)∗=(b˜,a˜). Letting f11=∑i=1peii=(Ip000),f22=∑i=p+12peii=(000Ip)f12=∑i=1peip+i=(0Ip00),f21=∑i=1pep+ii=(00Ip0). We have𝒜0=Mp(𝒟)f11⊕Mp(𝒟)f22,𝒜1=Mp(𝒟)f12⊕Mp(𝒟)f21,f11∗=f22,f22∗=f11. Hencef12∗=(f11f12f22)∗=f11f12∗f22,f12∗=cf12,forsomec∈Mp(𝒟). For any a∈Mp(𝒟),(af12)∗=(af11f12)∗=cf12a˜f22=ca˜f12. While(af12)∗=(f12(af22))∗=a˜f11cf12=a˜cf12. Therefore, c∈Z(Mp(𝒟)). Moreover f12∗∗=−f12=(cf12)∗=c˜cf12 implies c˜c=−Ip. So c=α∈K with αα˜=−1. Similarly f21∗=df21, d∈Z(Mp(𝒟)). Butf22=f11∗=(f12f21)∗=−f21∗f12∗=−dcf21f12=−dcf22 which implies −dc=1, and hence d=−c−1=−α−1=α˜. Therefore,(af12)∗=a˜f21∗=a˜α˜f21 or(abcd)∗=(d˜αb˜α˜c˜a˜), for a,b,c,d∈Mp(𝒟). The converse is
easy to check.

The proof of the next result is the same as [2,
Proposition 14].

Theorem 4.4.

If
𝒜=Mp+q(𝒟0),p,q>0, is a central simple superalgebra over a field K, and ∗ is a
pseudo-superinvolution on 𝒜, with 𝒜0=A1⊕A2,A1=Mp(𝒟0),A2=Mq(𝒟0),𝒜1=ℬ=ℬ1+ℬ2, and (𝒜0,∗|𝒜0) is not simple
then (A1,∗|A1) and (A2,∗|A2) are simple and ℬi are irreducible 𝒜0-bimodules with ℬ1∗=ℬ2 and ℬ2∗=ℬ1 satisfying the
hypothesis of Theorem 4.2 then ∗ is given by(abcd)∗=(a˜c˜−b˜d˜), where a∈Mp(𝒟0), d∈Mq(𝒟0), and ~ is an
involution on Mp(𝒟0), Mq(𝒟0), and where b˜∈Mq,p(𝒟0) for all b∈Mp,q(𝒟0), and c˜∈Mp,q(𝒟0) for all c∈Mq,p(𝒟0).

Conversely (4.14) defines a
pseudo-superinvolution on Mp+q(𝒟0).

ScharlauW.RacineM. L.Primitive superalgebras with superinvolutionRacineM. L.ZelmanovE. I.Simple Jordan superalgebras with semisimple even partLamT. Y.WallC. T. C.Graded Brauer groups