Existence of Pseudo-Superinvolutions of the First Kind

Abstract

?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 (𝑑_𝛿)^𝐽2=(−1)^𝛿𝑑_𝛿) 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=𝑀_𝑝+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={𝑎_𝛼∈𝒜_𝛼|𝑎_𝛼𝑏_𝛽=(−1)^𝛼𝛽𝑏_𝛽𝑎_𝛼forall𝑏_𝛽∈𝒜_𝛽}, 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≅𝑀_𝑛(𝒟), where 𝒟=𝒟₀+𝒟₁ is a finite dimensional associative division superalgebra over 𝐾, that is, all nonzero elements of 𝒟_𝛼, 𝛼=0,1, are invertible, and 𝑉=𝑉₀+𝑉₁ is an 𝑛-dimensional 𝒟-superspace.

If 𝒟₁={0}, the grading of 𝑀_𝑛(𝒟) is induced by that of 𝑉=𝑉₀+𝑉₁, 𝒜=𝑀_𝑝+𝑞(𝒟), 𝑝=dim_𝒟𝑉₀,𝑞=dim_𝒟𝑉₁, so 𝑝+𝑞 is a nontrivial decomposition of 𝑛. While if 𝒟₁≠{0}, then the grading of 𝑀_𝑛(𝒟) is given by (𝑀_𝑛(𝒟))_𝛼=𝑀_𝑛(𝒟_𝛼), 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=𝑀_𝑛(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=𝑀_𝑛(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=𝑀_𝑝+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=𝑀_𝑛(𝒟), where 𝒟₁≠{0} or 𝒜=𝑀_𝑝+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.

(1)If 𝒜=𝑀_𝑛(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.(2)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).(3)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=𝑀_𝑝+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 Definitions

Definition 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 𝑚_𝛼𝑟_𝛽∈𝑀_𝛼+𝛽 for any 𝑚_𝛼∈𝑀_𝛼,𝑟_𝛽∈𝑅_𝛽,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 ℎ_𝛾∶𝑀→𝑁, 𝛾∈𝐙₂, such that 𝑀_𝛼ℎ_𝛾⊆𝑁_𝛼+𝛾 and (𝑚_𝛼𝑟_𝛽)ℎ_𝛾=(𝑚_𝛼ℎ_𝛾)𝑟_𝛽,∀𝑚_𝛼∈𝑀_𝛼,𝑟_𝛽∈𝑅_𝛽,𝛼,𝛽∈𝐙₂. (2.1)

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 𝑏_𝛽∘𝑐_𝛾∶=(−1)^𝛽𝛾𝑐_𝛾𝑏_𝛽,𝑏_𝛽∈𝑅_𝛽,𝑐_𝛾∈𝑅_𝛾. (2.2)

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)], (2.3) where the multiplication on 𝒜⊗_𝐾ℬ is induced by(𝑎_𝛼⊗𝑏_𝛽)(𝑐_𝛾⊗𝑑_𝛿)=(−1)^𝛽𝛾𝑎_𝛼𝑐_𝛾⊗𝑏_𝛽𝑑_𝛿,𝑎_𝛼∈𝒜_𝛼,𝑐_𝛾∈𝒜_𝛾,𝑏_𝛽∈ℬ_𝛽,𝑑_𝛿∈ℬ_𝛿. (2.4) 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 𝒞_𝛾∶={𝑐_𝛾∈End_𝛾𝑀∣𝑐_𝛾𝑟_𝛼=(−1)^𝛼𝛾𝑟_𝛼𝑐_𝛾∀𝑟_𝛼∈𝑅_𝛼,𝛼∈ℤ₂}. (2.5)

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⊗_𝐾End_𝐾𝑉≅ℬ⊗_𝐾End_𝐾𝑊 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 𝑀_𝑝+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 𝒟

Theorem 3.1 (Division Superalgebra Theorem [3]). 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.
(i)𝒟=𝒟₀=ℰ, and 𝒟₁={0}.(ii)𝒟=ℰ⊗_𝐾𝐾[𝑢],𝑢²=𝜆∈𝐾^×,𝒟₀=ℰ⊗𝐾1,𝒟₁=ℰ⊗𝐾𝑢.(iii)𝒟=ℰ or 𝑀₂(ℰ), 𝑢∈𝒟 such that 𝑢²=𝜆∈𝐾/𝐾², 𝒟₀=𝐶_𝒟(𝑢),𝒟₁=𝑆_𝒟(𝑢), where 𝐶_𝒟(𝑢)={𝑑∈𝒟∣𝑑𝑢=𝑢𝑑},𝑆_𝒟(𝑢)={𝑑∈𝒟|𝑑𝑢=−𝑢𝑑}, moreover, in the second case, 𝑢=(⁰¹_𝜆0) 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=𝑀_𝑛(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(𝑣_𝛼,𝑤_𝛽)_𝜈∈𝒞_{𝛼+𝛽+𝜈},(𝑐_𝛾𝑣_𝛼,𝑤_𝛽)_𝜈=𝑐_𝛾(𝑣_𝛼,𝑤_𝛽)_𝜈,(𝑣_𝛼,𝑤_𝛽𝑐_𝛾)_𝜈=(𝑣_𝛼,𝑤_𝛽)_𝜈𝑐_𝛾 (3.1) for all 𝑣_𝛼∈𝑉_𝛼, 𝑤_𝛽∈𝑊_𝛽, and 𝑐_𝛾∈𝒞_𝛾. The bilinear pairing (,)_𝜈 is nondegenerate if(𝑣_𝛼,𝑊)_𝜈={0}⇒𝑣_𝛼=0,(𝑉,𝑤_𝛽)_𝜈={0}⇒𝑤_𝛽=0. (3.2) 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 via𝑐_𝛾𝑤_𝛽∶=(−1)^𝛽𝛾𝑤_𝛽𝑐_𝛾. (3.3) An element 𝑎_𝛼∈End_𝒞(𝑉)_𝛼 is said to have an adjoint𝑎^∗_𝛼∈End_𝒞^∘(𝑊)_𝛼 if(𝑣_𝛽𝑎_𝛼,𝑤_𝛿)_𝜈=(−1)^𝛼𝛿(𝑣_𝛽,𝑤_𝛿𝑎^∗_𝛼)_𝜈,∀𝑣_𝛽∈𝑉_𝛽,𝑤_𝛿∈𝑊_𝛿. (3.4) 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 action𝑑_𝛿𝑤_𝛽∶=(−1)^𝛿𝛽𝑤_𝛽𝑑^𝜎_𝛿,𝑑_𝛿∈𝒟_𝛿,𝑤_𝛽∈𝑊_𝛽. (3.5) Thus, (,)_𝜈∶𝑉×𝑊→𝒟 is a pseudo-sesquilinear pairing of (left) 𝒟 superspaces, that is,(𝑑_𝛿𝑣_𝛼,𝑤_𝛽)_𝜈=𝑑_𝛿(𝑣_𝛼,𝑤_𝛽)_𝜈,(𝑣_𝛼,𝑑_𝛿𝑤_𝛽)_𝜈=(−1)^𝛽𝛿(𝑣_𝛼,𝑤_𝛽)_𝜈𝑑^𝜎_𝛿,(𝑣_𝛼𝑑_𝛿,𝑤_𝛽)_𝜈=(−1)^{𝛿(𝛽+𝜈+1)}(𝑣_𝛼,𝑤_𝛽)_𝜈𝑑_𝛿 (3.6) for all 𝑣_𝛼∈𝑉_𝛼,𝑤_𝛽∈𝑊_𝛽,𝑑_𝛿∈𝒟_𝛿. 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, and𝛿_𝜈=√−1𝜈=1,1𝜈=0, (3.7) an 𝜖-Hermitian pseudo-superform is a pseudo-sesquilinear pairing satisfying(𝑣_𝛼,𝑤_𝛽)_𝜈=(−1)^𝛼(𝛽+1)𝜖𝛿_𝜈(𝑤_𝛽,𝑣_𝛼)_𝜈,∀𝑣_𝛼∈𝑉_𝛼,𝑤_𝛽∈𝑉_𝛽. (3.8)

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=𝑀_𝑛(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⟺𝑎_𝛼𝑅𝑏_𝛽≠{0}∀0≠𝑎_𝛼∈𝑅_𝛼,0≠𝑏_𝛽∈𝑅_𝛽. (3.9)

Theorem 3.2. If a central simple superalgebra?𝒜=𝑀_𝑛(𝒟)≅End_𝒟(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.

Proof [2, Lemma 5]. 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 𝑒^∗₀=𝑒₀, then ∗|_𝒟=⁻ is a pseudo- superinvolution on 𝒟, and for 𝑣_𝛼=𝑒₀𝑎_𝛼∈𝐼_𝛼, 𝑤_𝛽=𝑒₀𝑏_𝛽∈𝐼_𝛽, define(𝑣_𝛼,𝑤_𝛽)₀∶=𝑒₀𝑎_𝛼(𝑒₀𝑏_𝛽)^∗=𝑒₀𝑎_𝛼𝑏^∗_𝛽𝑒₀∈𝒟_𝛼+𝛽. (3.10) One checks that for all 𝑑_𝛿∈𝒟_𝛿, 𝑣_𝛼∈𝐼_𝛼, 𝑤_𝛽∈𝐼_𝛽, (𝑑_𝛿𝑣_𝛼,𝑤_𝛽)₀=𝑑_𝛿(𝑣_𝛼,𝑤_𝛽)₀,(𝑣_𝛼,𝑑_𝛿𝑤_𝛽)₀=(−1)^𝛽𝛿(𝑣_𝛼,𝑤_𝛽)₀𝑑_𝛿,(𝑣_𝛼,𝑤_𝛽)₀=(−1)^𝛼(𝛽+1)(𝑤_𝛽,𝑣_𝛼)₀, (3.11) 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 𝑎^∗_𝛼=𝜖𝑎_𝛼, 𝜖=±1 such that 𝑎_𝛼𝐼≠{0}, then 𝑎_𝛼𝐼=𝐼, so by [2, Lemma 5], there exists an idempotent 𝑓₀∈𝐼₀ such that 𝑎_𝛼𝑓₀=𝑎_𝛼 and 𝐼=𝑓₀𝒜. Thus, 𝑓₀𝑎_𝛼=𝑎_𝛼 and𝑎_𝛼=𝜖𝑎^∗_𝛼=𝜖(𝑓₀𝑎_𝛼)^∗=𝜖𝑎^∗_𝛼𝑓^∗₀=𝑎_𝛼𝑓^∗₀=(𝑎_𝛼𝑓₀)𝑓^∗₀. (3.12) Again the proof of [2, Lemma 5] shows that 𝑒₀=𝑓₀𝑓^∗₀∈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 𝑎^∗_𝛼=𝜖𝑎_𝛼∈𝐼_𝛼, 𝜖=±1, then 𝑎_𝛼𝐼={0}.
We will show that if 𝑏_𝛽𝑏^∗_𝛽≠0 for some 𝑏_𝛽∈𝐼_𝛽, then 𝐼^∗𝐼={0}. Indeed, by [2, Lemma 2], 𝑏_𝛽𝑏^∗_𝛽≠0 implies that {0}≠𝑏_𝛽𝑏^∗_𝛽𝒜⊆𝐼. Therefore, 𝑏_𝛽𝑏^∗_𝛽𝒜=𝐼 and 𝒜𝑏_𝛽𝑏^∗_𝛽=𝐼^∗. Since 𝑏_𝛽𝑏^∗_𝛽∈𝐼 is 𝜖-symmetric, 𝐼^∗𝐼=𝒜𝑏_𝛽𝑏^∗_𝛽𝐼={0}.
We claim that 𝑎^∗_𝛼𝑎_𝛼=0 for all 𝑎_𝛼∈𝐼_𝛼. Let 0≠𝑎_𝛼∈𝐼_𝛼, by [2, Lemma 5] 𝐼=𝑎_𝛼𝒜=𝑒₀𝒜 and 𝒜𝑒₀=𝒜𝑎_𝛼 is a minimal left superideal. If 𝑏_𝛽𝑏^∗_𝛽=0 for all 𝑏_𝛽∈𝑎_𝛼𝒜_𝛼+𝛽, then we are done. Otherwise, by the preceding argument,{0}=𝐼^∗𝐼=𝒜𝑎^∗_𝛼𝑎_𝛼𝒜∀𝑎_𝛼∈𝐼_𝛼. (3.13) Thus, 𝑎^∗_𝛼𝑎_𝛼=0, since 𝑎_𝛼=𝑎_𝛼𝑟₀ for some 𝑟₀∈𝒜₀ which implies that 𝑎^∗_𝛼𝑎_𝛼=𝑟^∗₀𝑎^∗_𝛼𝑎_𝛼𝑟₀∈𝒜𝑎^∗_𝛼𝑎_𝛼𝒜={0}.
From now on, we let 𝐼 be a minimal right superideal of 𝒜 such that 𝑎^∗_𝛼𝑎_𝛼=0?for all 𝑎_𝛼∈𝐼_𝛼. As in [2, Lemma 5], 𝐼=𝑒₀𝒜=𝑒₀𝒜₀+𝑒₀𝒜₁ and hence we have 𝑒₀𝒜𝑒^∗₀≠{0} by primeness. Therefore 𝑒₀𝒜_𝜈𝑒^∗₀≠{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 𝑒₀𝒜₁𝑒^∗₀≠{0}, since 𝑒^∗₀𝒜𝑒^∗₀=(𝑒₀𝒜𝑒₀)^∗ is a division superalgebra, 𝑒₀𝒜₀𝑒^∗₀⊇𝑒₀𝒜₁𝑒^∗₀𝒜₁𝑒^∗₀≠{0}. We may therefore assume that if 𝜈=1, then 𝒟₁={0}.
Assume 𝑒₀𝒜_𝜈𝑒^∗₀≠{0}. If for some 𝑟_𝜈∈𝒜_𝜈, 𝑟^∗_𝜈=𝛿_𝜈𝑟_𝜈, then (𝑒₀𝑟_𝜈𝑒^∗₀)^∗=𝛿_𝜈𝑒₀𝑟_𝜈𝑒^∗₀. If for all 𝑟_𝜈∈𝒜_𝜈, 𝑟^∗_𝜈−𝛿_𝜈𝑟_𝜈≠0, then
if 𝜈=1, then we have(𝑒₀(𝑟^∗_𝜈−𝛿_𝜈𝑟_𝜈)𝑒^∗₀)^∗=𝑒₀((−1)^𝜈𝑟_𝜈−𝛿_𝜈𝑟^∗_𝜈)𝑒^∗₀=𝑒₀(−𝑟_𝜈−𝛿_𝜈𝑟^∗_𝜈)𝑒^∗₀=−𝛿_𝜈𝑒₀(𝑟^∗_𝜈−𝛿_𝜈𝑟_𝜈)𝑒^∗₀, (3.14) if 𝜈=0, then we have(𝑒₀(𝑟^∗_𝜈−𝛿_𝜈𝑟_𝜈)𝑒^∗₀)^∗=𝑒₀(𝑟_𝜈−𝛿_𝜈𝑟^∗_𝜈)𝑒^∗₀=−𝑒₀(𝑟^∗_𝜈−𝛿_𝜈𝑟_𝜈)𝑒^∗₀=−𝛿_𝜈𝑒₀(𝑟^∗_𝜈−𝛿_𝜈𝑟_𝜈)𝑒^∗₀. (3.15) Thus in all cases, we can choose 𝑡_𝜈≠0∈𝒜_𝜈 such that(𝑒₀𝑡_𝜈𝑒^∗₀)^∗=𝜖𝛿_𝜈𝑒₀𝑡_𝜈𝑒^∗₀,𝜖=±1. (3.16) Since 𝑒^∗₀𝒜𝑒₀𝑡_𝜈𝑒^∗₀≠{0}, by primeness, and since 𝑒^∗₀𝒜₀𝑒^∗₀ is a division algebra, one can choose 𝑠_𝜈∈𝒜_𝜈 such that𝑒^∗₀𝑠_𝜈𝑒₀𝑡_𝜈𝑒^∗₀=𝑒^∗₀. (3.17) Applying ∗,𝑒₀=(−1)^𝜈2𝑒₀𝑡^∗_𝜈𝑒^∗₀𝑠^∗_𝜈𝑒₀=(−1)^𝜈2𝑒₀𝑡^∗_𝜈𝑒^∗₀𝑠^∗_𝜈𝑒₀. (3.18) Therefore,𝑒^∗₀𝑠_𝜈𝑒₀=𝑒^∗₀𝑠_𝜈((−1)^𝜈𝜖𝛿_𝜈𝑒₀𝑡_𝜈𝑒^∗₀𝑠^∗_𝜈𝑒₀)=(−1)^𝜈𝜖𝛿_𝜈(𝑒^∗₀𝑠_𝜈𝑒₀𝑡_𝜈𝑒^∗₀)𝑠^∗_𝜈𝑒₀=(−1)^𝜈𝜖𝛿_𝜈𝑒^∗₀𝑠^∗_𝜈𝑒₀. (3.19) If 𝜈=1, then 𝑒^∗₀𝑠^∗_𝜈𝑒₀=𝜖𝛿_𝜈(𝑒^∗₀𝑠_𝜈𝑒₀). Thus(𝑒^∗₀𝑠_𝜈𝑒₀)^∗=𝜖𝛿_𝜈(𝑒^∗₀𝑠_𝜈𝑒₀). (3.20) If 𝜈=0, then 𝑒^∗₀𝑠^∗_𝜈𝑒₀=𝜖𝛿_𝜈(𝑒^∗₀𝑠_𝜈𝑒₀). Thus(𝑒^∗₀𝑠_𝜈𝑒₀)^∗=𝜖𝛿_𝜈(𝑒^∗₀𝑠_𝜈𝑒₀). (3.21) So in all cases, we have(𝑒^∗₀𝑠_𝜈𝑒₀)^∗=𝜖𝛿_𝜈(𝑒^∗₀𝑠_𝜈𝑒₀). (3.22) We therefore have𝑒^∗₀𝑠_𝜈𝑒₀𝑡_𝜈𝑒^∗₀=𝑒^∗₀,𝑒₀𝑡_𝜈𝑒^∗₀𝑠_𝜈𝑒₀=𝑒₀,(𝑒₀𝑡_𝜈𝑒^∗₀)^∗=𝜖𝛿_𝜈𝑒₀𝑡_𝜈𝑒^∗₀,(𝑒^∗₀𝑠_𝜈𝑒₀)^∗=𝜖𝛿_𝜈𝑒^∗₀𝑠_𝜈𝑒₀. (3.23) For 𝑣_𝛼=𝑒₀𝑎_𝛼∈𝐼_𝛼, 𝑤_𝛽=𝑒₀𝑏_𝛽∈𝐼_𝛽,𝑣_𝛼𝑤^∗_𝛽=𝑒₀𝑎_𝛼𝑏^∗_𝛽𝑒^∗₀=𝑒₀𝑎_𝛼𝑏^∗_𝛽𝑒^∗₀𝑠_𝜈𝑒₀𝑡_𝜈𝑒^∗₀. (3.24) Define(𝑣_𝛼,𝑤_𝛽)_𝜈∶=𝑒₀𝑎_𝛼𝑏^∗_𝛽𝑒^∗₀𝑠_𝜈𝑒₀∈𝑒₀𝒜_{𝛼+𝛽+𝜈}𝑒₀=𝒟_{𝛼+𝛽+𝜈}. (3.25) By the last claim, (𝑣_𝛼,𝑣_𝛼)_𝜈∶=𝑒₀𝑎_𝛼𝑎^∗_𝛼𝑒^∗₀𝑠_𝜈𝑒₀=0, for all 𝑣_𝛼∈𝐼_𝛼. If (𝑣_𝛼,𝐼)_𝜈={0},𝑒₀𝑎_𝛼𝒜𝑒^∗₀𝑠_𝜈𝑒₀={0}, (3.26) and since 𝑒^∗₀𝑠_𝜈𝑒₀≠0,𝑒₀𝑎_𝛼=0,byprimeness. (3.27) Similarly, (𝐼,𝑤_𝛽)_𝜈={0} implies 𝑤_𝛽=0 and (,)_𝜈 is nondegenerate. If 𝑑_𝛿∈𝒟_𝛿, (𝑑_𝛿𝑣_𝛼,𝑤_𝛽)_𝜈=𝑑_𝛿(𝑣_𝛼,𝑤_𝛽)_𝜈. Moreover(𝑣_𝛼,𝑑_𝛿𝑤_𝛽)_𝜈=(𝑒₀𝑎_𝛼,𝑑_𝛿𝑒₀𝑏_𝛽)_𝜈=(−1)^𝛿𝛽𝑒₀𝑎_𝛼𝑏^∗_𝛽𝑒^∗₀𝑑^∗_𝛿𝑒^∗₀𝑠_𝜈𝑒₀=(−1)^𝛿𝛽𝑒₀𝑎_𝛼𝑏^∗_𝛽𝑒^∗₀𝑠_𝜈𝑒₀𝑡_𝜈𝑒^∗₀𝑑^∗_𝛿𝑒^∗₀𝑠_𝜈𝑒₀=(−1)^𝛿𝛽(𝑣_𝛼,𝑤_𝛽)_𝜈𝑒₀𝑡_𝜈𝑒^∗₀𝑑^∗_𝛿𝑒^∗₀𝑠_𝜈𝑒₀=(−1)^𝛿𝛽(𝑣_𝛼,𝑤_𝛽)_𝜈𝑑_𝛿, (3.28) where𝑑_𝛿∶=𝑒₀𝑡_𝜈𝑒^∗₀𝑑^∗_𝛿𝑒^∗₀𝑠_𝜈𝑒₀. (3.29) For 𝑑_𝛿∈𝒟_𝛿,𝑑_𝛿=𝑒₀𝑡_𝜈𝑒^∗₀(𝑒₀𝑡_𝜈𝑒^∗₀𝑑^∗_𝛿𝑒^∗₀𝑠_𝜈𝑒₀)^∗𝑒^∗₀𝑠_𝜈𝑒₀=(−1)^𝜈2+𝛿𝑒₀𝑡_𝜈𝑒^∗₀𝑠^∗_𝜈𝑒₀𝑑_𝛿𝑒₀𝑡^∗_𝜈𝑒^∗₀𝑠_𝜈𝑒₀=(−1)^𝜈2+𝛿𝜖𝛿_𝜈𝑒₀𝑑_𝛿𝜖𝛿_𝜈𝑒₀=(−1)^𝜈2+𝛿(𝛿_𝜈)²𝑑_𝛿=(−1)^𝛿𝑑_𝛿. (3.30) For 𝑐_𝛾∈𝒟_𝛾 and 𝑑_𝛿∈𝒟_𝛿,𝑐_𝛾𝑑_𝛿=𝑒₀𝑡_𝜈𝑒^∗₀(𝑐_𝛾𝑑_𝛿)^∗𝑒^∗₀𝑠_𝜈𝑒₀=(−1)^𝛾𝛿𝑒₀𝑡_𝜈𝑒^∗₀𝑑^∗_𝛿𝑐^∗_𝛾𝑒^∗₀𝑠_𝜈𝑒₀=(−1)^𝛾𝛿𝑒₀𝑡_𝜈𝑒^∗₀𝑑^∗_𝛿𝑒^∗₀𝑠_𝜈𝑒₀𝑡_𝜈𝑒^∗₀𝑐^∗_𝛾𝑒^∗₀𝑠_𝜈𝑒₀=(−1)^𝛾𝛿𝑑_𝛿𝑐_𝛾. (3.31) Thus “−” is a pseudo-superinvolution of 𝒟 and (,)_𝜈 is a nondegenerate pseudo-sesquilinear superform on 𝐼 whose adjoint is ∗. Finally,(𝑣_𝛼,𝑤_𝛽)_𝜈=𝑒₀𝑡_𝜈𝑒^∗₀(𝑒₀𝑎_𝛼𝑏^∗_𝛽𝑒^∗₀𝑠_𝜈𝑒₀)^∗𝑒^∗₀𝑠_𝜈𝑒₀=(−1)^{𝛼𝛽+𝛽}(−1)^{𝜈(𝛼+𝛽)}𝑒₀𝑡_𝜈𝑒^∗₀𝑠^∗_𝜈𝑒₀𝑏_𝛽𝑎^∗_𝛼𝑒^∗₀𝑠_𝜈𝑒₀=(−1)^{𝛼𝛽+𝛽}(−1)^{𝜈(𝛼+𝛽)}𝜖𝛿_𝜈𝑒₀𝑏_𝛽𝑎^∗_𝛼𝑒^∗₀𝑠_𝜈𝑒₀=(−1)^{𝛼𝛽+𝛽}(−1)^{𝜈(𝛼+𝛽)}𝜖𝛿_𝜈(𝑤_𝛽,𝑣_𝛼)_𝜈. (3.32) If 𝜈=0, then (𝑣_𝛼,𝑤_𝛽)₀=(−1)^{𝛼𝛽+𝛽}𝜖𝛿₀(𝑤_𝛽,𝑣_𝛼)₀, and hence(𝑤_𝛽,𝑣_𝛼)₀=(−1)^{𝛼𝛽+𝛽}𝜖𝛿₀(𝑣_𝛼,𝑤_𝛽)₀. (3.33) Thus (,)₀ is 𝜖-Hermitian pseudo-superform. If 𝜈=1, then we have assumed that 𝒟₁={0} and therefore (𝑣_𝛼,𝑤_𝛼)₁=0, for all 𝑣_𝛼,𝑤_𝛼∈𝐼_𝛼. Hence the right-hand side is 0 unless 𝛼+𝛽=1. Thus for all 𝑣_𝛼∈𝐼_𝛼, 𝑤_𝛽∈𝐼_𝛽,(𝑣_𝛼,𝑤_𝛽)_𝜈=(−1)^{𝜈+𝛼𝛽+𝛽}𝜖𝛿_𝜈(𝑤_𝛽,𝑣_𝛼)_𝜈=(−1)^{𝛼𝛽+𝛽}𝜖(−𝛿_𝜈)(𝑤_𝛽,𝑣_𝛼)_𝜈. (3.34) Thus(𝑤_𝛽,𝑣_𝛼)₁=(−1)^{𝛼𝛽+𝛽}𝜖𝛿₁(𝑣_𝛼,𝑤_𝛽)₁ (3.35) and (,)₁ is an 𝜖-Hermitian pseudo-superform.

If 𝒜=𝑀_𝑛(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 𝑎_𝛼∈𝒟_𝛼 such that𝑥^𝐽2=𝑎_𝛼𝑥𝑎_𝛼⁻¹∀𝑥∈𝒟. (3.36) Now, 𝑢^𝐽∈𝑍(𝒟₀)=𝐾(𝑢) implies that 𝑢^𝐽=𝑐+𝑑𝑢 for some 𝑐,𝑑∈𝐾, and 𝑢^𝐽𝑣^𝐽=(𝑣𝑢)^𝐽=(−𝑢𝑣)^𝐽=−𝑣^𝐽𝑢^𝐽 implies that (𝑐+𝑑𝑢)𝑣^𝐽=−𝑣^𝐽(𝑐+𝑑𝑢)=−(𝑐−𝑑𝑢)𝑣^𝐽, thus 𝑐+𝑑𝑢=−𝑐+𝑑𝑢 implies 𝑐=0, and hence 𝑢^𝐽=𝑑𝑢, 𝑑∈𝐾. Moreover, (𝑢²)^𝐽=(𝑢^𝐽)² implies that 𝑢²=𝑑²𝑢², so 𝑑=1 or 𝑑=−1, which means that 𝑢^𝐽=𝑢 or 𝑢^𝐽=−𝑢. So, in all cases 𝑢^𝐽2=𝑢, thus 𝑢^𝐽2=𝑎_𝛼𝑢𝑎_𝛼⁻¹=𝑢 implies that 𝛼=0, and hence 𝑎_𝛼=𝑎₀∈𝒟₀.
Case(1): if 𝑢^𝐽=𝑢, then 𝒟₀≈𝒟^∘₀ implies that 𝒟₀ has an involution of the first kind, so by [1, Chapter 8, Theorem 8.2], 𝑎₀𝑎^𝐽₀=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 𝑥^𝐼=(1+𝑎₀/𝛼)⁻¹𝑥^𝐽(1+𝑎₀/𝛼), an easy computation shows that 𝐼 is an involution of the first kind on 𝒟₀, since 𝑢^𝐼=𝑢, and hence 𝑥^𝐼=(1+𝑎₀/𝛼)⁻¹𝑥^𝐽(1+𝑎₀/𝛼) for all 𝑥∈𝒟 defines a 𝐾-antiautomorphism of the first kind on 𝒟, such that 𝑥^𝐼2=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), 𝑥^ℎ2=𝛼𝑥𝛼⁻¹ for all 𝑥∈𝒟, and 𝑢^ℎ=𝑢, and moreover, 𝛼𝛼^ℎ=𝛼^ℎ𝛼∈𝐾(𝑢). Suppose that 𝛼=𝑐+𝑑𝑢, where 𝑐,𝑑∈𝐾, then 𝑣^ℎ3=(𝑣^ℎ2)^ℎ=(𝛼𝑣𝛼⁻¹)^ℎ=(𝛼⁻¹)^ℎ𝑣^ℎ𝛼^ℎ, and 𝑣^ℎ3=(𝑣^ℎ)^ℎ2=𝛼𝑣^ℎ𝛼⁻¹, implies that 𝛼𝑣^ℎ𝛼⁻¹=(𝛼⁻¹)^ℎ𝑣^ℎ𝛼^ℎ, thus 𝛼^ℎ𝛼𝑣^ℎ(𝛼^ℎ𝛼)⁻¹=𝑣^ℎ, so, 𝛼^ℎ𝛼∈𝑍(𝒟)=𝐾. Therefore,(𝑐+𝑑𝑢)^ℎ(𝑐+𝑑𝑢)=(𝑐+𝑑𝑢)²=𝑐²+2𝑐𝑑𝑢+𝑑²𝑢²∈𝐾, (3.37) 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, 𝑣^ℎ2=(𝑑𝑢)𝑣(𝑑𝑢)⁻¹=−𝑣(𝑑𝑢)(𝑑𝑢)⁻¹=−𝑣, thus ℎ is a 𝐾-pseudo-superinvolution on 𝒟.
Case(2): if 𝑢^𝐽=−𝑢, then ∗∶𝒟→𝒟 defined by 𝑥^∗=𝑣𝑥^𝐽𝑣⁻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(𝑣^𝐽)⁻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≈𝒟^∘₀, so 𝒟₀ has an involution of the first kind (say 𝐽). Therefore, if ∗∶𝒟→𝒟 is defined by (𝑎+𝑏𝑢)^∗=𝑎^𝐽+𝛼𝑏^𝐽𝑢, where 𝑎,𝑏∈𝒟₀, then ∗ is a pseudo-superinvolution on 𝒟, since(𝑎+𝑏𝑢)^∗∗=(𝑎^𝐽+𝛼𝑏^𝐽𝑢)^∗=𝑎+𝛼²𝑏𝑢=𝑎−𝑏𝑢,(𝑎𝑢𝑏𝑢)^∗=(𝑎𝑏𝑢²)^∗=(𝑎𝑏)^∗(𝑢²)^∗=−(𝑎𝑏)^∗(𝑢^∗)²=−(𝑏^∗𝑢^∗)(𝑎^∗𝑢^∗)=−(𝑏𝑢)^∗(𝑎𝑢)^∗. (3.38)

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.
(1)If ∗ is a pseudo-superinvolution on 𝒟, then we can not choose 𝑢∈𝒟₁ such that 𝑢^∗=𝑢 or 𝑢^∗=−𝑢.(2)If − is an involution of 𝒟₀, then the superalgebra 𝒟 has a pseudo-superinvolution ∗ extending − given by(𝑎+𝑏𝑢)^∗=𝑎+𝛼𝑏𝑢. (3.39)

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+𝛼𝑏𝑢 (3.40) defines a pseudo-superinvolution on the superalgebra 𝒟=𝒟₀⊗𝐾[𝑢], extending “−,” such that (𝑢²)^∗=𝑢².

4. Existence of Pseudo-Superinvolution on 𝒜=𝑀_𝑝+𝑞(𝒟₀)

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, (4.1) where (𝐴_𝑖,∗|_𝐴𝑖) are simple and 𝐵_𝑖 are irreducible 𝐴-bimodules with𝐵^∗₁=𝐵₂,𝐵^∗₂=𝐵₁, (4.2) such that𝐴₁𝐵₁=𝐵₁=𝐵₁𝐴₂,𝐴₂𝐵₂=𝐵₂=𝐵₂𝐴₁,𝐵₁𝐵₂=𝐴₁,𝐵₂𝐵₁=𝐴₂,{0}=𝐴₂𝐵₁=𝐴₁𝐵₂=𝐵₁𝐴₁=𝐵₂𝐴₂=𝐵₁𝐵₁=𝐵₂𝐵₂. (4.3)

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=𝑀_𝑝+𝑞(𝒟), 𝒜₀=𝑀_𝑝(𝒟)⊕𝑀_𝑞(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=𝑀_𝑝+𝑞(𝒟₀), where?𝒜₀=𝑀_𝑝(𝒟₀)+𝑀_𝑞(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,𝑀_𝑝(𝒟₀)?has an involution?∼?and?(𝒜,∗)?is isomorphic to?𝑀_2𝑝(𝒟₀)?with the pseudo-superinvolution?∗?given by?𝑎𝑏𝑐𝑑^∗=^{0𝑥002𝑑𝑐}𝑑𝛼𝑏^{0𝑥002𝑑𝑐}_{0𝑥002𝑑𝑐}𝛼𝑐^{0𝑥002𝑑𝑐}𝑎^{0𝑥002𝑑𝑐}, (4.4) for 𝑎,𝑏,𝑐,𝑑∈𝑀_𝑝(𝒟₀), and 𝛼∈𝐾 such that 𝛼𝛼^{0𝑥002𝑑𝑐}=−1.
Conversely if 𝑀_𝑝(𝒟₀) has an involution ∼ then (4.4) defines a pseudo-superinvolution on the simple superalgebra 𝒜=𝑀_𝑝+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 𝑀_𝑝(𝒟) has an involution ^0𝑥0𝑎^02𝑑𝑐=𝑎^𝑡, 𝑡 the transpose. Since (𝒜₀,∗|_𝒜0) is simple, 𝑀_𝑞(𝒟) is anti-isomorphic to 𝑀_𝑝(𝒟) and 𝑝=𝑞. Up to isomorphism, (𝒜₀,∗|_𝒜0) is given by (𝑀_𝑝(𝒟)⊕𝑀_𝑝(𝒟),∗) with (𝑎,𝑏)^∗=(𝑏^{0𝑥002𝑑𝑐},𝑎^{0𝑥002𝑑𝑐}). Letting 𝑓₁₁=∑^𝑝_𝑖=1𝑒_𝑖𝑖=𝐼_𝑝000,𝑓₂₂=∑^2𝑝_𝑖=𝑝+1𝑒_𝑖𝑖=000𝐼_𝑝𝑓₁₂=∑^𝑝_𝑖=1𝑒_{𝑖𝑝+𝑖}=0𝐼_𝑝00,𝑓₂₁=∑^𝑝_𝑖=1𝑒_{𝑝+𝑖𝑖}=𝐼00_𝑝0. (4.5) We have𝒜₀=𝑀_𝑝(𝒟)𝑓₁₁⊕𝑀_𝑝(𝒟)𝑓₂₂,𝒜₁=𝑀_𝑝(𝒟)𝑓₁₂⊕𝑀_𝑝(𝒟)𝑓₂₁,𝑓^∗₁₁=𝑓₂₂,𝑓^∗₂₂=𝑓₁₁. (4.6) Hence𝑓^∗₁₂=(𝑓₁₁𝑓₁₂𝑓₂₂)^∗=𝑓₁₁𝑓^∗₁₂𝑓₂₂,𝑓^∗₁₂=𝑐𝑓₁₂,forsome𝑐∈𝑀_𝑝(𝒟). (4.7) For any 𝑎∈𝑀_𝑝(𝒟),(𝑎𝑓₁₂)^∗=(𝑎𝑓₁₁𝑓₁₂)^∗=𝑐𝑓𝑎₁₂^{0𝑥002𝑑𝑐}𝑓₂₂=𝑐𝑎^{0𝑥002𝑑𝑐}𝑓₁₂. (4.8) While(𝑎𝑓₁₂)^∗=(𝑓₁₂(𝑎𝑓₂₂))^∗=𝑎^{0𝑥002𝑑𝑐}𝑓₁₁𝑐𝑓₁₂=𝑎^{0𝑥002𝑑𝑐}𝑐𝑓₁₂. (4.9) Therefore, 𝑐∈𝑍(𝑀_𝑝(𝒟)). Moreover 𝑓^∗∗₁₂=−𝑓₁₂=(𝑐𝑓₁₂)^∗=𝑐^{0𝑥002𝑑𝑐}𝑐𝑓₁₂ implies ^0𝑥0𝑐^02𝑑𝑐𝑐=−𝐼_𝑝. So 𝑐=𝛼∈𝐾 with 𝛼𝛼^{0𝑥002𝑑𝑐}=−1. Similarly 𝑓^∗₂₁=𝑑𝑓₂₁, 𝑑∈𝑍(𝑀_𝑝(𝒟)). But𝑓₂₂=𝑓^∗₁₁=(𝑓₁₂𝑓₂₁)^∗=−𝑓^∗₂₁𝑓^∗₁₂=−𝑑𝑐𝑓₂₁𝑓₁₂=−𝑑𝑐𝑓₂₂ (4.10) which implies −𝑑𝑐=1, and hence 𝑑=−𝑐⁻¹=−𝛼⁻¹=𝛼^{0𝑥002𝑑𝑐}. Therefore,(𝑎𝑓₁₂)^∗=𝑎^{0𝑥002𝑑𝑐}𝑓^∗₂₁=𝑎^{0𝑥002𝑑𝑐}𝛼^{0𝑥002𝑑𝑐}𝑓₂₁ (4.11) or𝑎𝑏𝑐𝑑^∗=^{0𝑥002𝑑𝑐}𝑑𝛼𝑏^{0𝑥002𝑑𝑐}_{0𝑥002𝑑𝑐}𝛼𝑐^{0𝑥002𝑑𝑐}𝑎^{0𝑥002𝑑𝑐}, (4.12) for 𝑎,𝑏,𝑐,𝑑∈𝑀_𝑝(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=𝑀_𝑝+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=𝑀_𝑝(𝒟₀),𝐴₂=𝑀_𝑞(𝒟₀),𝒜₁=ℬ=ℬ₁+ℬ₂, (4.13) and (𝒜₀,∗|_𝒜0) is not simple then (𝐴₁,∗|_𝐴1) and (𝐴₂,∗|_𝐴2) are simple and ℬ_𝑖 are irreducible 𝒜₀-bimodules with ℬ^∗₁=ℬ₂ and ℬ^∗₂=ℬ₁ satisfying the hypothesis of Theorem 4.2 then ∗ is given by𝑎𝑏𝑐𝑑^∗=^{0𝑥002𝑑𝑐}𝑎𝑐^{0𝑥002𝑑𝑐}−𝑏^{0𝑥002𝑑𝑐}𝑑^{0𝑥002𝑑𝑐}, (4.14) where 𝑎∈𝑀_𝑝(𝒟₀), 𝑑∈𝑀_𝑞(𝒟₀), and ∼ is an involution on 𝑀_𝑝(𝒟₀), 𝑀_𝑞(𝒟₀), and where ^0𝑥0𝑏^02𝑑𝑐∈𝑀_𝑞,𝑝(𝒟₀) for all?𝑏∈𝑀_𝑝,𝑞(𝒟₀), and ^0𝑥0𝑐^02𝑑𝑐∈𝑀_𝑝,𝑞(𝒟₀)? for all 𝑐∈𝑀_𝑞,𝑝(𝒟₀).
Conversely (4.14) defines a pseudo-superinvolution on 𝑀_𝑝+𝑞(𝒟₀).