#### Abstract

Let be a field of characteristic not and let be central simple superalgebra over , and let be superinvolution on . Our main purpose is to classify the group of automorphisms and inner automorphisms of (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 is nothing but a -graded associative ring. A -graded ideal of an associative super ring is called a superideal of . An associative super ring is simple if it has no nontrivial superideals. Let be an associative super ring with ; then is said to be a division super ring if all nonzero homogeneous elements are invertible; that is, every has an inverse necessarily in .

Let be a field of characteristic not ; an associative -graded -algebra is a finite dimensional central simple superalgebra over a field , if is the center of , where and the only superideals of are (0) and itself. We say that is the set of all homogeneous elements of .

Finite dimensional central simple associative superalgebras over a field are isomorphic to , where is a finite dimensional associative division superalgebra over ; that is, all nonzero elements of , , are invertible, and is an -dimensional -superspace. If , the grading of is induced by that of , , , and , so is a nontrivial decomposition of . Meanwhile, if , then the grading of is given by , .

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

Theorem 1 (Division Superalgebra Theorem [1, 2]). If is a finite dimensional associative division superalgebra over a field , then exactly one of the following holds, where throughout denotes a finite dimensional associative division algebra over :(i), and .(ii), , , and .(iii) or ; such that    if , , where and , for some quadratic Galois extension with Galois automorphism . Moreover, in the second case, if and does not embed in .

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

In  Racine described all types of superinvolutions on . It appears that if is a superinvolution on such that is simple algebra, then and is conjugate to the transpose involution. Otherwise, is conjugate to the orthosymplectic involution.

In  we proved that if is a finite dimensional central simple associative superalgebra over a field of characteristic not 2 such that has a superinvolution of the first kind, then , where is a division algebra over .

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

Let be a separable quadratic field extension over with Galois group , where and and (, if ). We recall a theorem of Albert-Reihm on the existence of -involution (involution of the second kind) which states that finite dimensional central simple algebra over has a -involution if and only if the corestriction of splits over . 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 is homogeneous element in , then is an inner automorphism on , with conjugation by Also , where is a central simple superalgebra of any type, means the set of all automorphisms on commuting with a given superinvolution defined on , and means the set of all inner automorphisms on commuting with

In this paper we examine the characterization of automorphisms and inner automorphisms on , where is a central simple superalgebra of any type, which commute with a given superinvolution defined on . 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 be any -superalgebra; we define the map by

This map, , is a superalgebra automorphism, called the sign automorphism, since for all and . The automorphism has order 2, if (unless ), and if

Lemma 3. Let be an odd superalgebra over a field with a superinvolution , and let be the sign automorphism. Then

Proof. If , then by [4, Theorem  3.2] Skolem-Noether theorem forces that or is inner but not both of them, so, or

Now, let be a superinvolution on an odd central simple superalgebra of any kind and let Then for some invertible But is inner, so Therefore, , , and which implies that So, we can choose to be , and hence for some

Now, . Let Let be a relation on defined as follows: Then one can easily show that is an equivalence relation on , and , where , is a group with the well-defined operation

Theorem 4. Let be an odd central simple superalgebra over a field with a superinvolution ; then .

Proof. Let such that ; then So, is an onto homomorphism such that . Therefore, .

Let be an even central simple superalgebra over , where is a nontrivial grading division superalgebra. Then by [1, Division Superalgebra Theorem] , where and and is the centralizer of in

Let be a superinvolution on , and let Then by the Skolem-Noether theorem there exists an invertible element such that , . Since , , we have for some If is inner, then for some invertible element which implies that , , and therefore Let ; if , then But , so centralizes ; thus This is a contradiction and therefore Therefore, for any such that is inner, , for some invertible element , and hence

Corollary 5. Let be an even central simple superalgebra over a field , where and is a nontrivial grading division superalgebra, and let be a superinvolution on . Then

Proof. If , then for any such that we get which implies that is inner and so for some invertible in . Thus But if , then it is easy to check that Therefore

Theorem 6. Let , where , and let be a superalgebra with a superinvolution Then .

Proof. Let such that Then So, is an onto homomorphism, where . Therefore, .

Next we introduce this example to show that may equal with for any .

Example 7. Let , let , and let , , where , , and . Then is a quaternion division superalgebra over field . Let be a superinvolution on defined by  , ; ; ; then Now, , since But , where , because if where , then where and Hence, Since , the highest power of in the right-hand side is even, but the highest power of in the left-hand side is odd, a contradiction.

Now we try to classify the group of automorphisms of central simple superalgebras , and , where is a division algebra over and

First let , where is a division algebra over , and let be any superinvolution on If , then, by [1, Isomorhpism Theorem], for some invertible element and thus,which implies that , , and therefore , where If is a superinvolution of the first kind, thenHence , which implies that and therefore Moreover, if is a superinvolution of the second kind on , then

Theorem 8. Let , where is a division algebra over , and let be any superinvolution on If is a superinvolution of the first kind, then , where is the group defined in Theorem 4. If is a superinvolution of the second kind, then , where and the equivalence relation on is defined by for some

Proof. If is a superinvolution of the first kind, then is a group isomorphism.
Similarly, if is a superinvolution of the second kind, then it is easy to check that is a group under the well-defined operation , and so is a group isomorphism. Thus in all cases we get .

Second let , where is a division algebra over and Let be any superinvolution on of any kind. If , then by [1, Isomorhpism Theorem] , for some invertible element in , since , the homogeneous invertible elements in are the invertible elements in . Thus , and, therefore, if , then , where is an invertible element in . Now, if , thenTherefore, , , and hence , where Thus it is easy to check that , where

#### 3. Examples of Group of Automorphisms of Even Superalgebras

In this section we classify in detail the groups of automorphisms of superalgebras , and , where or , where is the Hamiltonian quaternion algebra and is the field of real numbers.

Let , where or , and let be a superinvolution of the first kind on such that is simple; then by [1, Proposition  13] is defined by where is the involution on induced by . Then .

If , then by replacing by we get

If , then by replacing by we get

So .

Theorem 9. In the situation described above, let Then , where is the connected component of the identity in .

Proof. By [8, page 593], , where and is the connected component of the identity. Let Then is onto group homomorphism because if , then for some invertible element since is the connected component of the identity. Then it is easy to check that () since .
If , then by replacing by we get
If , then by replacing by we get Therefore, and Also it is easy to check that But So, or Therefore, , or , since . Therefore

In the second case, let , where or , and let be a superinvolution of the first kind on such that is not simple; then by [1, Proposition  14] is defined by where () and () and is the standard involution on .

Since is a superinvolution of the first kind we have . Similarly, .

Theorem 10. In the situation described above, let Then , where

Proof. Since is semisimple and , we get . If , then So . Therefore, is a group homomorphism and To see this let such that ; then or (since , and if and only if and (); since we get Therefore, , , which implies that , where and and hence since and Therefore, .

Theorem 11. As in the theorem above, if , then for a fixed invertible matrix such that

Proof. Since , then , which implies that . Let such that thenTherefore, ; thus, by Theorem 10, Now, Therefore,

For the case , , where (it is easy to check that ) and is the connected component of the identity in Now, let (), where ; then is a group homomorphism and where . Therefore,

Theorem 12. In the situation described above, if , then for a fixed invertible matrix , such that and is the connected component of the identity in .

Proof. Consider since Let such that ; thenTherefore, and so Now, Therefore,

#### Conflict of Interests

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