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 [3] 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 [1] 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 [4] 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 [5] 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 [7] 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, .
We will start with the case .
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.