In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first introduced by Max Koecher in his Jordan algebraic approach to Hermitian symmetric spaces and bounded symmetric domains of tube type. Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compact subgroup. Both groups act transitively on the compactification. The theory has been extended to cover all Hermitian symmetric spaces using the theory of Jordan pairs or Jordan triple systems. Koecher obtained the results in the more general case directly from the Jordan algebra case using the fact that only Jordan pairs associated with period two automorphisms of Jordan algebras are required.
Definitions
Let A be a unital Jordan algebra over a field k of characteristic ≠ 2.See: For a in A define the Jordan multiplication operator on A by
and the quadratic representation Q(a) by
It satisfies
the fundamental identity
the commutation or homotopy identity
where
In particular if a or b is invertible then
It follows that A with the operations Q and R and the identity element defines a quadratic Jordan algebra, where a quadratic Jordan algebra consists of a vector space A with a distinguished element 1 and a quadratic map of A into endomorphisms of A, a ↦ Q(a), satisfying the conditions:
("fundamental identity") ("commutation or homotopy identity"), where
The Jordan triple product is defined by
so that
There are also the formulas
For y in A the mutation Ay is defined to the vector space A with multiplication
If Q(y) is invertible, the mutual is called a proper mutation or isotope.
Quadratic Jordan algebras
Let A be a quadratic Jordan algebra over a field k of characteristic ≠ 2. Following , a linear Jordan algebra structure can be associated with A such that, if L(a) is Jordan multiplication, then the quadratic structure is given by Q(a) = 2L(a)2 − L(a2).
Firstly the axiom Q(a)R(b,a) = R(a,b)Q(a) can be strengthened to
Indeed, applied to c, the first two terms give
Switching b and c then gives
Now let
Replacing b by a and a by 1 in the identity above gives
In particular
The Jordan product is given by
so that
The formula above shows that 1 is an identity. Defining a2 by a∘a = Q(a)1, the only remaining condition to be verified is the Jordan identity
In the fundamental identity
Replace a by a + t1, set b = 1 and compare the coefficients of t2 on both sides:
Setting b = 1 in the second axiom gives
and therefore L(a) must commute with L(a2).
Inverses Let A be a unital Jordan algebra over a field k of characteristic ≠ 2. An element a in a unital Jordan algebra A is said to be invertible if there is an element b such that ab = 1 and a2b = a.See:
Properties.See:
If and , then . The Jordan identity can be polarized by replacing by and taking the coefficient of . This gives
Taking or and or shows that commutes with and commutes with . Hence . Applying gives . Hence . Conversely if and , then the second relation gives . So both and are invertible. The first gives so that and are each other's inverses. Since commutes with it commutes with its inverse . Similarly commutes with . So and .
Indeed, if is invertible then the above implies is invertible with inverse . Any inverse b satisfies , so . Conversely if is invertible let . Then . The fundamental identity then implies that and are each other's inverses so that .
This follows from the formula .
Suppose that . Then by the fundamental identity is invertible, so is invertible.
This is an immediate consequence of the fundamental identity and the fact that is invertible if and only and are invertible.
In the commutation identity , set with . Then and . Since commutes with , .
If and commute, then implies . Conversely suppose that is invertible with inverse . Then . Moreover commutes with and hence its inverse . So it commutes with .
The algebra is commutative and associative, so if is an inverse there and . Conversely leaves invariant. So if it is bijective on it is bijective there. Thus lies in .
Elementary properties of proper mutations
In fact See: multiplication in the algebra Ay is given by
so by definition is commutative. It follows that
with
If e satisfies , then taking a = 1 gives
Taking a = e gives
so that L(y) and L(e) commute. Hence y is invertible and e = y−1.
Now for y invertible set
Then
Moreover,
Finally
since
Hence
Thus is a unital quadratic Jordan algebra. It therefore corresponds to a linear Jordan algebra with the associated Jordan multiplication operator M(a) given by
This shows that the operators satisfy the Jordan identity so that the proper mutation or isotope is a unital Jordan algebra. The correspondence with quadratic Jordan algebras shows that its quadratic representation is given by .
Nonunital mutations The definition of mutation also applies to non-invertible elements y. If A is finite-dimensional over R or C, invertible elements a in A are dense, since invertibility is equivalent to the condition that det Q(a) ≠ 0. So by continuity the Jordan identity for proper mutations implies the Jordan identity for arbitrary mutations. In general the Jordan identity can be deduced from Macdonald's theorem for Jordan algebras because it involves only two elements of the Jordan algebra. Alternatively, the Jordan identity can be deduced by realizing the mutation inside a unital quadratic algebra.See:
For a in A define a quadratic structure on A1 = A ⊕ k by
It can then be verified that is a unital quadratic Jordan algebra. The unital Jordan algebra to which it corresponds has Ay as an ideal, so that in particular Ay satisfies the Jordan identity. The identities for a unital quadratic Jordan algebra follow from the following compatibility properties of the quadratic map and the squaring map
.
Hua's identity Let A be a unital Jordan algebra. If a, b and a – b are invertible, then Hua's identity holds:
In particular if x and 1 – x are invertible, then so too is 1 – x−1 with
To prove the identity for x, set . Then . Thus commutes with and . Since , it also commutes with and . Since , also commutes with and .
It follows that . Moreover, since . So commutes with and hence . Thus has inverse .
Now let be the mutation of A defined by a. The identity element of is . Moreover, an invertible element c in A is also invertible in with inverse .
Let in . It is invertible in A, as is . So by the special case of Hua's identity for x in
Bergman operator If A is a unital Jordan algebra, the Bergman operator is defined for a, b in A by
If a is invertible then
while if b is invertible then
In fact if a is invertible
and similarly if b is invertible.
More generally the Bergman operator satisfies a version of the commutation or homotopy identity:
and a version of the fundamental identity:
There is also a third more technical identity:
Quasi-invertibility Let A be a finite-dimensional unital Jordan algebra over a field k of characteristic ≠ 2.In the main application in , A is finite dimensional. In that case invertibility of operators on A is equivalent to injectivity or surjectivity. The general case is treated in and . For a pair with and invertible define
In this case the Bergman operator defines an invertible operator on A and
In fact
Moreover, by definition is invertible if and only if is invertible. In that case
Indeed,
The assumption that be invertible can be dropped since can be defined only supposing that the Bergman operator is invertible. The pair is then said to be quasi-invertible. In that case is defined by the formula
If is invertible, then for some . The fundamental identity implies that . So by finite-dimensionality is invertible. Thus is invertible if and only if is invertible and in this case
In fact
so the formula follows by applying to both sides.
As before is quasi-invertible if and only if is quasi-invertible; and in that case
If k = R or C, this would follow by continuity from the special case where and were invertible. In general the proof requires four identities for the Bergman operator:
In fact applying to the identity yields
The first identity follows by cancelling and . The second identity follows by similar cancellation in
.
The third identity follows by applying the second identity to an element d and then switching the roles of c and d. The fourth follows because
.
In fact is quasi-invertible if and only if is quasi-invertible in the mutation . Since this mutation might not necessarily unital this means that when an identity is adjoint becomes invertible in . This condition can be expressed as follows without mentioning the mutation or homotope:
In fact if is quasi-invertible, then satisfies the first identity by definition. The second follows because . Conversely the conditions state that in the conditions imply that is the inverse of . On the other hand, for in . Hence is invertible.
Equivalence relation Let A be a finite-dimensional unital Jordan algebra over a field k of characteristic ≠ 2. Two pairs with invertible are said to be equivalent if is invertible and .
This is an equivalence relation, since if is invertible so that a pair is equivalent to itself. It is symmetric since from the definition . It is transitive. For suppose that is a third pair with invertible and . From the above
is invertible and
As for quasi-invertibility, this definition can be extended to the case where and are not assumed to be invertible.
Two pairs are said to be equivalent if is quasi-invertible and . When k = R or C, the fact that this more general definition also gives an equivalence relation can deduced from the invertible case by continuity. For general k, it can also be verified directly:
The relation is reflexive since is quasi-invertible and . The relation is symmetric, since . The relation is transitive. For suppose that is a third pair with quasi-invertible and . In this case
so that is quasi-invertible with
The equivalence class of is denoted by .
Structure groups
Let be a finite-dimensional complex semisimple unital Jordan algebra. If is an operator on , let be its transpose with respect to the trace form. Thus , , and . The structure group of A consists of g in such that
They form a group . The automorphism group Aut A of A consists of invertible complex linear operators g such that L(ga) = gL(a)g−1 and g1 = 1. Since an automorphism g preserves the trace form, g−1 = gt.
The structure group is closed under taking transposes g ↦ gt and adjoints g ↦ g*. The structure group contains the automorphism group. The automorphism group can be identified with the stabilizer of 1 in the structure group. If a is invertible, Q(a) lies in the structure group. If g is in the structure group and a is invertible, ga is also invertible with (ga)−1 = (gt)−1a−1. The structure group Γ(A) acts transitively on the set of invertible elements in A. Every g in Γ(A) has the form g = h Q(a) with h an automorphism and a invertible.
The complex Jordan algebra A is the complexification of a real Euclidean Jordan algebra E, for which the trace form defines an inner product. There is an associated involution on which gives rise to a complex inner product on . The unitary structure group Γu(A) is the subgroup of Γ(A) consisting of unitary operators, so that . The identity component of is denoted by . It is a connected closed subgroup of .
The stabilizer of 1 in Γu(A) is Aut E. Every g in Γu(A) has the form g = h Q(u) with h in Aut E and u invertible in A with u* = u−1.