The Krein Cartan Decomposition of the Operator Algebra

Introduction

The Krein form on $\mathbb{B}$ induces on the algebra of $\mathbb{C}$-linear operators of $\mathbb{B}$ — a copy of $M_4(\mathbb{C})$ — an adjoint, an involution, a Lie algebra and a symmetric space, and the whole of the indefinite operator theory of this category is written in those terms. This article sets up that structure: the three adjoints of the operator algebra (the transpose of the bilinear form, the Hermitian adjoint, the Krein adjoint), the three involutions they define and the three classical Lie algebras that are their fixed spaces — the complex orthogonal algebra, $\mathfrak{u}(4)$ and $\mathfrak{u}(1,3)$; the Krein decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ into $J$-skew and $J$-self-adjoint operators; the involution $T\mapsto(T^{\dagger})^{-1}$ of the group and the reason it is not a Cartan involution; and the Cartan involution of $U(1,3)$ itself, whose symmetric space is the complex hyperbolic space of The Krein Level Sets and the Hyperbolic Structure. The group is The Krein Isometry Group and Its $J$-Contractions; the individual operators are J-Self-Adjoint and J-Unitary Operators on the Biquaternion Algebra.

Conventions. $\mathbb{B}\cong\mathbb{C}^{4}$ with coefficient basis $e_0,e_1,e_2,e_3$; the Hermitian form has Gram matrix $\mathrm{I}_4$, the bilinear form $\mathrm{I}_4$ and the Krein form $E=\mathrm{diag}(1,-1,-1,-1)$; $J={}^{\natural}$ with matrix $E$; the Krein adjoint of an operator $M$ is $M^{\dagger}=JM^{*}J$, where $M^{*}$ is the conjugate transpose; and $\mathfrak{g}=\mathrm{End}_{\mathbb{C}}(\mathbb{B})$ is the real Lie algebra of all $\mathbb{C}$-linear operators, of real dimension $32$.

The Three Adjoints of the Operator Algebra

Theorem (the three involutions). On $\mathfrak{g}$ the three pairings of The Three Pairings of the Biquaternion Algebra define three adjoints,

$$ B(M\tilde{P},\tilde{Q})=B(\tilde{P},M^{\mathsf T}\tilde{Q}), \qquad \langle M\tilde{P},\tilde{Q}\rangle=\langle\tilde{P},M^{*}\tilde{Q}\rangle, \qquad [M\tilde{P},\tilde{Q}]=[\tilde{P},M^{\dagger}\tilde{Q}], $$

namely the transpose $M^{\mathsf T}$, the Hermitian adjoint $M^{*}$ and the Krein adjoint $M^{\dagger}=JM^{*}J$; in the coefficient basis these are the transpose, the conjugate transpose and the conjugate transpose conjugated by $E$. Each gives an involution of $\mathfrak{g}$,

$$ \theta_{\mathsf T}(M)=-M^{\mathsf T},\qquad \theta_{*}(M)=-M^{*},\qquad \theta_{J}(M)=-M^{\dagger}=-JM^{*}J . $$

Proof. The three adjoints exist because the three forms are non-degenerate with invertible Gram matrices $\mathrm{I}_4,\mathrm{I}_4,E$, and the formulas in the basis are the standard ones for a bilinear, a sesquilinear and an $E$-sesquilinear form. The maps are $\mathbb{R}$-linear involutions of $\mathfrak{g}$.

Theorem (the three fixed algebras). The fixed algebras of the three involutions are the classical Lie algebras

$$ \mathfrak{g}^{\theta_{\mathsf T}}=\{M:M^{\mathsf T}=-M\}=\mathfrak{so}_4(\mathbb{C}), \qquad \mathfrak{g}^{\theta_{*}}=\{M:M^{*}=-M\}=\mathfrak{u}(4), $$ $$ \mathfrak{g}^{\theta_{J}}=\{M:M^{\dagger}=-M\}=\mathfrak{u}(1,3), $$

of real dimensions $12$, $16$ and $16$; each is closed under the commutator bracket, and the three are the Lie algebras of the isometry groups $O_4(\mathbb{C})$, $U(4)$ and $U(1,3)$ of the three pairings.

Proof. The fixed conditions are the defining equations $\mathfrak{so}_n$, $\mathfrak{u}(n)$ and $\mathfrak{u}(p,q)$; closure under the bracket is the standard fact for the fixed space of an involution that is an automorphism of the bracket, which each is, because $M\mapsto-M^{\mathsf T}$, $M\mapsto-M^{*}$ and $M\mapsto-M^{\dagger}$ satisfy $(MN)^{\sigma}=M^{\sigma}N^{\sigma}$ up to the sign pattern of a Lie algebra involution.

The Krein Decomposition

Definition. The Krein-decomposition of $\mathfrak{g}$ is the eigenspace decomposition of the involution $\theta_{J}$,

$$ \mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}, \qquad \mathfrak{k}=\{M:M^{\dagger}=-M\}=\mathfrak{u}(1,3), \qquad \mathfrak{p}=\{M:M^{\dagger}=M\}=J\cdot\mathrm{Herm}_4 . $$

Theorem (the decomposition and the bracket pattern). The decomposition is direct, $\dim_{\mathbb{R}}\mathfrak{k}=\dim_{\mathbb{R}}\mathfrak{p}=16$, the relations

$$ [\mathfrak{k},\mathfrak{k}]\subseteq\mathfrak{k}, \qquad [\mathfrak{k},\mathfrak{p}]\subseteq\mathfrak{p}, \qquad [\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{k}, \qquad \{\mathfrak{p},\mathfrak{p}\}\subseteq\mathfrak{p} $$

hold, with $\{\cdot,\cdot\}$ the anticommutator, so $\mathfrak{k}$ is a Lie subalgebra, $\mathfrak{p}$ is a Jordan subalgebra, and the pair $(\mathfrak{k},\mathfrak{p})$ is a Lie–Jordan pair attached to the Krein form.

Proof. $\mathfrak{k}$ and $\mathfrak{p}$ are the $\mp1$-eigenspaces of an involution, so they are direct and of equal dimension $32/2=16$. Write $M^{\epsilon}$ for an element with $M^{\dagger}=\epsilon M$, $\epsilon=\pm1$, and use $(AB)^{\dagger}=B^{\dagger}A^{\dagger}$. For $M,N\in\mathfrak{k}$ one has $(MN)^{\dagger}=(-N)(-M)=NM$, so $[M,N]^{\dagger}=NM-MN=-[M,N]$ and $[M,N]\in\mathfrak{k}$. For $M\in\mathfrak{k}$ and $N\in\mathfrak{p}$ one has $(MN)^{\dagger}=N(-M)=-NM$ and $(NM)^{\dagger}=(-M)N=-MN$, so $[M,N]^{\dagger}=(MN)^{\dagger}-(NM)^{\dagger}=MN-NM=[M,N]$ and $[M,N]\in\mathfrak{p}$. For $M,N\in\mathfrak{p}$ one has $(MN)^{\dagger}=NM$, so $[M,N]^{\dagger}=NM-MN=-[M,N]$ and $[M,N]\in\mathfrak{k}$. Finally $\{M,N\}^{\dagger}=NM+MN=\{M,N\}$ for $M,N\in\mathfrak{p}$.

Remark (the $J$-skew algebra is $J$ times the unitary algebra). As vector spaces

$$ \mathfrak{u}(1,3)=J\cdot\mathfrak{u}(4)=\{JM:M\in\mathfrak{u}(4)\}, $$

since $(JM)^{\dagger}=JM^{*}$ is in $\mathfrak{u}(1,3)$ exactly when $M$ is skew-Hermitian; the identification is not a Lie algebra isomorphism, because $J$ is not central.

The Involution of the Group

Definition. On the group $GL(\mathbb{B})$ of invertible operators define

$$ \theta_{J}(T)=\bigl(T^{\dagger}\bigr)^{-1}. $$

Theorem (the fixed group is $U(1,3)$). The map $\theta_{J}$ is an involution with fixed group exactly the Krein isometry group $U_{J}(\mathbb{B})\cong U(1,3)$; its differential at the identity is the involution $\theta_{J}(M)=-M^{\dagger}$ of $\mathfrak{g}$ and its fixed algebra is $\mathfrak{k}=\mathfrak{u}(1,3)$.

Proof. $\theta_{J}^{2}(T)=\bigl((T^{\dagger})^{-1}\bigr)^{\dagger}{}^{-1}=(T^{\dagger\dagger})^{-1}{}^{-1}=T$, using $(S^{-1})^{\dagger}=(S^{\dagger})^{-1}$; the fixed condition $\theta_{J}(T)=T$ is $T^{\dagger}T=\mathrm{id}$, the definition of $J$-unitarity, and the group is $U(1,3)$.

Theorem (the involution is not Cartan). The fixed group $U_{J}(\mathbb{B})\cong U(1,3)$ of $\theta_{J}$ is non-compact, so $\theta_{J}$ is not a Cartan involution of $GL(\mathbb{B})$: the fixed group of a Cartan involution is compact by definition.

Proof. $U(1,3)$ contains the boosts, which are unbounded (The Krein Isometry Group and Its $J$-Contractions).

The Cartan Involution of $U(1,3)$

Theorem (the Cartan involution). The restriction of the definite involution

$$ \theta_{0}(T)=(T^{*})^{-1} $$

to the Krein isometry group is a Cartan involution of $U(1,3)$; its fixed group is the maximal compact subgroup $U(1)\times U(3)$, and its differential $\theta_{0}(M)=-M^{*}$ splits

$$ \mathfrak{u}(1,3)=\mathfrak{k}_{0}\oplus\mathfrak{p}_{0}, \qquad \mathfrak{k}_{0}=\mathfrak{u}(1)\oplus\mathfrak{u}(3), \qquad \dim_{\mathbb{R}}\mathfrak{k}_{0}=10,\quad \dim_{\mathbb{R}}\mathfrak{p}_{0}=6 . $$

Proof. For $T\in U(1,3)$, that is $T^{*}ET=E$, the operator $T^{*}$ is again in $U(1,3)$ and $\theta_{0}$ preserves the group; its fixed elements are the isometries that are unitary for $\langle\cdot,\cdot\rangle$, i.e. $U(4)\cap U(1,3)=U(1)\times U(3)$. The differential is $-M^{*}$, whose fixed algebra on $\mathfrak{u}(1,3)$ is $\mathfrak{u}(1)\oplus\mathfrak{u}(3)$ of real dimension $1+9=10$, leaving $\mathfrak{p}_{0}$ of dimension $16-10=6$; the fixed group is compact, which is the definition of a Cartan involution.

Corollary (the symmetric space). The quotient

$$ U(1,3)\,/\,\bigl(U(1)\times U(3)\bigr) $$

is a Riemannian symmetric space of real dimension $6$, and it is the complex hyperbolic space $\mathbb{CH}^{3}$ of The Krein Level Sets and the Hyperbolic Structure, realised as the space of maximal positive definite subspaces, that is the open unit ball of $\mathbb{C}^{3}$.

Proof. The quotient of a Lie group by a maximal compact subgroup is a symmetric space of non-compact type, of dimension $\dim\mathfrak{p}_{0}=6$; the identification with the space of positive lines and with the ball is the parametrisation of the maximal positive definite subspaces. The Cartan decomposition of the group is

$$ U(1,3)=\bigl(U(1)\times U(3)\bigr)\cdot\exp(\mathfrak{p}_{0}), $$

the compact part followed by the hyperbolic part, whose exponentials are the boosts and their conjugates.

The Trace Form and the Killing Form

Theorem (the trace form). The pairing $(M,N)\mapsto\operatorname{Tr}(MN)$ is a non-degenerate complex bilinear form on $\mathfrak{g}$, invariant under the three adjoints up to conjugation; restricted to $\mathfrak{u}(1,3)$ the definite form $(M,N)\mapsto\operatorname{Re}\operatorname{Tr}(MN)$ has signature $(6,10)$ on the real vector space of dimension $16$, positive on $\mathfrak{p}_{0}$ and negative on $\mathfrak{k}_{0}$.

Proof. The trace form is non-degenerate on $M_4(\mathbb{C})$, and $\operatorname{Tr}(MN)=\operatorname{Tr}(NM)$ makes it invariant under the adjoint action. For $\mathfrak{u}(p,q)$ the trace form is proportional to the Killing form, whose signature is $(2pq,p^{2}+q^{2})$, which for $p=1,q=3$ is $(6,10)$; this is the same statement as the Cartan decomposition of the previous section.

Theorem (the Jordan cone). The $J$-self-adjoint part $\mathfrak{p}=J\cdot\mathrm{Herm}_4$ is a Jordan algebra under the anticommutator, and it contains the $J$-positive cone $\mathcal{P}_{J}=J\cdot\{S\ge0\}$ of Indefinite Positivity and the Krein Cone of the Biquaternion Algebra as its cone of squares.

Proof. The anticommutator of two $J$-self-adjoint operators is $J$-self-adjoint, as shown above, and the Jordan axiom follows from associativity of the product in $\mathfrak{g}$. The cone statement is the definition of $\mathcal{P}_{J}$.

Worked Examples

Three involutions of the identity-map operator. $M=\mathrm{id}$: $M^{\mathsf T}=M^{*}=M^{\dagger}=\mathrm{id}$, and $\theta(\mathrm{id})=-\mathrm{id}$ for each of the three involutions.

The fundamental symmetry. $M=J$: $J$ is the matrix $E$, self-adjoint for all three adjoints, and $\theta_{J}(J)=-J$; so $J\in\mathfrak{p}$, the $J$-self-adjoint part.

A $J$-skew element. $M=JL$, where $L$ is any definite-skew operator; $M^{\dagger}=-M$, so $M\in\mathfrak{k}=\mathfrak{u}(1,3)$.

A boost generator. The derivative at $t=0$ of the boost $T_{t}$ of The Krein Isometry Group and Its $J$-Contractions is the operator $H$ with $H e_0=e_1$, $H e_1=e_0$, $H e_2=H e_3=0$; it is $J$-self-adjoint, $H^{\dagger}=H$, hence an element of $\mathfrak{p}_{0}$, the hyperbolic part, and its exponential is the boost.

A compact generator. $M=L_{ie_0}$: skew-Hermitian and $J$-skew, an element of $\mathfrak{k}_{0}=\mathfrak{u}(1)\oplus\mathfrak{u}(3)$.

Summary

The algebra of $\mathbb{C}$-linear operators of $\mathbb{B}$ is $\mathfrak{g}=M_4(\mathbb{C})=GL_4(\mathbb{C})$, of real dimension $32$, and the three pairings of the algebra give its three adjoints: the transpose $M^{\mathsf T}$ for the bilinear form, the Hermitian adjoint $M^{*}$, and the Krein adjoint $M^{\dagger}=JM^{*}J$. The corresponding involutions $M\mapsto-M^{\sigma}$ have the fixed algebras $\mathfrak{so}_4(\mathbb{C})$, $\mathfrak{u}(4)$ and $\mathfrak{u}(1,3)$, of real dimensions $12$, $16$ and $16$. The Krein involution splits $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ into the $J$-skew operators $\mathfrak{k}=\mathfrak{u}(1,3)$ and the $J$-self-adjoint operators $\mathfrak{p}=J\cdot\mathrm{Herm}_4$, each of dimension $16$; $\mathfrak{k}$ is a Lie algebra, $\mathfrak{p}$ a Jordan algebra, and the bracket of $\mathfrak{k}$ with $\mathfrak{p}$ stays in $\mathfrak{p}$. On the group the involution $T\mapsto(T^{\dagger})^{-1}$ has fixed group the non-compact $U(1,3)$, so it is not a Cartan involution; the Cartan involution is the restriction of $T\mapsto(T^{*})^{-1}$, whose fixed group is the maximal compact $U(1)\times U(3)$, with $\dim\mathfrak{k}_{0}=10$ and $\dim\mathfrak{p}_{0}=6$, and whose symmetric space is the complex hyperbolic space $\mathbb{CH}^{3}$ of real dimension $6$. The trace form of $\mathfrak{u}(1,3)$ has signature $(6,10)$, positive on the hyperbolic part and negative on the compact part, and the $J$-positive cone sits inside the Jordan algebra $\mathfrak{p}$.

Summary of Notation

Symbol	Meaning
$\mathfrak{g}=\mathrm{End}_{\mathbb{C}}(\mathbb{B})\cong M_4(\mathbb{C})$	The operator algebra; real dimension $32$
$M^{\mathsf T}$, $M^{}$, $M^{\dagger}=JM^{}J$	The three adjoints
$\theta_{\mathsf T},\theta_{*},\theta_{J}$	The three involutions $M\mapsto-M^{\sigma}$
$\mathfrak{so}_4(\mathbb{C})$, $\mathfrak{u}(4)$, $\mathfrak{u}(1,3)$	The three fixed algebras; dimensions $12$, $16$, $16$
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$	The Krein decomposition
$\mathfrak{k}=\mathfrak{u}(1,3)$, $\mathfrak{p}=J\cdot\mathrm{Herm}_4$	The $J$-skew and $J$-self-adjoint parts
$\theta_{J}(T)=(T^{\dagger})^{-1}$	The group involution; fixed group $U(1,3)$, non-compact
$\theta_{0}(T)=(T^{*})^{-1}$, $U(1)\times U(3)$	The Cartan involution and its fixed group
$U(1,3)/(U(1)\times U(3))\cong\mathbb{CH}^{3}$	The symmetric space of real dimension $6$
$(6,10)$	The signature of the trace form of $\mathfrak{u}(1,3)$