Bi-Orthogonal Symmetry Groups

• Bi-orthogonal symmetry groups are groups that preserve two distinct orthogonal decompositions in both classical and quantum algebraic contexts.
• They are constructed via universal compact quantum groups using free product algebras and Van Daele–Wang unitaries to ensure dual filtration preservation.
• Their applications span noncommutative geometry, quantum symmetry breaking, and the classification of invariant subspaces under products of orthogonal groups.

A bi-orthogonal symmetry group encodes the structure-preserving automorphisms of mathematical objects that carry two distinct orthogonal decompositions—either in the context of classical group actions on fields such as real matrices, or as universal compact quantum groups acting on C*-algebras with multiple orthogonal filtrations. In the quantum setting, these groups universally capture all “bi-orthogonal” symmetries by requiring group actions to preserve both orthogonal filtrations simultaneously, with natural emergence in the study of noncommutative geometry and quantum symmetries. In the classical context, “bi-orthogonal” symmetry refers to the action of products of orthogonal groups (such as $O(N) \times O(M)$) on spaces with structured bilinear invariants.

1. Orthogonal Filtrations and Bi-Orthogonal Actions

Let $A$ be a unital C$^*$-algebra equipped with a faithful state $\varphi$. An orthogonal filtration is a decomposition $A = \bigoplus_{i \in I} V_i$ such that $V_0 = \mathbb{C}1_A$, $\varphi(a^*b) = 0$ for $a \in V_i$, $b \in V_j$, $i \neq j$, each $V_i$ is finite-dimensional, $V_i^* = V_{i^*}$ for some involution on $I$, and the span of all $V_i$ is dense in $A$.

A compact quantum group $G$ acts on $A$ by a unital $*$-homomorphism $\alpha: A \to A \otimes C(G)$, compatible with the quantum group structure. For a filtration-preserving action, $\alpha(V_i) \subset V_i \otimes C(G)$ for all $i$. In the bi-orthogonal scenario, two orthogonal filtrations $$A = \bigoplus_{i \in I} V_i = \bigoplus_{j \in J} W_j$$ (with respect to the same state) are given and $G$ acts bi-orthogonally if $\beta(V_i) \subset V_i \otimes C(G)$ and $\beta(W_j) \subset W_j \otimes C(G)$ for all $i,j$ (Banica et al., 2011).

2. Universal Construction of Bi-Orthogonal Quantum Symmetry Groups

The bi-orthogonal symmetry group of $(A; (V_i), (W_j))$, denoted $\mathrm{QISO}_{\mathrm{bi}}(A; (V_i), (W_j))$, is the universal compact quantum group acting bi-orthogonally on $A$. It is constructed as follows:

• For each $V_i$, select an orthonormal basis and build the Van Daele–Wang universal unitary algebra $A_u(Q^{(i)})$; similarly, for each $W_j$, build $A_u(R^{(j)})$.
• Form the algebraic free product $$D = \ast_{i \in I} A_u(Q^{(i)}) \ast_{j \in J} A_u(R^{(j)})$$.
• Define a formal coaction $\delta: A \to A \otimes D$ determined by the bases.
• Impose $*$-homomorphism and multiplicativity relations necessary for the action, and quotient by the resulting Hopf $*$-ideal to obtain $$C(\mathrm{QISO}_{\mathrm{bi}}(A)) = D / I_{\mathrm{bi}}$$.
• This construction yields a well-defined compact quantum group that acts as the universal bi-orthogonal symmetry group (Banica et al., 2011).

3. Classical Symmetry Breaking by Bi-Fundamentals and Orthogonal Groups

In the context of symmetry breaking by real bi-fundamental Higgs fields transforming in the $(N, M)$ of $O(N) \times O(M)$, the group action is defined by independent real orthogonal rotations on the rows and columns of an $N \times M$ matrix $\Phi$. The most general renormalizable $O(N) \times O(M)$-invariant potential is:

$$V[\Phi] = -\mu^2 \operatorname{Tr}(\Phi\Phi^T) + \frac{1}{2}\lambda_1 (\operatorname{Tr} \Phi \Phi^T)^2 + \frac{1}{2}\lambda_2 \operatorname{Tr}(\Phi \Phi^T)^2$$

The vacuum structure is classified by complete block-diagonalization under group action, reducing the problem to the case where $\langle\Phi\rangle$ consists of $K$ identical $1 \times 1$ blocks (for the case with two quartic couplings), each block corresponding to a symmetry breaking pattern. The unbroken subgroup is $O(N-K) \times O(M-K) \times O(K)_{\text{diag}}$ (Schellekens, 2017).

4. Key Examples: Free Group Algebras and Matrix Quantum Groups

A central example involves the reduced C$^*$-algebra of the free group $\mathbb{F}_n$. Orthogonal filtrations using word-length and block-length (number of alternating blocks in the free product decomposition) yield an explicit bi-orthogonal quantum symmetry group. The resulting group, denoted $K_n^+$, is a compact matrix quantum group whose function algebra $A_k(n)$ is generated by a $2n \times 2n$ matrix $$U = (U_{i\sigma, j\tau})_{i\sigma, j\tau \in \{\pm1\} \times \{1,\dots,n\}}$$. The relations are

1. $$U_{i\sigma, j\tau} = U_{\overline{\imath}\,\overline{\sigma}, \,\overline{\jmath}\, \overline{\tau}}$$ (symmetry under opposite pairs)
2. Each $U_{i\sigma, j\tau}$ is a normal partial isometry
3. The matrix $(U_{i\sigma, j\tau} U_{i\sigma, j\tau}^*)$ is “magic” (orthogonal projections summing to $1$ per row and column)

The coproduct is given by $$\Delta(U_{i\sigma, j\tau}) = \sum_{k\rho} U_{i\sigma, k\rho} \otimes U_{k\rho, j\tau}$$ (Banica et al., 2011).

5. Representation Theory and Structural Properties

The representation theory of $K_n^+$ is classified by a hyperoctahedral category of non-crossing partitions $D_\circ$. Irreducible representations correspond to half-labeled partitions, and their fusion is governed by partition concatenation. This leads to exponential growth in the number of irreducibles, reflecting “free” quantum group behavior. Structural properties include:

• Coamenability: If all constituent Van Daele–Wang algebras are coamenable, the free product and its quotients (such as $K_n^+$) inherit coamenability.
• Haagerup and rapid decay properties: If each factor has the Haagerup property and rapid decay, so does the full bi-orthogonal symmetry group.
• Growth of irreducibles: Groups such as $K_n^+$ exhibit exponential growth in irreducible representations, inherited from the core structure of $$H_n^+ = \mathrm{QISO}(C^*(\mathbb{F}_n), \{\text{word length}\})$$ (Banica et al., 2011).

6. Relation to Single-Filtration Quantum Symmetry Groups

There is always an inclusion $$\mathrm{QISO}_{\mathrm{bi}}(A) \subset \mathrm{QISO}(A, \{V_k\})$$; for the free group algebra, $K_n^+ \subset H_n^+$. This inclusion is strict for $n \geq 2$, demonstrating that preservation of an additional filtration (e.g., block-length) strictly reduces the quantum symmetry group relative to the single-filtration case (Banica et al., 2011).

Context	Object	Symmetry Group
Classical (real matrix)	$N \times M$ real $\Phi$	$O(N) \times O(M)$
Quantum (C$^*$-algebra)	$(A; (V_i), (W_j))$	$\mathrm{QISO}_{\mathrm{bi}}(A; (V_i), (W_j))$
Free group C$^*$-alg.	$C^*(\mathbb{F}_n)$, 2 filtrations	$K^+_n$ (bi-orthogonal compact matrix quantum group)

7. Summary of Symmetry Breaking Patterns and Absolute Minima

For $O(N) \times O(M)$-invariant potentials with real bi-fundamental fields:

• Vacuum extrema are classified by integer $K = 1, 2, \dots, \min(N, M)$, each corresponding to $K$-dimensional identity blocks in $\Phi$.
• The residual subgroup is $O(N-K) \times O(M-K) \times O(K)_{\text{diag}}$.
• The ground state (absolute minimum) corresponds to $K=1$ if $\lambda_2 < 0$ (maximal breaking), or $K = K_\text{max}$ if $\lambda_2 > 0$ (block-diagonal filling).

Table: Absolute Minima and Residual Subgroup for $O(N)\times O(M)$

$K$	Vacuum block $\,\Phi \sim v \diag(1_K,0)\,$	Residual subgroup $H$
$1$	One $1\times 1$ block	$O(N-1) \times O(M-1) \times O(1)_\text{diag}$
$K_\text{max}$	$K_\text{max}$ $1\times 1$ blocks	$$O(N-K_\text{max}) \times O(M-K_\text{max}) \times O(K_\text{max})_\text{diag}$$

Intermediate $K$ yield saddle points or local minima, never the global minimum (Schellekens, 2017).

• Symmetry breaking and bi-fundamental Higgs fields: (Schellekens, 2017)
• Quantum symmetry groups with bi-orthogonal filtrations: (Banica et al., 2011)