Bilinear Algebraic Semigroups: Structure & Applications

• Bilinear algebraic semigroups are algebraic structures defined by a bilinear multiplication that adheres to semigroup properties and appears in various mathematical settings.
• They provide a framework linking representation theory, random matrix ensembles, and orthogonal polynomial systems to analyze spectral properties and eigenvalue statistics.
• Their applications in the Langlands program and differential Galois theory offer powerful methods for examining automorphic forms, combinatorial decompositions, and control systems.

A bilinear algebraic semigroup is an algebraic structure defined by a set equipped with a bilinear multiplication operation possessing semigroup properties, often realized in contexts bridging representation theory, algebraic geometry, random matrix theory, and automorphic forms. The notion arises naturally in the framework of the global Langlands program and has profound connections to spectral theory, combinatorics, and symmetry groups.

1. Structural Definition and Langlands Program Context

Bilinear algebraic semigroups are constructed from two symmetric families of algebraic or transcendental objects, referred to as “quanta,” with the multiplication operation formalized as a right-by-left product (often realized for matrix groups as $$GL_{2n}(F_v\times F_v)\simeq T^t_{2n}(F_v)\times T_{2n}(F_v)$$). Here, $F_v$ denotes an extension field (ramified, pseudounramified, or compact) or its toroidal compactification.

In the global Langlands program, these structures encode the “bisemiobject” structure: each bisemiobject admits representation in a bisemisheaf $M^{(2n)}_{v_R}\otimes M^{(2n)}_{v_L}$, which serves as the functional module, providing automorphic or “quantized” representations parameterized by conjugacy class representatives (“bisections”) corresponding to a specified number of quanta.

The bilinear multiplication on such bisemiobjects is intimately linked to the action of a differential bioperator $(D_R\otimes D_L)$, which operates on the right and left parts, respectively, producing eigenbifunctions and eigenbivalues; these encode spectral data essential in quantum mechanical models and statistical physics (Pierre, 2011).

2. Bilinear Semigroups in Random Matrices and Orthogonal Polynomial Ensembles

The bilinear algebraic semigroup framework provides a geometric reinterpretation of random matrix ensembles, specifically when generalizing to BGUE/BGOE (bilinear Gaussian unitary/orthogonal ensembles). In this setting, the ensemble consists of bilinear matrices of the form $G=TG^T\times TG$, where $TG\in T_r$ are “triangular” matrices and $TG^T\in T^t_r$.

The probability distribution for the eigenvalues in this bilinear context becomes

$$d\mu^{\rm BGUE}_r(x) \propto \exp\left(-r\sum_{i=1}^r x_i^2\right) \prod_{i<j}(x_i - x_j)^2\, dx_1\cdots dx_r.$$

The determinantal structure and underlying orthogonal polynomials $P_i(x)$ are induced by Hecke operators corresponding to subsemilattices of biquanta, and the roots of these polynomials encode the (pseudo)unramified eigenbivalues of the operator $(D_R\otimes D_L)$ acting on bisemisheaves.

This construction establishes a profound link between algebraic–geometric symmetries and the statistical properties of random matrices, reinterpreting the standard joint probability density via representations of the bilinear semigroup (Pierre, 2011).

3. Differential Bilinear Galois Semigroup and Its Representations

Central to the theory is the notion of a differential bilinear Galois semigroup, defined as the semidirect product of the automorphism semigroups $Aut_k(\phi_R())\times Aut_k(\phi_L())$ acting on functional objects and their duals. The representation spaces are birationally equivalent to matrix semigroups: $$\text{Rep}^{(2k)}\left(Aut_k(\phi_R())\times Aut_k(\phi_L())\right)\simeq GL_r\left(\phi_R^{(2k)}\times\phi_L^{(2k)}\right),$$ where the matrices encode transformations on biquantized lattices.

The symmetry described by this semigroup dictates the structure of bisemisheaves and bisections, underlying automorphic forms, and transferring directly to the spectral analysis of random matrices and to the local and global statistics of L-functions and zeta zeros.

4. Spectral Problems: Eigenbivalues and Statistical Spacings

The eigenbivalue problem for bilinear algebraic semigroups is formulated as

$$(D_R\otimes D_L)\, \phi(G^{(1)}(F_v\times F_v)) = ER(j)\, \phi(G^{(1)}(F_v\times F_v)),\quad 1\leq j\leq r,$$

with $ER(j)$ interpreted as energy levels of biquanta. The consecutive spacings $\delta ER(j)=ER(j+1)-ER(j)$ correspond to infinitesimal Lie algebra generators (specifically for $gl_1(F^{(nr)}_{v^1})$ in the pseudounramified setting) and measure the “addition” of free biquanta in a spectral lattice.

This spectral interpretation extends directly to the Riemann zeta function, where the nontrivial zeros $s = \tfrac{1}{2} \pm i\gamma_j$ are realized as square roots of eigenbivalues

$\gamma_j = \sqrt{E^{(nr)}R,L(j)},$

such that the spacings $\delta \gamma_j$ coincide with the spectral gaps in the bilinear ensemble. Proposition 3.27 establishes a bijective correspondence between $\{\delta E(j)\}_j$ in BGUE and $\{\delta\gamma_j\}_j$, revealing a deep symmetry underpinning both random matrix statistics and the distribution of zeta zeros (Pierre, 2011).

5. Semigroup Structure in Representation Theory and Algebraic Geometry

Beyond the analytic and spectral implications, bilinear algebraic semigroups play a critical role in tensor product decomposition for Lie algebras—for instance, the presentation of semigroup algebras $Q_T(sl_3(\mathbb{C}))$ encodes the combinatorics of $sl_3(\mathbb{C})$ tensor products. These semigroups underpin toric degenerations of the Grassmannian $Gr_3(\mathbb{C}^n)$ and model combinatorial invariants relevant for moduli spaces and coordinate ring calculations (Manon et al., 2012).

Combinatorial bialgebras constructed via commutative semigroups (with the coproduct $\Delta_S$ defined by multiplicative decompositions) serve as algebraic realizations of these structures, admitting dual bases (via the PBW theorem) and Schützenberger factorization, facilitating explicit resolutions of the identity and mirroring the bilinear decomposition of algebraic semigroups (Deneufchâtel, 2012).

6. Structural Properties: Regularity and Classification

Algebraic semigroups, including bilinear variants, satisfy strong $\pi$-regularity: for each $x$ in the semigroup, a power $x^n$ enters the unit group of a corner $eSe$ associated with an idempotent $e$,

$x^n \in H_e,\quad e^2 = e.$

This structural stabilization ensures a group-like behavior within every semigroup and is preserved in the bilinear context under mild algebraic-geometric conditions (Brion et al., 2012).

Complete and commutative algebraic semigroups display rich geometric stratification via their idempotent scheme $E(S)$, which, when $S$ is irreducible and commutative, is shown to be finite and encoded by the faces of a rational polyhedral convex cone, with the minimal containing subsemigroup realized as an affine toric variety (Brion, 2013).

7. Connections to Control Theory, Operads, and Extended Structures

Extended associative semigroups (EAS) and their linearizations (lEAS) introduce braid-type compatibility conditions for bilinear maps $\Phi: A\otimes A\to A\otimes A$, realizing these structures as variations of the braid equation and revealing deep links to bialgebras, Hopf algebras, and operad parametrization. Nondegenerate finite commutative EAS split as semidirect/direct products of groups, demonstrating that group-theoretic building blocks underlie many practical instances of bilinear algebraic semigroups (Foissy, 2021).

In the context of dynamical systems, the existence of invariant cones for semigroup actions in exterior powers $\Lambda^k\mathbb{R}^d$ is determined by the flag type $\mathcal{O}(S)$, and the absence thereof underpins the controllability of associated bilinear control systems (Castelani et al., 2021).

Conclusion

Bilinear algebraic semigroups represent a fundamental algebraic structure at the intersection of representation theory, algebraic geometry, random matrix theory, and analytic number theory. Through their realization in the global Langlands program, their spectral correspondence with random matrix ensembles, and their deep symmetry properties manifested in differential bilinear Galois semigroups, they provide a unifying algebraic-geometric language for spectral statistics, combinatorial decompositions, tensor invariants, and automorphic forms. Ongoing developments continue to extend their reach into combinatorial bialgebras, operad theory, algebraic control, and the resolution of undecidability problems in algebraic contexts, thereby reinforcing their centrality in modern mathematical research.