Algebra representation
View article: A conceptual derivation of the dual Steenrod algebra
A conceptual derivation of the dual Steenrod algebra Open
In this note I give a conceptual proof of the fact that the mod 2 dual Steenrod algebra corepresents the group scheme of strict automorphisms of the formal additive group over $${\mathbb {F}}_2$$ . Contrary to existing proofs, it does …
View article: Linear Algebra
Linear Algebra Open
View article: Linear Algebra
Linear Algebra Open
View article: Stated Skeins and DAHAs
Stated Skeins and DAHAs Open
Skein algebras of surfaces quantize character varieties of topological surfaces, and in low genus, these quantizations are often related to algebras arising in representation theory. For example, Terwilliger defined a universal $SL_2$ sphe…
View article: Categorical Realizations of Lie Algebra Deformations
Categorical Realizations of Lie Algebra Deformations Open
This paper explores the intricate relationship between Lie algebra deformations and categorical structures, proposing novel frameworks for their categorical realization. Lie algebras, fundamental algebraic objects in mathematics and physic…
View article: Algebraic and Geometric Spectra: Unifying Matrix Canonical Forms
Algebraic and Geometric Spectra: Unifying Matrix Canonical Forms Open
Matrix canonical forms are fundamental tools in linear algebra, offering simplified representations of linear operators or matrices that facilitate analysis and computation. While forms like the Jordan Canonical Form and the Rational Canon…
View article: Depth 2 inclusions of simple $C^*$-algebras and their weak $C^*$-Hopf algebra symmetries
Depth 2 inclusions of simple $C^*$-algebras and their weak $C^*$-Hopf algebra symmetries Open
Let $B \subset A$ be a depth $2$ inclusion of simple unital $C^*$-algebras with a conditional expectation of index-finite type. We show that the second relative commutant $B' \cap A_1$ carries a canonical structure of a weak $C^*$-Hopf alg…
View article: Character Identities Between Affine and Virasoro Vertex Operator Algebra Modules
Character Identities Between Affine and Virasoro Vertex Operator Algebra Modules Open
The affine vertex operator algebras for $\mathfrak{sl}_2$ and the Virasoro minimal models are related by Drinfeld-Sokolov reduction and by the Goddard-Kent-Olive coset construction. In this work, we propose another connection based on cert…
View article: Canonical Clifford Structures on Tensor Products of Exterior Algebras
Canonical Clifford Structures on Tensor Products of Exterior Algebras Open
This paper explores the construction of canonical Clifford structures on tensor products of exterior algebras. Clifford algebras and exterior algebras are fundamental algebraic structures with broad applications in geometry, physics, and c…
View article: The Rational Algebraic Realization of $\mathfrak{su}(2)$ via Dihedron Algebra: The Isomorphism Map for Universal Hyperbolic Geometry
The Rational Algebraic Realization of $\mathfrak{su}(2)$ via Dihedron Algebra: The Isomorphism Map for Universal Hyperbolic Geometry Open
This report details the successful construction of an explicit, linear map $\phi$ that establishes a Lie algebra isomorphism between the Universal Hyperbolic Geometry Lie algebra ($\mathfrak{uhg} \cong \mathfrak{so}(2,1)$) and the compact …
View article: Categorical Realizations of Lie Algebra Deformations
Categorical Realizations of Lie Algebra Deformations Open
This paper explores the intricate relationship between Lie algebra deformations and categorical structures, proposing novel frameworks for their categorical realization. Lie algebras, fundamental algebraic objects in mathematics and physic…
View article: A structural classification of algebras with graded involution and quadratic codimension growth
A structural classification of algebras with graded involution and quadratic codimension growth Open
The theory of algebras with polynomial identities has developed significantly, with special attention devoted to the classification of varieties according to the asymptotic behavior of their codimension sequences. This sequence is a fundam…
View article: Canonical Clifford Structures on Tensor Products of Exterior Algebras
Canonical Clifford Structures on Tensor Products of Exterior Algebras Open
This paper explores the construction of canonical Clifford structures on tensor products of exterior algebras. Clifford algebras and exterior algebras are fundamental algebraic structures with broad applications in geometry, physics, and c…
View article: The Identity of Generating Invariants: Universal Hyperbolic Geometry and su(3) Quartic Casimirs
The Identity of Generating Invariants: Universal Hyperbolic Geometry and su(3) Quartic Casimirs Open
Version 2.0 (Structural Refinement and Definitive Conclusion) This is the updated and final version of the manuscript establishing the $\mathbb{Q}$-Identity between Universal Hyperbolic Geometry (UHG) and the …
View article: Algebraic and Geometric Spectra: Unifying Matrix Canonical Forms
Algebraic and Geometric Spectra: Unifying Matrix Canonical Forms Open
Matrix canonical forms are fundamental tools in linear algebra, offering simplified representations of linear operators or matrices that facilitate analysis and computation. While forms like the Jordan Canonical Form and the Rational Canon…
View article: A structural classification of algebras with graded involution and quadratic codimension growth
A structural classification of algebras with graded involution and quadratic codimension growth Open
The theory of algebras with polynomial identities has developed significantly, with special attention devoted to the classification of varieties according to the asymptotic behavior of their codimension sequences. This sequence is a fundam…
View article: The Rational Algebraic Realization of $\mathfrak{su}(2)$ via Dihedron Algebra: The Isomorphism Map for Universal Hyperbolic Geometry
The Rational Algebraic Realization of $\mathfrak{su}(2)$ via Dihedron Algebra: The Isomorphism Map for Universal Hyperbolic Geometry Open
This report details the successful construction of an explicit, linear map $\phi$ that establishes a Lie algebra isomorphism between the Universal Hyperbolic Geometry Lie algebra ($\mathfrak{uhg} \cong \mathfrak{so}(2,1)$) and the compact …
View article: The Tripartite Algebra of Coherence: T₇, anti-T₇ and the Monstrous Algebra as a Unified Structure
The Tripartite Algebra of Coherence: T₇, anti-T₇ and the Monstrous Algebra as a Unified Structure Open
This work introduces the Tripartite Algebra of Coherence, a unified mathematical-ontological framework composed of the T₇ algebra (positive regime), the anti-T₇ algebra (negative regime), and the Monstrous algebra (reciprocal deformation r…
View article: Instantons on the Blown-up Surface and the Affine Vertex Algebra
Instantons on the Blown-up Surface and the Affine Vertex Algebra Open
Vafa-Witten observed that Yoshioka's blow-up formula for the Euler characteristics of rank $r$ instantons on an algebraic surface coincides with the character of the Wess-Zumino-Witten model for $\mathrm{SU}(r)$ at level $1$, and raised th…
View article: The Rational Algebraic Realization of $\mathfrak{su}(2)$ via Dihedron Algebra: The Isomorphism Map for Universal Hyperbolic Geometry
The Rational Algebraic Realization of $\mathfrak{su}(2)$ via Dihedron Algebra: The Isomorphism Map for Universal Hyperbolic Geometry Open
This technical report details the explicit construction and symbolic verification of the $\mathbb{Q}$-linear Lie algebra isomorphism $\phi: \mathfrak{uhg} \to \mathbf{C}_b(\mathbb{Q}) \otimes \mathfrak{su}(2)$. Th…
View article: The Tripartite Algebra of Coherence: T₇, anti-T₇ and the Monstrous Algebra as a Unified Structure
The Tripartite Algebra of Coherence: T₇, anti-T₇ and the Monstrous Algebra as a Unified Structure Open
This work introduces the Tripartite Algebra of Coherence, a unified mathematical-ontological framework composed of the T₇ algebra (positive regime), the anti-T₇ algebra (negative regime), and the Monstrous algebra (reciprocal deformation r…
View article: Instantons on the Blown-up Surface and the Affine Vertex Algebra
Instantons on the Blown-up Surface and the Affine Vertex Algebra Open
Vafa-Witten observed that Yoshioka's blow-up formula for the Euler characteristics of rank $r$ instantons on an algebraic surface coincides with the character of the Wess-Zumino-Witten model for $\mathrm{SU}(r)$ at level $1$, and raised th…
View article: On (super)symmetrizing forms and Schur elements of cyclotomic Hecke-Clifford algebras
On (super)symmetrizing forms and Schur elements of cyclotomic Hecke-Clifford algebras Open
In this paper, we introduce Schur elements for supersymmetrizing superalgebras. We show that the cyclotomic Hecke-Clifford algebra $\mathcal{H}^f_{c}(n)$ is supersymmetric if $f=f^{(\mathtt{0})}_{\underline{Q}}$ and, symmetric if $f=f^{(\m…
View article: Universal Algebra of Non-Classical Set Operations
Universal Algebra of Non-Classical Set Operations Open
This paper explores the application of universal algebra to the study of non-classical set theories, providing a unified framework for analyzing operations in fuzzy, intuitionistic, and paraconsistent contexts. By abstracting set-theoretic…
View article: The Universal Algebra of Logical Connectives
The Universal Algebra of Logical Connectives Open
This paper explores the profound relationship between universal algebra and the study of logical connectives. It posits that universal algebra provides a robust and unifying framework for analyzing the structural properties of diverse logi…
View article: Derived Classification of Division Algebras
Derived Classification of Division Algebras Open
This paper introduces a novel framework for the classification of division algebras through the lens of derived categories. While classical classifications, such as Frobenius's theorem for finite-dimensional associative real division algeb…
View article: Homotopical Foundations for Linear Algebra
Homotopical Foundations for Linear Algebra Open
This paper explores the establishment of homotopical foundations for linear algebra, moving beyond the traditional set-theoretic and categorical approaches to incorporate concepts from homotopy theory. We propose a framework where vector s…
View article: Non-Commutative Diagonalization: Generalized Canonical Forms for Operator Algebras
Non-Commutative Diagonalization: Generalized Canonical Forms for Operator Algebras Open
The diagonalization of matrices is a cornerstone of linear algebra, providing a canonical form that simplifies the study of linear operators. However, this procedure is fundamentally limited to sets of commuting operators. This paper addre…
View article: Homotopy Theoretic Foundations of Commutative Algebra
Homotopy Theoretic Foundations of Commutative Algebra Open
This paper explores the burgeoning field connecting homotopy theory with commutative algebra, providing a foundational framework for understanding algebraic structures through a homotopical lens. Traditional commutative algebra often relie…
View article: Homotopical Foundations for Linear Algebra
Homotopical Foundations for Linear Algebra Open
This paper explores the establishment of homotopical foundations for linear algebra, moving beyond the traditional set-theoretic and categorical approaches to incorporate concepts from homotopy theory. We propose a framework where vector s…