The paper connects Brauer algebra homology to symmetric group homology.
problem Understanding the homology of Brauer algebras.
method Interpreting Brauer algebras as Tor-groups and comparing to symmetric group homology.
result Isomorphism of homology groups under specific conditions.
The faithfulness of the orthogonal group case of Brauer's representation of the Brauer centralizer algebras restricted to their Temperley-Lieb subalgebras, which was established by Vaughan Jones, is here proved in a new, elementary and self-contained, manner.
New dg-algebras generalize Brauer graph algebras, with applications to stability conditions and quadratic differentials.
problem Generalizing Brauer graph algebras to new dg-algebras.
method Derived categories, mixed-angulations of surfaces, stability conditions, and quadratic differentials.
result Spaces of stability conditions on derived categories of these algebras are described in terms of spaces of quadratic differentials.
We show that for complex analytic K3 surfaces any torsion class in H^2(X,O_X^*) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surfa…
The paper explores handlebody versions of various diagram algebras.
problem None explicitly stated, but related to algebraic structures.
method Study of handlebody versions of classical diagram algebras and reformulation of cellular algebras.
result All mentioned algebras are part of the reformulated cellular algebra theory.
Homology of partition algebras matches symmetric group homology under certain conditions.
problem Understanding homology of partition algebras and comparing it to symmetric groups.
method Inductive resolution and high acyclicity arguments, parallel to earlier work on Brauer algebras.
result Homology of partition algebras is isomorphic to symmetric group homology under specific conditions.
Given a compact Riemann surface X and a semisimple affine algebraic group G defined over C, there are moduli spaces of Higgs bundles and of connections associated to (X,G). We compute the Brauer group of the smooth locus of these varieties.
It is shown that the multiplicative monoids of Temperley-Lieb algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the…
Study K-theory of Etesi C∗-algebras to understand smooth manifolds.
problem Understanding smooth manifolds through K-theory of Etesi C∗-algebras. method Calculate topological and smooth invariants of manifolds using K-theory of Etesi C∗-algebras. result Smoothings of a manifold form a torsion abelian group isomorphic to the Brauer group of a number field.
New monoids tied to symmetric group and Jones/Brauer monoids discovered.
problem Understanding monoids attached to symmetric group and related monoids.
method Introduced ramified monoids and tied-like monoids, provided presentations.
result Found new monoids that cannot be described as ramified monoids.
Two number fields are said to be Brauer equivalent if there is an isomorphism between their Brauer groups that commutes with restriction. In this paper we prove a variety of number theoretic results about Brauer equivalent number fields (e.g., they must have the same signature). These results are then applied to the ge…
Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally construct bijections between central simple algebras, maximal orders, various Galois co…
The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones,…
It is shown that the multiplicative monoids of Brauer's centralizer algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself, and where, moreover, a kind of symmetry involving the self-adjoint functor is satisfied. As in a previous paper, of …
Let M be a compact 3-manifold and Γ=π1(M). Work of Thurston and Culler--Shalen established the SL2(C) character variety X(Γ) as fundamental tool in the study of the geometry and topology of M. This is particularly the case when M is the exterior of a hyperbolic knot K in S3. The mai…
We develop the theory of resolvent degree, introduced by Brauer \cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic…
Compactifies Minkowski space using unitary matrices.
problem Compactifying Minkowski space for quantum field theories.
method Using Cayley transform and unitary group $\U(2)$.
result Defines interesting backgrounds for quantum field theories.
Study deformation invariants of projective surfaces using twisted sheaves and Azumaya algebras.
problem Deformation invariants of projective surfaces with specific cohomology conditions.
method Virtual intersection numbers on moduli spaces of stable twisted sheaves and Azumaya modules.
result Invariants do not depend on the choice of Brauer-Severi variety or Azumaya algebra.
The paper contains a survey of train constructions for infinite symmetric groups and related groups. For certain pairs (a group G, a subgroup K), we construct categories, whose morphisms are two-dimensional surfaces tiled by polygons and colored in a certain way. A product of morphisms is a gluing of combinatorial …
Characterizes group-equivariant neural networks for three groups.
problem Understanding equivariant neural networks for orthogonal, special orthogonal, and symplectic groups.
method Characterized all possible group-equivariant neural networks for three groups.
result Found spanning sets of matrices for learnable, linear equivariant layer functions.
An algorithm for efficient computation of equivariant neural network layers.
problem Efficiently computing with Brauer's group equivariant neural network layers.
method Category theoretic constructions and Kronecker product matrices.
result Significant reduction in computational cost compared to naive implementation.
New algebraic structure derived from Hopf algebra and Drinfel'd twist.
problem Developing a new algebraic structure from existing mathematical concepts.
method Extending L∞-algebra to a Hopf algebra, twisting with Drinfel'd twist, and identifying Hopf morphisms and braided morphisms. result Braided L∞-algebra is derived from the process. Study of cluster and skein algebras for surfaces, showing their connection.
problem Understanding algebraic structures of curve algebras on surfaces.
method Generalization and explicit definition of maps between cluster and skein algebras.
result Explicit maps between cluster and skein algebras, showing their close relationship.
Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.
problem Investigating conditions for pseudo-Riemannian algebraic Ricci solitons on 4D Lie algebras.
method Analyzing the algebraic Ricci soliton equation for each 4D Lie algebra.
result Complete description of pseudo-Riemannian algebraic Ricci solitons in dimension four.
Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we …
New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.
problem Classifying Hsiang algebras and understanding their properties.
method Introducing quasicomposition and tripling constructions to study Hsiang algebras.
result The triple of a quasicomposition algebra is an exceptional Hsiang algebra.
A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define…
Study biderivations in complete Leibniz algebras, extending Lie algebra results.
problem Defining and studying biderivations in complete Leibniz algebras.
method Analyze biderivations according to two definitions, provide conditions for biderivations, and compare symmetric and skew-symmetric biderivations.
result Necessary and sufficient conditions for biderivations in Leibniz algebras are provided.
The paper extends a theorem to Lie-Rinehart algebras and provides new decompositions of universal enveloping algebras.
problem Understanding universal enveloping algebras of Lie-Rinehart algebras.
method Extending a theorem to left Hopf algebroids and applying it to universal enveloping algebras of Lie-Rinehart algebras.
result Provides a crossed product decomposition of universal enveloping algebras for curved and flat connections.
The paper classifies Lie algebras with special operators.
problem Classifying 3D Lie algebras with regular semisimple algebraic Nijenhuis operators.
method Described all Nijenhuis eigenbases for each 3D Lie algebra.
result Different answers in real and complex cases, some Lie algebras admit operators, others do not.
Symmetric spaces' connections form Lie admissible triple algebras.
problem Understanding the algebraic structure of symmetric spaces' connections.
method Analyzing the connection as a binary operator on tangent bundle sections, identifying Lie admissibility constraints.
result Connection algebra of symmetric spaces is a Lie admissible triple algebra.
The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.
problem Characterizing and understanding k-para-Kähler Lie algebras.
method Generalization of para-Kähler Lie algebras to k-para-Kähler Lie algebras, introduction of new structures, determination of Lie algebras.
result Determination of all k-symplectic Lie algebras of dimension (k+1) and six-dimensional 2-para-Kähler Lie algebras.
The paper investigates gradings of complex simple Lie algebras, focusing on ∣3∣-gradings and their algebraic structures.
problem Investigating the algebraic structure of ∣3∣-gradings of complex simple Lie algebras. method Completely determining the possible reductive algebras n0 and proving the uniqueness of a specific free nilpotent Lie algebra. result The only free nilpotent Lie algebra of step 3 that appears as the negative part of a ∣3∣-grading is the usual ∣3∣-grading of the exceptional Lie algebra g2. New algebra pong algebra computed for knot Floer homology.
problem Computing A-infinity structure on knot Floer homology.
method Introduced differential graded algebra, pong algebra.
result Computed A-infinity structure on pong algebra's homology.
Study resolves conjecture linking two algebraic structures on surfaces.
problem Compatibility of skein and cluster algebra structures on surfaces.
method Established compatibility between skein and cluster algebras of surfaces.
result Cluster algebra of positive genus surfaces is not finitely generated.
Study on pre-Lie structures for semisimple Lie algebras over C.
problem Admissibility of pre-Lie structures in semisimple Lie algebras.
method Examined properties of anti-flexible algebras (AFAs), computed Lie-admissibility criteria, and provided examples.
result Explicit counterexample of an AFA admissible by sl(2, C).
Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.
problem Characterizing and understanding post-Lie algebras and their associated structures.
method Utilizes Manin triples and generalized Hessian Lie groups to define and characterize post-Lie algebras with nondegenerate symmetric invariant bilinear forms.
result Establishes a bialgebra theory for post-Lie algebras via the Manin triple approach, including new algebraic structures like pp-post-Lie algebras.
Characterizes G2-structures on Lie algebras with non-trivial center.
problem Classifying Lie algebras with G2-structures.
method Analyzing Lie algebras with non-trivial center, using contactization and symplectic properties.
result Six unimodular Lie algebras with non-trivial center admit closed G2-structures.
Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms o…
This paper presents results on the framization of some knot algebras, defined by the authors. We explain the motivations of the concept of framization, coming from the Yokonuma--Hecke algebras, as well as recent results on the framization of the Temperley--Lieb algebra. Finally, we propose framizations for other knot a…
For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and non-existence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algeb…
Similarity algebra extends algebraic structures with quantitative bounds.
problem Exact algebraic structures with strict axioms.
method Framework for approximate algebraic and Lie structures with ε-estimates. result Similarity structures converge to classical algebraic objects as εightarrow0. Extends Loday-Quillen-Tsygan theorem to bornological Lie algebra homology.
problem Calculating Lie algebra homology of gauge algebras using cyclic homology.
method Extends proof to bornological Lie algebra homology of Fréchet and LF-algebras, prepares statements about homological algebra of topological vector spaces.
result Constructs a spectral sequence to calculate stable part of bornological Lie algebra homology of gauge algebras.
New algebra for twice-punctured torus curves.
problem Constructing a new algebra for skein theory.
method Using Heegaard dual of Iwahori--Hecke operator, Dehn twists are represented.
result Automorphisms correspond to Dehn twists on the twice-punctured torus.
We define a Poisson Algebra called the {\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functio…
New braided Frobenius algebras created from specific Hopf algebras.
problem Creating new algebraic structures from Hopf algebras.
method Heap operation and Yang-Baxter operator on tensor product.
result Heap operation induces a braiding compatible with Frobenius operations.
Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables us to explore Poisson algebras in the realm of non-associative algebras. We stud…