Tangent Lie algebras of automorphism groups of free algebras

In recent work with Harman, we introduced a new notion of measure for oligomorphic groups, and showed how they can be used to produce interesting tensor categories. Determining the measures for an oligomorphic group is an important and difficult combinatorial problem, which has only been solved in a handful of cases. The purpose of this paper is to solve this problem for a certain infinite family of oligomorphic groups, namely, the automorphism group of the n -colored circle (for each n \ge 1 ).

Many fundamental mathematical structures, such as the rationals or the random graph, are homogeneous, meaning that local isomorphisms extend to global automorphisms. Such structures arise as limits of classes of finite structures and encode these classes in a single object. This viewpoint has proved fruitful in model theory, universal algebra, and computer science, with applications to constraint satisfaction, automata theory, and verification. Homogeneous structures have rich automorphism groups, which makes them interesting for topological dynamics. For many applications, however, automorphism groups do not store enough information about the homogeneous structure, and one must instead consider polymorphism clones. Universal algebra has recently achieved major results for polymorphism clones on finite structures, culminating in the 2017 resolution of the Feder–Vardi dichotomy conjecture. An analogous conjecture for homogeneous structures remains open despite growing structural insights.

A quasigroup is a pair ( Q , * ) where Q is a non-empty set and * is a binary operation on Q such that for every ( u , v ) ∈ Q 2 there exists a unique ( x , y ) ∈ Q 2 such that u * x = v = y * u . Let q be an odd prime power, let 𝔽 q denote the finite field of order q , and let ℛ q denote the set of non-zero squares in 𝔽 q . Let ( a , b ) ∈ 𝔽 q 2 be such that { a b , ( a - 1 ) ( b - 1 ) } ⊆ ℛ q . Let 𝒬 a , b denote the quadratic quasigroup ( 𝔽 q , * a , b ) where * a , b is defined by x * a , b y = x + a ( y - x ) if y - x ∈ ℛ q , x + b ( y - x ) otherwise . The operation table of a quadratic quasigroup is a quadratic Latin square. Recently, it has been determined exactly when two quadratic quasigroups are isomorphic and the automorphism group of any quadratic quasigroup has been determined. In this paper, we extend these results. We determine exactly when two quadratic quasigroups are isotopic and we determine the autotopism group of any quadratic quasigroup. In the process, we count the number of 2 × 2 subsquares in quadratic Latin squares.

Representation theory of the group of automorphisms of a finite rooted tree

Algebraic curves with a large cyclic automorphism group

Abstract In 2016 Tafazolian et al. [31] introduced new families of $$\mathbb {F}_{q^{2n}}$$ F q 2 n -maximal function fields $$\mathcal {Y}_{n,s}$$ Y n , s and $$\mathcal {X}_{n,s,a,b}$$ X n , s , a , b arising as subfields of the first generalized GK function field (GGS). In this way the authors found new examples of maximal function fields that are not isomorphic to subfields of the Hermitian function field. In this paper we construct analogous function fields $$\tilde{\mathcal {Y}}_{n,s}$$ Y ~ n , s and $$\tilde{\mathcal {X}}_{n,s,a,b}$$ X ~ n , s , a , b as subfields of the second generalized GK function field (BM) and determine their automorphism groups. Using that the automorphism group is an invariant under isomorphism, we show that the function fields $$\tilde{\mathcal {Y}}_{n,s}$$ Y ~ n , s and $${\mathcal {Y}}_{n,s}$$ Y n , s , as well as $$\tilde{\mathcal {X}}_{n,s,a,b}$$ X ~ n , s , a , b and $$\mathcal {X}_{n,s,a,b}$$ X n , s , a , b , are not isomorphic unless m / s divides $$q^2-q+1$$ q 2 - q + 1 and 3 divides n . In other words, the difference between the BM and GGS function fields can be found again at the level of the subfields that we consider.

In this paper, we describe explicitly the structure of the derivation algebra and automorphism group of the symplectic oscillator Lie algebra [Formula: see text] ([Formula: see text]), where [Formula: see text] is the symplectic Lie algebra and [Formula: see text] is the [Formula: see text]-dimensional Heisenberg algebra.

We give a concise presentation for the group of pure symmetric outer automorphisms of a given splitting of a free product G 1 ∗ … ∗ G n . These are the (outer) automorphisms which preserve the conjugacy classes of the free factors G i . This is achieved by considering the action of these automorphisms on a particular subcomplex of “Outer Space,” which we show to be simply connected. We then apply a theorem of K. S. Brown to extract our presentation.

We prove that the automorphism group of an affine, cubic surface with equation xyz=g(x,y) contains \mathbb{Z} as a finite index subgroup. These equations were first studied by Jacobsthal (1939) and Mordell (1952).

On the characterization of affine toric varieties by their automorphism group

Analyzing the structure of the automorphism groups of graphs, and investigating the properties of graphs that are constructed from algebraic structures are two important research topics in algebraic graph theory. Blending these two aspects of study, an algebraic intersection graph, called the invariant intersection graph of a graph, has been introduced in the literature. In this paper, we study certain properties of the invariant intersection graphs of graphs, and obtain some structural characterizations of these graphs, based on the automorphism group of the graph on which the invariant intersection graph is constructed.

Let p , l be distinct prime numbers. A tripod-degree over p at l is defined to be an l -adic unit obtained by forming the image, by the l -adic cyclotomic character, of some continuous automorphism of the geometrically pro- l fundamental group of a split tripod over a finite field of characteristic p . The notion of a tripod-degree plays an important role in the study of the geometrically pro- l anabelian geometry of hyperbolic curves over finite fields, e.g., in the theory of cuspidalizations of the geometrically pro- l fundamental groups of hyperbolic curves over finite fields. In the present paper, we study the tripod-degrees. In particular, we prove that, under a certain condition, the group of tripod-degrees over p at l coincides with the closed subgroup of the group of l -adic units topologically generated by p . As an application of this result, we also conclude that, under a certain condition, the natural homomorphism from the group of automorphisms of the split tripod to the group of outer continuous automorphisms of the geometrically pro- l fundamental group of the split tripod that lie over the identity automorphism of the absolute Galois group of the basefield is surjective.

Abstract Jabłonowski [‘On biquandle-based invariant of immersed surface-links, Yoshikawa oriented fifth move, and ribbon 2-knots’, Preprint, 2025, arXiv:2505.14724] proved that the knot quandles of Suciu’s n -knots, which share isomorphic knot groups, are mutually nonisomorphic, and Yasuda [‘Knot quandles distinguish Suciu’s ribbon knots’, Preprint, 2025, arXiv:2508.15129] later gave a different proof. We present yet another proof of this result by analysing the conjugacy classes of certain automorphisms of the free group of rank two.

Generalized Quasi-dihedral Group as Automorphism Group of Riemann Surfaces

We describe a relationship between the Lie algebra $\mathfrak{sl}_4(\mathbb C)$ and the hypercube graphs. Consider the $\mathbb C$-algebra $P$ of polynomials in four commuting variables. We turn $P$ into an $\mathfrak{sl}_4(\mathbb C)$-module on which each element of $\mathfrak{sl}_4(\mathbb C)$ acts as a derivation. Then $P$ becomes a direct sum of irreducible $\mathfrak{sl}_4(\mathbb C)$-modules $P = \sum_{N\in \mathbb N} P_N$, where $P_N$ is the $N$th homogeneous component of $P$. For $N\in \mathbb N$ we construct some additional $\mathfrak{sl}_4(\mathbb C)$-modules ${\rm Fix}(G)$ and $T$. For these modules the underlying vector space is described as follows. Let $X$ denote the vertex set of the hypercube $H(N,2)$, and let $V$ denote the $\mathbb C$-vector space with basis $X$. For the automorphism group $G$ of $H(N,2)$, the action of $G$ on $X$ turns $V$ into a $G$-module. The vector space $V^{\otimes 3} = V \otimes V \otimes V$ becomes a $G$-module such that $g(u \otimes v \otimes w)= g(u) \otimes g(v) \otimes g(w)$ for $g\in G$ and $u,v,w \in V$. The subspace ${\rm Fix}(G)$ of $V^{\otimes 3}$ consists of the vectors in $V^{\otimes 3}$ that are fixed by every element in $G$. Pick $\varkappa \in X$. The corresponding subconstituent algebra $T$ of $H(N,2)$ is the subalgebra of ${\rm End}(V)$ generated by the adjacency map $\sf A$ of $H(N,2)$ and the dual adjacency map ${\sf A}^*$ of $H(N,2)$ with respect to $\varkappa$. In our main results, we turn ${\rm Fix}(G)$ and $T$ into $\mathfrak{sl}_4(\mathbb C)$-modules, and display $\mathfrak{sl}_4(\mathbb C)$-module isomorphisms $P_N \to {\rm Fix}(G) \to T$. We describe the $\mathfrak{sl}_4(\mathbb C)$-modules $P_N$, ${\rm Fix}(G)$, $T$ from multiple points of view.

In this paper, we investigate quadratic residue codes of prime length [Formula: see text] over [Formula: see text] where [Formula: see text] or [Formula: see text] These codes are defined via their generating idempotents, and we examine their extended versions. Furthermore, we demonstrate that the extended quadratic residue codes over [Formula: see text] possess large automorphism groups, a property that can be effectively exploited in decoding algorithms.

Abstract Let F n F_{n} be a free group of rank 𝑛. An SL ⁢ ( 2 ) \mathrm{SL}(2) character of F n F_{n} means the trace of an SL ⁢ ( 2 ) \mathrm{SL}(2) representation of F n F_{n} . Let 𝐾 be a field of characteristic 0. The automorphism group Aut ⁢ ( F n ) \mathrm{Aut}(F_{n}) of F n F_{n} naturally acts on the commutative 𝐾-algebra of the SL ⁢ ( 2 ) \mathrm{SL}(2) characters. Then the augmentation ideal J n + J_{n}^{+} of the commutative 𝐾-algebra generated by the SL ⁢ ( 2 ) \mathrm{SL}(2) characters is an Aut ⁢ ( F n ) \mathrm{Aut}(F_{n}) -invariant. The main purpose of this paper is to study the structure of the graded quotient ( J n + ) k / ( J n + ) k + 1 (J_{n}^{+})^{k}/(J_{n}^{+})^{k+1} as an Aut ⁢ ( F n ) \mathrm{Aut}(F_{n}) -module.

Abstract Given a finite group 𝐺 and a conjugacy class of involutions 𝑋 of 𝐺, we define the commuting involution graph C ⁢ ( G , X ) \mathcal{C}(G,X) to be the graph with vertex set 𝑋 and x , y ∈ X x,y\in X adjacent if and only if x ≠ y x\neq y and x ⁢ y = y ⁢ x xy=yx . In this paper, the automorphism group of the graph C ⁢ ( G , X ) \mathcal{C}(G,X) is determined when G = PSL 2 ⁡ ( q ) G=\operatorname{PSL}_{2}(q) .

For a symplectic vector space V , a projective subvariety Z ⊂ ℙ V is a Legendrian variety if its affine cone Z ^ ⊂ V is Lagrangian. In addition to the classical examples of subadjoint varieties associated to simple Lie algebras, many examples of nonsingular Legendrian varieties have been discovered which have positive-dimensional automorphism groups. We give a characterization of subadjoint varieties among such Legendrian varieties in terms of the isotropy representation. Our proof uses some special features of the projective third fundamental forms of Legendrian varieties and their relation to the lines on the Legendrian varieties.