Papers

This paper characterizes Cohen-Macaulay algebras in the sense of Auslander and Reiten [Progr. Math. 95(1991)] within the following two classes of finite dimensional algebras: (i) truncated path algebras and (ii) Nakayama algebras. For the former, our characterization in terms of ordinary quivers shows that if a Cohen-Macaulay truncated path algebra $\Lambda$ is connected, then $\Lambda$ or $D(\Lambda)$ gives a dualizing module, where $D$ denotes the standard duality. For the latter, we show that every Cohen-Macaulay Nakayama algebra $\Lambda$ is Iwanaga-Gorenstein. In particular, $\Lambda$ always gives a dualizing module. Consequently, there is no non-trivial Cohen-Macaulay algebra among truncated path and Nakayama algebras.

We investigate set-theoretic solutions of the Yang–Baxter equation from a categorical perspective. The paper has two main objectives. On the one hand, several existing constructions and classification results in the literature are recast in categorical terms, thereby showing that certain hypotheses previously imposed on such solutions are in fact superfluous. On the other hand, the categorical approach leads naturally to new classes of Yang-Baxter solutions. Our approach consists in identifying suitable subcategories of the category $\mathcal{YB}$ of Yang-Baxter solutions and providing categorical characterisations of the solutions they encode. As an application, we prove that the category of Płonka bi-magmas is isomorphic to the subcategory of $\mathcal{YB}$ consisting of solutions of the Baaj-Long-Skandalis equation. Along the way, we introduce and study a new class of Lyubashenko type solutions and examine the extent to which non-degenerate bijective solutions of the braid equation arise from skew left braces.

We study how local coherenAAce of abelian categories behaves under gluing. For AB5 locally finitely presentable abelian categories, we give conditions on the gluing functors under which local coherence is transferred along a gluing $(\mathcal{A}, \mathcal{B}, \mathcal{C})$: from $\mathcal{A}$ and $\mathcal{C}$ to $\mathcal{B}$, and conversely from $\mathcal{B}$ to $\mathcal{A}$ or $\mathcal{C}$. The coherence theorem for augmented homotopes of Bondal and Zhdanovskiy is recovered as a special case, the coherence theorem for trivial extensions of Fossum, Griffith, and Reiten is a companion result in the same direction. We also introduce quasiprojective pairs of adjoint functors and prove a coherence criterion that generalises the criterion of Gomez Pardo and Martinez Hernandez to arbitrary Grothendieck categories. Finally, we extend Stenström's characterisation of coherence via FP-injective modules to compactly assembled Grothendieck categories: such a category is coherently assembled if and only if the class of CM-injective objects, defined through compact morphisms, is closed under filtered colimits. Along the way we revisit the various equivalent conditions for the coherence of a ring, collecting them in a single place. The proofs make use of the compactly assembled categories of Efimov.

Let $A=\mathbb{Q}[t^{\frac{1}{2^n}}:n\geq 0]$ and consider $A[u,u^{-1};\sigma]$, a skew Laurent polynomial ring, where $\sigma(t^q)=t^{2q}$. We study the kernel of derived extension of scalars along $ A[u,u^{-1};\sigma]\longrightarrow\mathbb{Q}[u,u^{-1}], $ the quotient by $\mathfrak{m}A[u,u^{-1};\sigma]$, with $\mathfrak{m}=(t^{\frac{1}{2^n}}:n\geq 0)$. A finite free resolution of $\mathfrak{m}A[u,u^{-1};\sigma]$ over $ A[u,u^{-1};\sigma]\otimes_{\mathbb{Q}} \bigl(A[u,u^{-1};\sigma]\bigr)^{\mathrm{op}} $ proves smoothness. A specialization inequality for matrix ranks supplies a nonzero exact colimit-preserving functor that vanishes on every compact object, proving failure of compact generation.

We give an explicit countably infinite orthomodular lattice generated by three specified elements and a direct argument that it admits no injective homomorphism into any $\sigma$-complete orthomodular lattice. In particular, it does not embed into any set-sized complete orthomodular lattice. The source is a generated subalgebra of a cofinite reduced product of finite-dimensional real subspace lattices. A nonzero finite term $a$ is orthogonal to the orbit of a terminal line under every finite word in three Sasaki projections, while another finite term $X$ satisfies $\pi_a\pi_X(t)=a$. If the image of this countable orbit has a supremum in a target orthomodular lattice, Sasaki adjunction forces that supremum to commute with the generator images. This forces the image of $a$ to vanish. The proof imposes no atomicity, modularity, Hilbert representation, cardinality bound on the target, or preservation of source infinitary operations. All target lemmas are derived from the orthomodular axioms.

We introduce Rota-Baxter operators on Lie affgebras by an identity independent of the choice of origin and prove that they induce descendant Lie affgebra structures. We establish a bijection between ordered affine Lie factorizations and idempotent operators split at a chosen idempotent fixed point. A cokernel obstruction determines the existence of this point, while invariance under the structural endomorphisms characterizes splitting. We also determine when weighted descendants satisfy the affine Jacobi identity and when operators of nonzero weight give affine doubles. For canonical Lie affgebras, we describe all affine Rota-Baxter operators by Lie subalgebra decompositions and translation vectors, and classify them up to automorphism in dimensions at most two.

In this article, we present the diagonalization of skew-Hermitian dual quaternion matrices in the sense of unitary equivalence. For a unit quaternion $\phi$, we also obtain the diagonalization result of $\phi$-Hermitian dual quaternion matrices in the sense of unitary $\phi$-congruence, which generalizes the Autonne–Takagi factorization of symmetric complex matrices and $\phi$-Hermitian quaternion matrices.

We study the completed Jacobi algebras of $x^3+xy^3+p(y)$ for $\operatorname{ord}_yp=n\ge4$. The explicit normal-form morsification, the filtered length and centre proofs, and the cyclic terminal identities give finite free semisimple deformations. For every parameter of finite order we construct a polynomial curve neighbourhood and a local projective small contraction; the resulting geometric GV and Donaldson–Thomas calculations identify the block multiplicities. The one-dimensional limit $E_{6,\infty}$ is studied separately as an order: we determine its normalization, conductor, radical-idealizer chain, and maximal overorders. The canonical chain yields a nine-dimensional finite quotient whose Ringel dual is isomorphic to its opposite. An expanded construction gives a choice-independent $22$-dimensional quotient and a $40$-dimensional Ringel dual; a corner quotient relates the two finite constructions. The length and centre formulas are proved by a filtered comparison with a homogeneous matrix order and by terminal identities at the exceptional endpoint.

We give an explicit example of a 9-dimensional 5-step nilpotent complex Lie algebra, generated by two elements, that is isomorphic to its complex conjugate but has no real form. In contrast to the earlier 10-dimensional examples, our proof is completely elementary and very short.

A Hsiang cubic is a cubic form on $\mathbb{R}^{n}$ whose zero set is a minimal cone; equivalently, it is the cubic form of a commutative nonassociative metrized algebra satisfying a single identity of degree four. We show that the intrinsic geometry of such an algebra, encoded in its idempotents and their Peirce decompositions, is the geometry of a Chebyshev-type extremal problem on the sphere. A Hsiang cubic equioscillates on $S^{n-1}$: its maximum set is the set of normalized idempotents, and on suitable great circles through an idempotent it restricts to the Chebyshev polynomial $T_3$. For the spherical analogue of the problem of least deviation, the minimization of $\|p\|_\infty^2/\|p\|_2^2$ over harmonic cubics, a sharp lower bound based on Kellogg's inequality shows that Cartan's isoparametric cubics are the global minimizers in dimensions $5$, $8$, $14$ and $26$. In general, we compute the ratio of every Hsiang cubic in terms of the Killing form and characterize stationarity by a Kolmogorov criterion: it amounts to a positive measure on the idempotent variety. Such measures are given explicitly in several cases which together cover every Hsiang cubic known to exist; in particular, all of them are stationary. These measures give quadrature identities on the idempotent variety whose moments are invariants of the algebra; they are spherical designs exactly in the Einstein (Killing metrized) case, i.e.\ when the Killing form is proportional to the inner product. Distinguished low-dimensional Hsiang cubics are strict local minimizers, whereas a polar Hsiang cubic is a stationary non-minimizer.

Polynomial first-order projective differential moment invariants are investigated. The main result establishes an exact correspondence between such moment invariants and symmetric polynomial multipoint relative projective differential invariants of weight $-1$. This correspondence describes each homogeneous component with respect to moment degree and allows previously obtained results on relative projective invariants to be used for systematic construction. It is also proved that passing from the formal algebra of moments to their integral definitions introduces no additional algebraic relations among the constructed invariants. For low moment degrees, the absence of nontrivial invariants of degrees $1$ and $2$ is established, and a complete description of degree $3$ invariants is obtained. The ten cubic differential moments are naturally identified with the coefficients of a ternary cubic, connecting the construction with classical invariant theory. In particular, the Aronhold invariants $S$ and $T$ yield projective differential moment invariants of degrees $4$ and $6$. One of the known moment invariants of Wang et al. is shown to be proportional to $S$. The algebra of polynomial moment invariants is proved not to be finitely generated. The field of rational moment invariants coincides with the field of fractions of the polynomial invariant algebra. For the explicit constructions, compact multipoint representations define the corresponding moment polynomials without requiring their full expansions. These representations provide a basis for algorithmic construction and software computation of the invariants.

Let $\Lambda_n$ be the set of all eigenvalues of standardized Laplacian matrices of order $n$. In 2005, a polygon $S_n\subseteq\Lambda_n$ was constructed, and it was conjectured that $\Lambda_n=S_n$. We observe that $S_n$ is a zonotope, which gives a short proof of $S_n\subseteq\Lambda_n$, and we describe $\Lambda_n$ as a union, over all vectors $v$, of intersections of zonotopes $Z_i(v)$ built from the ratios $v_j/v_i$. For a vector whose components are the $n$ distinct $n$th roots of unity, every $Z_i(v)$ equals $S_n$. We then show that the conjecture is false for $n=7$: the directed $6$-cycle with one vertex duplicated has an eigenvalue lying $2.1\cdot10^{-4}$ beyond one edge of $S_7$. For the analogous digraph of order $n$, the corresponding eigenvalue lies at signed distance $\delta_n\sim\frac{\pi^2}{2n^4}$ beyond the supporting line of the analogous edge of $S_n$, so the construction yields counterexamples for all sufficiently large $n$; numerically, it does so for every $7\le n\le300$. We also prove that $S_m\subseteq S_n$ whenever $m\mid n$, with $S_m\setminus\{0,1\}$ lying in the interior of $S_n$ for $n\ge3m$.

We introduce the global obstruction property (GOP) for Artin algebras: for every finitely generated module of infinite projective dimension, two distinct syzygies have a common nonprojective indecomposable direct summand. We prove that GOP implies the generalized Auslander–Reiten condition, and hence the Auslander–Reiten conjecture. We give criteria for GOP to hold for pullbacks over semisimple algebras and construct an example which is not syzygy-finite. For each Artin algebra $A$, we construct an algebra $\mathcal{G}(A)$ satisfying GOP by using triangular matrix extensions. We show that $\text{fin.dim}\mathcal{G}(A)=\text{fin.dim} A+2$ and that $A$ and $\mathcal{G}(A)$ are singularly equivalent. Finally, we give a pair of derived equivalent finite-dimensional algebras for which GOP holds on only one side.

The rank of the tensor of $3\times 3$ matrix multiplication over the field with two elements is at most $23$ by Laderman's algorithm, and Rudich and Rousseau recently proved that it is at least $22$. We prove that it equals $23$. Hence Laderman's algorithm uses the fewest multiplications among all bilinear algorithms over $\mathbb{F}_2$ and among all bilinear algorithms with integer coefficients. The proof uses the substitution method in the form developed by D'Ambrosio, Wang and Yang et al., which bounds the number of first factors of a decomposition that lie in a given subspace of the space of first factors. We sharpen these bounds for $111$ of Wang's $496$ classes of subspaces, constrain the ranks of the first factors through a $27\times 27$ flattening of the tensor, and list, up to symmetry, all sets of $22$ first factors that satisfy these constraints. An exact completion search shows that none of them is the set of first factors of a decomposition. The computation emits certificates, which are checked in the Lean 4 proof assistant. In one formal proof the largest checks are evaluated as compiled code, and in a second one the kernel evaluates every check, so that its final theorem depends only on the three standard axioms of Lean. The Lean proofs build on the formalization of Rudich and Rousseau and import their table of bounds and their reduction for block algorithms.

For a complex simple Lie algebra $\mathfrak{g}$ with compact conjugation $\tau$, Killing form $B$, and Hermitian form $H(x, y) = -B(x, \tau y)$, it is shown that $H([x,y],[x,y]) \leq (h^{\vee})^{-1}H(x,x)H(y,y)$, where $h^{\vee}$ is the dual Coxeter number, with equality exactly for $H$-orthogonal pairs in $\tau$-stable long-root subalgebras isomorphic to $\mathfrak{sl}(2, \mathbb{C})$. Equivalently, the comass of the Cartan $3$-form with respect to $H$ equals its comass on the compact real form: the spectral norm of the Cartan $3$-form does not increase under complexification. For the Frobenius norm on the image of a representation of Dynkin index $\ell$ the optimal constant is $2/\ell$; this contains the Böttcher–Wenzel and Bloch–Iserles inequalities, extends the latter to complex skew-symmetric matrices, and improves them substantially for the exceptional Lie algebras of types $F_{4}$, $E_{6}$, $E_{7}$, and $E_{8}$. The proof uses the Nahm algebra of $\mathfrak{g}$ with the conjugation induced by $\tau$: critical points of the relevant function correspond to $\tau$-twisted idempotents, and its Hessian is governed by $\tau$-twisted multiplication operators, whose anticommutation with multiplication by $i$ turns the one-sided second order condition at a maximum into a two-sided spectral bound. Subalgebras isomorphic to $\mathfrak{sl}(2, \mathbb{C})$ of Dynkin index $j$ give critical points with critical value $1/(jh^{\vee})$, whose Morse indices are computed, and the set of maximum points is a nondegenerate critical manifold. Geometrically, the maximal complex sectional curvature of a compact simple Lie group equals its maximal sectional curvature.

Inspired from some works on constructions of algebraic structures from other ones, we purpose to give some construction theorems on BiHom-Poisson conformal superalgebras which unify the notion of the Z2 -graded Poisson algebras, the BiHom-Poisson algebras and their conformal setting. So, we introduce BiHom-Poisson conformal superalgebras and show that they are closed under twisting. We then prouve that any regular BiHom-associative conformal algebra leads to BiHom-Poisson conformal algebra via the commutator bracket. Then, we point out that any BiHom-Poisson conformal superalgebra together with Rota-Baxter operator give rises to another BiHom-Poisson conformal superalgebra. Next, we show that tensor prod- uct of any BiHom-associative superalgebra and any BiHom-Poisson conformal superalgebra is also a BiHom-Poisson conformal superalgebra. Contruction from Hom-post-Poisson conformal superalgebras is also given.

We construct a finite-dimensional Auslander-regular algebra $R$ with global dimension three, which is right diagonal but not left diagonal; moreover, any simple right $R$-module with grade $i$ has projective dimension $i$, but there is a simple left $R$-module with grade one and projective dimension two. It answers a question raised by Iyama negatively.

We develop a structure theory of finite solvable Lie conformal algebras. It is proved that every such algebra admits a Cartan subalgebra, and any two Cartan subalgebras are conjugate by a finite product of exponentials of nilpotent zero modes of elements of the derived algebra. Applied to finite vertex algebras, this yields inner conjugacy of Cartan subalgebras, answering a question of D'Andrea and Marchei. The nilradical and solvable radical of an arbitrary finite Lie conformal algebra are shown to be saturated and invariant under ordinary and conformal derivations. For finite free Lie conformal algebras, we prove a simultaneous diagonalization theorem for conformal tori and characterize the existence of a nonzero torus by explicit identities for constant weights. An example demonstrates that maximal conformal tori of a finite nilpotent Lie conformal algebra can have different ranks, so that conjugacy fails in general. The totally saturated null-filiform Lie conformal algebras admitting a nonzero conformal torus are classified. For each algebra in this classification, we prove that maximal conformal tori have rank one and that the maximal solvable extension with the prescribed nilradical is unique (up to isomorphism) and is the semidirect product with a maximal conformal torus.

Classical Schur–Weyl duality plays a fundamental role in the representation theory of general linear and symmetric groups. In this paper, we develop an affine Schur–Weyl theory for the Laurent polynomial loop semigroup. Let $\hat G_K(n)$ denote the Laurent polynomial loop semigroup inside the loop group $GL_n(K((t)))$ over a field $K$, and let $\hat S_K(n,r)$ be the affine Schur algebra. The natural action of $\hat G_K(n)$ on the affine tensor space $\Omega_K^{\otimes r}$ induces an algebra homomorphism $K\hat G_K(n)\rightarrow \hat S_K(n,r).$ We prove that this homomorphism is surjective for every field $K$ satisfying $|K|>r$. Furthermore, we establish the double centralizer property for affine Schur algebras over unique factorization domains for all $n\geq2$. For $n\geq2$ and any field $K$ with $|K|>r$, this yields the corresponding centralizer property for $\hat G_K(n)$. In contrast, for $n=1$ and $r\geq2$, the corresponding natural homomorphism is not surjective over any field. For $n\geq 2$, we use the double centralizer theorem to determine the center of the affine Schur algebra and, when $K$ is a field with $|K|>r$, show that $\hat\zeta_{r,K}$ maps the center of $K\hat G_K(n)$ onto it. Over an algebraically closed field $K$ of arbitrary characteristic, we prove that every finite-dimensional irreducible $\hat S_K(n,r)$-module is a tensor product of evaluation modules. Moreover, we obtain explicit character formulas expressing these irreducible characters as products of characters of irreducible classical Schur algebra modules. Finally, we study polynomial representations of $\hat G_K(n)$ over algebraically closed fields $K$, and classify its finite-dimensional irreducible polynomial representations. For finite fields $\mathbb F_q$ with $q>r$, this yields a classification in the defining-characteristic setting.

We prove that the three-element flat semiring $S_7$ is strongly nonfinitely based within the class of additively idempotent semirings. More generally, let $\mathcal N$ be the subvariety of $\Var(S_7)$ defined by $x^2y\approx x^2$. For every finite ai-semiring $A$, each variety $\mathcal W$ satisfying $\mathcal N\subseteq\mathcal W\subseteq\Var(A)$ has infinite axiomatic rank.

Let $\delta=xz-yt$. We give an explicit tame factorization of the Anick automorphism $(x+\delta t,y+\delta z,z,t)$ of $F[x,y,z,t]$. After a linear change of coordinates, the factorization consists of sixteen elementary automorphisms, all fixing $z$. The proof uses a commutator construction with variables already present in the algebra. A complete list of the elementary factors is included in Section 5. The explicit polynomial factorization identities are defined over $\ZZ[1/2]$; no conclusion in characteristic $2$ is asserted.

We resolve the remaining cases of a longstanding problem on the lower and upper Lie nilpotency indices of modular group algebras. Equality of the two indices has been known in characteristics greater than three since 1992; we establish it in the remaining characteristics two and three, thereby completing the problem for finite groups of prime-power order over finite fields. The proof gives a uniform argument for every prime and introduces a terminal restricted Poincare-Birkhoff-Witt construction that converts the lower-Lie problem into a chronological reachability argument, with no structural hypothesis on the derived subgroup.

Let $W$ be an algebra with involution over a field of characteristic zero. We develop a theory of $W$-polynomial identities with involution for finite-dimensional $(W,*)$-algebras $A$, using the multiplier algebra with involution of $A$. We prove that the $(W,*)$-exponent of $A$ exists and coincides with the ordinary $*$-exponent. We then consider a four-dimensional subalgebra $M$ of the algebra of $4\times4$ upper triangular matrices, endowed with the reflection involution, and study two non-equivalent $(W,*)$-algebra structures on it. For both structures, we determine the corresponding $T_W^*$-ideals of $W$-polynomial identities with involution and compute the $(W,*)$-codimension sequences explicitly; for one of them, we also determine the complete $(W,*)$-cocharacter sequence. Finally, we prove that these two $W$-algebras with involution generate distinct varieties of almost polynomial growth.

We construct deformations of contraction algebras of type E to semisimple algebras. We consider the class given by the potential x^3+xy^3+p(y), where p(y) is a non-zero polynomial of order larger or equal to 4.

We show that the Fomin–Kirillov algebras $FK_n$ are infinite-dimensional for $n\geq6$. This answers a question posed by Fomin and Kirillov in 1997. To prove this, we demonstrate that the Nichols algebras supported on the conjugacy class of transpositions of the symmetric group are infinite-dimensional for both relevant 2-cocycles. For this, we develop a new method of partially semisimplifying a Nichols algebra along an automorphism of order $p$ in characteristic $p$.

We study a family of liftings $\mathfrak U(\lambda)$ of the super Jordan plane over the infinite cyclic group in characteristic zero. For a nonzero lifting parameter $\lambda$, we identify a natural subalgebra $\mathfrak B$ with the bosonization of $U(\mathfrak{osp}(1|2))$, also known as $sl_{-1}(2)$, and show that the lifting $\mathfrak U(\lambda)$ is obtained from $\mathfrak B$ by an Ore localization at a single distinguished element. This structural description provides a unified approach to the representation theory and ideal structure of $\mathfrak U(\lambda)$. We classify the finite-dimensional simple modules of $\mathfrak B$ and prove that every finite-dimensional $\mathfrak B$-module is completely reducible; the localization then yields the corresponding results for $\mathfrak U(\lambda)$. We determine the prime, primitive, and completely prime ideals of both algebras. Finally, we compute their classical quotient rings, showing in particular that the classical quotient ring of $\mathfrak U(\lambda)$ is a $2\times2$ matrix algebra over the skew field of fractions of the tensor product of its centre and the first Weyl algebra.

We construct categorical models and cluster characters for skew-symmetrizable cluster algebras admitting good ramified realizations. Goodness is equivalent to nondegenerate realizability. When a reddening sequence is given, it is enough to check sign coherence of the arrow types along that sequence. Triangular extensions with arbitrary connecting bimodule types preserve goodness when both component matrices admit reddening sequences. Consequently, every matrix obtained from one vertex by triangular extensions and mutations admits a good Jacobi-finite nondegenerate realization for every compatible positive integral symmetrizer. For a good Jacobi-finite nondegenerate realization, the monodromy $F$-polynomials satisfy mutation for all finite decorated representations, without a rigidity hypothesis. A full initial order determines valued universal presentations and their cluster characters. Under the reachability condition, these characters form a basis of the middle cluster algebra for arbitrary geometric coefficients, with frozen variables inverted, and contain all cluster monomials. The same family is an upper-cluster-algebra basis when the middle and upper algebras coincide. For a fixed Jacobian algebra and $\delta$-vector, the Newton polytope of the generic monodromy $F$-polynomial is independent of the order.

We develop the method of central extensions (the Skjelbred–Sund method) for nilpotent right alternative algebras over the field of complex numbers and apply it to obtain the algebraic classification of five-dimensional nilpotent right alternative algebras. More precisely, we classify, up to isomorphism, all complex five-dimensional nilpotent right alternative algebras that have no annihilator component (that is, which are not a direct sum of a smaller algebra and a one-dimensional algebra with zero product) and which are not $2$-step nilpotent. Every such algebra is a non-split central extension of a nontrivial nilpotent right alternative algebra of dimension three (by a two-dimensional space) or of dimension four (by a one-dimensional space). For each of the relevant three- and four-dimensional algebras we compute the second cohomology space, the automorphism group and its action on the second cohomology, and we determine all orbits that give non-split extensions. The resulting list consists of $124$ algebras and families of algebras; it contains $29$ algebras with two-dimensional annihilator and $95$ algebras with one-dimensional annihilator.

The iteration of $z\mapsto z^2+c$ over the octonions has been drawn for more than thirty years, and it is known that the octonionic Julia sets are bodies of revolution about the imaginary part of the constant. Real eight-dimensional composition algebras, however, are much richer than the octonions: beside the Hurwitz algebras there are the symmetric composition algebras, the para-octonions and the Okubo algebras, which have no unit and whose automorphism groups are $G_2$ and $PSU(3)$ respectively. In this work we study, to our knowledge for the first time, the quadratic dynamics over these algebras, and we do it in an experimental spirit: an atlas of 134 four-dimensional renderings, measured quarter by quarter, suggests the statements, which we then prove and test against new, predicted pictures. We show that the symmetries of the Julia sets of the Okubo algebra are governed by the root decomposition of $\mathfrak{su}(3)$ with respect to the plane of $1$ and $i$: for a constant in that plane the continuous symmetry is a maximal torus, which acts by rotations on three root planes and grows to $U(2)$ on the Weyl walls; the automorphism group of the symmetrized product is $PSU(3)\rtimes\mathbb{Z}_2$. We then prove that the Mandelbrot set with seed $0$ of every symmetric composition algebra is the pull-back of a single planar set through the invariants $n(c)$ and $n(c,c\diamond c)$: the tricorn in the compact case, the tricorn and a hyperbolic tricorn in the split case. The result is quite surprising since the para-octonionic and the Okubonic Mandelbrot sets turn out to be the same planar figure saturated by two different groups. Finally, we show that every bilateral map over a symmetric composition algebra is an octonionic quadratic map in disguise, and that at the Petersson angle $\theta=90^\circ$ the dynamics acquires symmetries that do not come from automorphisms.

The discriminant of a square matrix is defined as a discriminant of its characteristic polynomial. In this paper we provide a description of linear maps preserving the matrix discriminant assuming that the ground field is large enough. Our description generalizes the known results to the case where the field may have a finite cardinality and its characteristic may divide the size of considered matrices. To obtain this description, we extend the approach used in our previous work on the subject and, in addition to that, introduce new invariants for the action of all bijective linear maps preserving a given polynomial, which are on independent interest and could be applied to other problems in the theory of linear preservers.