arXiv subjects
Publications and source records attributed to Shan Ren.
Let $p$ be an odd prime and $\mathbb{F}_p$ be the prime field of order $p$. Consider a $2$-dimensional orthogonal group $G$ over $\mathbb{F}_p$ acting on the standard representation $V$ and the dual space $V^*$. We compute the invariant ring $\mathbb{F}_p[V\oplus V^*]^G$ via explicitly exhibiting a minimal generating set. Our method finds an application of $s$-invariants appeared in covariant theory of finite groups.
This article is a continuation of our recent work (Yin Chen and Runxuan Zhang, Shape enumerators of self-dual NRT codes over finite fields. SIAM J. Discrete Math. 38 (2024), no. 4, 2841-2854) in the setting of quantum error-correcting codes. We use algebraic invariant theory to study three weight enumerators of formally self-dual quantum codes over arbitrary finite fields. We derive a quantum analogue of Gleason's theorem, demonstrating that the weight enumerator of a formally self-dual quantum code can be expressed algebraically by two polynomials. We also show that the double weight enumerator of a formally self-dual quantum code can be expressed algebraically by five polynomials. We explicitly compute the complete weight enumerators of some special self-dual quantum codes. Our approach illustrates the potential of employing algebraic invariant theory to compute weight enumerators of self-dual quantum codes.
Using group actions and orbit-stabilizer methods, we study the geometry of isomorphism classes of finite-dimensional $ω$-Lie algebras over a field $\mathbb{K}$ of characteristic $\neq 2$ and establish a one-to-one correspondence between the set of isomorphism classes and the orbit space of a stabilizer of $ω$. We also apply techniques from computational ideal theory to explore the geometric structure of the affine variety of all 3-dimensional $ω$-Lie algebras over $\mathbb{K}$, showing that this variety is a 6-dimensional irreducible affine variety and a complete intersection. As an application, we derive a complete classification of all 3-dimensional $ω$-Lie algebras over an algebraically closed field of characteristic $\neq 2$, up to $ω$-Lie algebra isomorphism.
Consider the special linear group of degree $2$ over an arbitrary finite field, acting on the full space of $2 \times 2$-matrices by transpose. We explicitly construct a generating set for the corresponding modular matrix invariant ring, demonstrating that this ring is a hypersurface. Using a recent result on $a$-invariants of Cohen-Macaulay algebras, we determine the Hilbert series of this invariant ring, and our method avoids seeking the generating relation. Additionally, we prove that the modular matrix invariant ring of the group of upper triangular $2 \times 2$-matrices is also a hypersurface.
This article explores Rota-Baxter operators on finite-dimensional $ω$-Lie algebras over a field of characteristic not 2. We provide several methods for constructing left-symmetric algebras, $ω$-Lie algebras, and Hom-Lie algebras via compatible Rota-Baxter operators on a given $ω$-Lie algebra. We also study the geometric structures of compatible Rota-Baxter operators of weight $0$ and isometric Rota-Baxter operators of weight $1$ over the field of complex numbers. In particular, we prove that the affine variety of all isometric Rota-Baxter operators of weight $1$ on any finite-dimensional non-Lie complex simple $ω$-Lie algebra is $1$-dimensional. Furthermore, we show that for every $4$-dimensional non-Lie complex $ω$-Lie algebra, there always exists a nilpotent compatible Rota-Baxter operator of weight $0$ such that the induced Hom-Lie algebra is nonabelian but solvable.
We study Kupershmidt operators, Nijenhuis operators, and Kupershmidt-Nijenhuis structures on finite-dimensional pre-Malcev algebras over a field of characteristic zero. We construct several new families of complex pre-Malcev algebras that are not pre-Lie algebras in dimensions two, three and four. We use the compatibility of linear operators to establish connections between Kupershmidt operators, Nijenhuis operators, and Kupershmidt-Nijenhuis structures on pre-Malcev algebras. Moreover, we use a method from computational ideal theory to characterize the geometric structures of the varieties of Kupershmidt operators and Nijenhuis operators on a three-dimensional complex pre-Malcev algebra.
This article explores the structure theory of compatible generalized derivations of finite-dimensional $ω$-Lie algebras over a field $\mathbb{K}$. We prove that any compatible quasiderivation of an $ω$-Lie algebra can be embedded as a compatible derivation into a larger $ω$-Lie algebra, refining the general result established by Leger and Luks in 2000 for finite-dimensional nonassociative algebras. We also provide an approach to explicitly compute (compatible) generalized derivations and quasiderivations for all $3$-dimensional non-Lie complex $ω$-Lie algebras.
Let $\mathbb{F}_p$ be the prime field of order $p>0$ and $G$ be an elementary abelian $p$-group.For some $n$-dimensional cohyperplane $G$-representations $V$ over $\mathbb{F}_p$, we show that $\mathbb{F}_p[V\oplus V^*]^G$, the invariant ring of a vector and a covector is a complete intersection by exhibiting an explicit generating set (in fact, a SAGBI basis) and exposing all relations among the generators.
We study connections between skew-symmetric solutions of the classical Yang-Baxter equation (CYBE) and $\mathcal{O}$-operators of Malcev algebras. We prove that a skew-symmetric solution of the CYBE on a Malcev algebra can be interpreted as an $\mathcal{O}$-operator associated to the coadjoint representation. We show that this connection can be enhanced with symplectic forms when considering non-degenerate skew-symmetric solutions. We also show that $\mathcal{O}$-operators associated to a general representation could give skew-symmetric solutions of the CYBE on certain semi-direct products of Malcev algebras. We reveal the relationship between invertible $\mathcal{O}$-operators and compatible pre-Malcev algebra structures on a Malcev algebra. We finally obtain several analogous results on connections between the CYBE and $\mathcal{O}$-operators in the case of pre-Malcev algebras.