Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1611.01992

Simple finite-dimensional double algebras

M. E. Goncharov·P. S. Kolesnikov
View original source ↗Read source document ↗Export metadata ↓

Abstract

A double algebra is a linear space $V$ equipped with linear map $V\otimes V\to V\otimes V$. Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double algebras do not exist over an arbitrary field, and all simple finite-dimensional associative double algebras over an algebraically closed field are trivial. Over an arbitrary field, every simple finite-dimensional associative double algebra is commutative. A double algebra structure on a finite-dimensional space $V$ is naturally described by a linear operator $R$ on the algebra $\End V$ of linear transformations of~$V$. Double Lie algebras correspond in this sense to skew-symmetric Rota---Baxter operators, double associative algebra structures---to (left) averaging operators.

Explore related subjects

math.QAmath.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

M. E. Goncharov, P. S. Kolesnikov. 2016-11-07. Simple finite-dimensional double algebras. https://doi.org/10.1016/j.jalgebra.2017.04.020

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Factorization envelopes and enveloping vertex algebras

We develop a bornological version of Costello and Gwilliam's procedure for extracting vertex algebras from suitable prefactorization algebras on the complex plane. Using bornological complex analysis, we remove the discreteness condition imposed in their extraction theorem. We then construct, from a suitable Lie conformal algebra, a prefactorization algebra to which this extraction procedure applies, and prove that the resulting vertex algebra is isomorphic to the enveloping vertex algebra of the original Lie conformal algebra. Our construction uses a factorization envelope and extends the construction of Costello--Gwilliam in the affine vertex algebra case and that of Williams in the Virasoro vertex algebra case. Moreover, a super analogue yields new prefactorization algebras corresponding to vertex superalgebras, such as the Neveu--Schwarz vertex superalgebra, the $N=2$ vertex superalgebra, and the $N=4$ vertex superalgebra.

math.QA↗
arXiv2026

BiHom-L-R-smash biproduct and BiHom-Yetter-Drinfel'd-Long category

In this article, we first introduce the notion of BiHom-L-R-$\binom{m,n,p,q}{s,t,u,v}$-smash biproduct over a BiHom-Hopf algebra, denoted by $D\natural H$, where $m,n,p,q,s,t,u,v\in \mathbb{Z}$, and give the sufficient condition for $D\natural H$ to be a BiHom-bialgebra. Furthermore, we describe the concept of BiHom-$\binom{m,n,p,q}{s,t,u,v}$-Yetter-Drinfel'd-Long bimodule via BiHom-L-R-$\binom{m,n,p,q}{s,t,u,v}$-smash biproduct bialgebra, and prove that the category $\mathcal{LR}(H)(m,n,p,q)$ of BiHom-$\binom{m,n,p,q}{s,t,u,v}$-Yetter-Drinfel'd-Long bimodule is a strict braided monoidal category. Finally, for a finite-dimensional BiHom-Hopf algebra H, \(\mathcal{LR}(H)\binom{m,n,p,q}{s,t,u,v}\) is isomorphic to the BiHom-$\binom{s,t}{p,q}$-Yetter-Drinfel'd category \({}_{H\otimes H^*}^{H\otimes H^*}\mathcal{YD}\binom{s,t}{p,q}\) as braided monoidal categories.

math.QA↗
arXiv2026

On finite dimensionality of homology of subalgebras of vector fields

We show that finite tensor products of modules of tensor fields are Noetherian modules over any graded Lie subalgebra of finite codimension in the Lie algebra of polynomial vector fields on $\mathbb{R}^n$. As a corollary, we prove the conjecture of I.\,M. Gelfand, announced at the ICM in Nice in 1970, on the finite-dimensionality of the continuous cohomology of graded Lie subalgebras of finite codimension in the Lie algebra of formal vector fields $W_n$.

math.QA↗