Permutation Weights for A_r^(1) Lie Algebras

Fractional Super Lie Algebras and Groups

n^{th} root of a Lie algebra and its dual (that is fractional supergroup) based on the permutation group $S_n$ invariant forms are formulated in the Hopf algebra formalism. Detailed discussion of $S_3$-graided $sl(2)$ algebras is done.

广义弦图与权系统

Generalized chord diagrams and weight systems

本文将弦图上的权系统扩展至任意排列，提出推广Vassiliev关系的函数关系，证明$gl$-和$so$-权系统满足该关系，并分析了其性质及相关的排列Hopf代数。

Universal volume of groups and anomaly of Vogel's symmetry

We show that integral representation of universal volume function of compact simple Lie groups gives rise to six analytic functions on $CP^2$, which transform as two triplets under group of permutations of Vogel's projective parameters. This substitutes expected invariance under permutations of universal parameters by more complicated covariance. We provide an analytical continuation of these functions and particularly calculate their change under permutations of parameters. This last relation is universal generalization, for an arbitrary simple Lie group and an arbitrary point in Vogel's plan…

Permutation and Its Partial Transpose

Permutation and its partial transpose play important roles in quantum information theory. The Werner state is recognized as a rational solution of the Yang--Baxter equation, and the isotropic state with an adjustable parameter is found to form a braid representation. The set of permutation's partial transposes is an algebra called the "PPT" algebra which guides the construction of multipartite symmetric states. The virtual knot theory having permutation as a virtual crossing provides a topological language describing quantum computation having permutation as a swap gate. In this paper, permuta…

Special Weights and Roots for Finite Lie Algebras

With the introduction of special roots, we show the existence of some special weights with quite interesting properties for finite Lie algebras. We propose and discuss two statements which lead us to an explicit construction of these special weights and roots. It is seen that special weights provides us an alternative way of expressing the whole action of Weyl groups.

The decomposition of an arbitrary $2^w\times 2^w$ unitary matrix into signed permutation matrices

Birkhoff's theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. A similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices. Unitary matrices of dimension equal to a power of~2 (say $2^w$) deserve special attention, as they represent quantum qubit circuits. We investigate which subgroup of the signed permutation matrices suffices to decompose an arbitrary such matrix. It turns out to be a matrix group isomorphic to the extraspecial group {\bf E}$_{2^{2w+1}}^+$ of order $2^{2w+1}$. …

$3$-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on $3$-Lie algebras

In this paper, first we introduce the notions of relative Rota-Baxter operators of nonzero weight on $3$-Lie algebras and $3$-post-Lie algebras. A 3-post-Lie algebra consists of a 3-Lie algebra structure and a ternary operation such that some compatibility conditions are satisfied. We show that a relative Rota-Baxter operator of nonzero weight induces a $3$-post-Lie algebra naturally. Conversely, a $3$-post-Lie algebra gives rise to a new 3-Lie algebra, which is called the subadjacent 3-Lie algebra, and an action on the original 3-Lie algebra. Then we construct an $L_\infty$-algebra whose Maur…

Spatial random permutations and Poisson-Dirichlet law of cycle lengths

We study spatial permutations with cycle weights that are bounded or slowly diverging. We show that a phase transition occurs at an explicit critical density. The long cycles are macroscopic and their cycle lengths satisfy a Poisson-Dirichlet law.

Classification of infinite dimensional weight modules over the Lie superalgebra $sl(2/1)$

We give a classification of infinite dimensional indecomposable weight modules over the Lie superalgebra $sl(2/1)$.

Computation of partition functions of free fermionic solvable lattice models via permutation graphs

In this paper, we introduce a novel and general method for computing partition functions of solvable lattice models with free fermionic Boltzmann weights. The method is based on the ``permutation graph'' and the ``$F$-matrix'': the permutation graph is a generalization of the $R$-matrix, and the $F$-matrix is constructed based on the permutation graph. The method allows generalizations to lattice models that are related to Cartan types B and C. Two applications are presented: they involve an ice model related to Tokuyama's formula and another ice model representing a Whittaker function on the …

Representations up to homotopy from weighted Lie algebroids

Weighted Lie algebroids were recently introduced as Lie algebroids equipped with an additional compatible non-negative grading, and represent a wide generalisation of the notion of a VB -algebroid. There is a close relation between two term representations up to homotopy of Lie algebroids and VB - algebroids. In this paper we show how this relation generalises to weighted Lie algebroids and in doing so we uncover new and natural examples of higher term representations up to homotopy of Lie algebroids. Moreover, we show how the van Est theorem generalises to weighted objects.

Virtual Extension of Temperley--Lieb Algebra

The virtual knot theory is a new interesting subject in the recent study of low dimensional topology. In this paper, we explore the algebraic structure underlying the virtual braid group and call it the virtual Temperley--Lieb algebra which is an extension of the Temperley--Lieb algebra by adding the group algebra of the symmetrical group. We make a connection clear between the Brauer algebra and virtual Temperley--Lieb algebra, and show the algebra generated by permutation and its partial transpose to be an example for the virtual Temperley--Lieb algebra and its important quotients.