A. M. Gavrilik, N. Z. Iorgov
The nonstandard q-deformed algebras U'_q(so_n) are known to possess q-analogues of Gel'fand-Tsetlin type representations. For these q-algebras, all the Casimir elements (corresponding to basis set of Casimir elements of so(n)) are found, and their eigenvalues within irreducible representations are given explicitly.
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For each quantum superalgebra $U_q[osp(m|n)]$ with $m>2$, an infinite family of Casimir invariants is constructed. This is achieved by using an explicit form for the Lax operator. The eigenvalue of each Casimir invariant on an arbitrary irreducible highest weight module is also calculated.
We show that the quantum Casimir operators of the quantum linear group constructed in early work of Bracken, Gould and Zhang together with one extra central element generate the entire center of $\Uq$. As a by product of the proof, we obtain intriguing new formulae for eigenvalues of these quantum Casimir operators, which are expressed in terms of the characters of a class of finite dimensional irreducible representations of the classical general linear algebra.
We show how to construct, starting from a quasi-Hopf (super)algebra, central elements or Casimir invariants. We show that these central elements are invariant under quasi-Hopf twistings. As a consequence, the elliptic quantum (super)groups, which arise from twisting the normal quantum (super)groups, have the same Casimir invariants as the corresponding quantum (super)groups.
Harish-Chandra images of orthosymplectic Sugawara operators and Casimir elements
研究正交辛李超代数的西格尔 - 杉原向量，通过哈里什 - 钱德拉同态计算其像并扩展到相关算子和元素，还生成了高阶高丁哈密顿量等，附录回顾了一般线性李超代数类似结果。
We apply the techniques of quasideterminants to construct new families of Casimir elements for the general Lie superalgebra whose images under the Harish-Chandra isomorphism are respectively the elementary, complete and power sums supersymmetric functions
We present the eigenvalues of the Casimir invariants for the type I quantum superalgebras on any irreducible highest weight module.
Let $U_q(\hat{\cal G})$ be an infinite-dimensional quantum affine Lie algebra. A family of central elements or Casimir invariants are constructed and their eigenvalues computed in any integrable irreducible highest weight representation. These eigenvalue formulae are shown to absolutely convergent when the deformation parameter $q$ is such that $|q|>1$. It is proven that the universal R-matrix $R$ of $U_q(\hat{\cal G})$ satisfies the celebrated conjugation relation $R^\dagger=TR$ with $T$ the usual twist map. As applications, the braid generator is shown to be diagonalizable on arbitrary tenso…
Higher-order Casimir elements and hook partitions for quantum groups of types B, C and D
本文研究了B、C和D类型量子群的高阶Casimir元，通过钩分区与不可约特征标的关系，确定了中心的生成集，并揭示了Casimir元与钩分区之间的稳定性现象。
We categorify the Casimir element of the idempotented form of quantum sl(2).
It is shown, for any irreducible representation of $E_8$ Lie algebra, that eigenvalues of Casimir operators can be calculated in the form of invariant polinomials which are decomposed in terms of $A_8$ basis functions. The general method is applied for degrees 8,12 and 14 for which 2,8 and 19 invariant polinomials are obtained respectively. For each particular degree, these invariant polinomials can be taken to be $E_8$ basis functions in the sense that any Casimir operator of $E_8$ has always eigenvalues which can be expressed as linear superpositions of them. This can be investigated by show…
In 1986 Goldman introduced a Lie algebra structure on the linear span L of conjugacy classes of the fundamental group of a closed oriented surface. It is easy to see that the class e_1 of the trivial loop is a central element in L. We prove that any central element of L is a multiple of e_1 (a conjecture communicated to the author by Chas and Sullivan), and moreover that any central element of the Poisson algebra SL is a polynomial of e_1.
The aim of this paper is to give a set of central elements of the algebras $U'_q({\rm so}_m)$ and $U_q({\rm iso}_m)$ when q is a root of unity. They are surprisingly arise from a single polynomial Casimir element of the algebra $U'_q({\rm so}_3)$. It is conjectured that the Casimir elements of these algebras under any values of q (not only for q a root of unity) and the central elements for q a root of unity derived in this paper generate the centers of $U'_q({\rm so}_m)$ and $U_q({\rm iso}_m)$ when q is a root of unity.
For the last fifteen years quantum superalgebras have been used to model supersymmetric quantum systems. A class of quasi-triangular Hopf superalgebras, they each contain a universal $R$-matrix, which automatically satisfies the Yang--Baxter equation. Applying the vector representation to the left-hand side of a universal $R$-matrix gives a Lax operator. These are of significant interest in mathematical physics as they provide solutions to the Yang--Baxter equation in an arbitrary representation, which give rise to integrable models. In this thesis a Lax operator is constructed for the quantis…
In this paper we are dealing with the Reflection Equation algebra ${\cal M}(R)$, associated with a $GL_N$ type Hecke symmetry $R$. In this algebra we define the $q$-analogs of the partial derivatives $\partial_j^i$ in generators $m_i^j$ of ${\cal M}(R)$. The product $\hat L = MD$ of two matrices $M=\|m_i^j\|$ and $D=\|\partial_i^j\|$ turns out to be a generating matrix of a modified Reflection Equation algebra $\hat{\cal L}(R)$ which is similar to the universal enveloping algebra $U(gl_N)$ in many aspects. Central elements of the modified Reflection Equation algebra give rise to $q$-Casimir op…
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