h-deformation of Grassmann supergroup Gr$(1| 1)$

Noncommutative Supergeometry and Quantum Supergroups

This is a review of concepts of noncommutative supergeometry - namely Hilbert superspace, C*-superalgebra, quantum supergroup - and corresponding results. In particular, we present applications of noncommutative supergeometry in harmonic analysis of Lie supergroups, non-formal deformation quantization of supermanifolds, quantum field theory on noncommutative spaces; and we give explicit examples such as deformation of flat superspaces, noncommutative supertori, solvable topological quantum supergroups.

New Quantum Deformation of Osp(1;2)

The complete set of formulas describing the new quantum deformation of the OSP(1;2) supergroup is provided. A general ansatz is solved for the deformation of the Borel subalgebra of its dual quantum deformation of osp(1;2).

A three-parametric deformation of GL(1/1)

A three-parametric $R$-matrix satisfying a graded Yang-Baxter equation is introduced.This $R$-matrix allows us to construct new quantum supergroups which are deformations of the supergroup $GL(1/1)$ and the universal enveloping algebra $U[gl(1/1)]$.

Hopf algebra structure of Gr$_q(1| 1)$ related to GL$_q(1| 1)$

We show that the algebra of functions on the Grassmann supergroup Gr$_q(1| 1)$ has a (graded) Hopf algebra structure related to GL$_q(1| 1)$.

多参数量子超群、形变与特殊化

Multiparameter Quantum Supergroups, Deformations and Specializations

本文引入Yamane量子通用包络超代数的多参数版本，研究其扭形变与2-上循环形变、半经典极限以及形变的可交换性。

h-deformation of GL(1|1)

h-deformation of (graded) Hopf algebra of functions on supergroup GL(1|1) is introduced via a contration of GL_q (1|1). The deformation parameter h is odd (grassmann). Related differential calculus on h-superplane is presented.

Drinfeld twist for two-parametric deformation of gl(2) and sl(1/2)

Drinfeld twist is applied to the Lie algebra gl(2) so that a two-parametric deformation of it is obtained, which is identical to the Jordanian deformation of the gl(2) obtained by Aneva et al. The same twist element is applied to deform the Lie superalgebra sl(1/2), since the gl(2) is embedded into the sl(1/2). By making use of the FRT-formalism, we construct a deformation of the Lie supergroup $ SL(1/2). $

Two-parametric extension of h-deformation of GL$(1| 1)$

The two-parametric quantum deformation of the algebra of coordinate functions on the supergroup GL$(1| 1)$ via a contraction of GL$_{p,q}(1| 1)$ is presented. Related differential calculus on the quantum superplane is introduced.

多参数量子一般线性超群

Multiparameter quantum general linear supergroup

该研究引入一般线性超群的单参数量子化与多参数量子化，通过构造线性对偶、2 - 上循环变形等方法得到相关代数，给出多项式版本，与马尼恩的代数比较，并给出类似庞加莱 - 伯克霍夫 - 维特定理，阐述了多参数量子化的构建及相关性质。

Classification of the quantum deformation of the superalgebra $GL(1|1)$

We present a classification of the possible quantum deformations of the supergroup $GL(1|1)$ and its Lie superalgebra $gl(1|1)$. In each case, the (super)commutation relations and the Hopf structures are explicitly computed. For each $R$ matrix, one finds two inequivalent coproducts whether one chooses an unbraided or a braided framework while the corresponding structures are isomorphic as algebras. In the braided case, one recovers the classical algebra $gl(1|1)$ for suitable limits of the deformation parameters but this is no longer true in the unbraided case.

h-deformation of Gr(2)

The $h$-deformation of functions on the Grassmann matrix group $Gr(2)$ is presented via a contraction of $Gr_q(2)$. As an interesting point, we have seen that, in the case of the $h$-deformation, both R-matrices of $GL_h(2)$ and $Gr_h(2)$ are the same.

Two deformations of a fermionic solution to pentagon equation

Two novel fermionic - expressed in terms of Grassmann--Berezin calculus of anticommuting variables - solutions of pentagon equation are proposed, both being deformations of the known solution related to the affine group.

Relations in Grassmann Algebra Corresponding to Three- and Four-Dimensional Pachner Moves

New algebraic relations are presented, involving anticommuting Grassmann variables and Berezin integral, and corresponding naturally to Pachner moves in three and four dimensions. These relations have been found experimentally - using symbolic computer calculations; their essential new feature is that, although they can be treated as deformations of relations corresponding to torsions of acyclic complexes, they can no longer be explained in such terms. In the simpler case of three dimensions, we define an invariant, based on our relations, of a piecewise-linear manifold with triangulated bound…

Covariant Differential and Integral Calculi for Lattice (l,q)-deformed Fields

Using the Hecke $\hat R$-matrix, we give a definition of the lattice $(l,q)$-deformed $n$-component boson and Grassmann fields. Here $l$ is a deformation parameter for the commutation relations of "values" of these fields in two arbitrary lattice sites and $q$ is a deformation parameter for $n$-component $q$-boson or $q$-Grassmann variable. In framework of the Wess-Zumino approach to the noncommutative differential calculus the commutation relations between differentials and derivatives of these fields are determined. The $SL_q(n,C)$-invariant generalization of the Berezin integration for the …

Deformation of a 3 -> 3 Pachner move relation capturing exotic second homologies

A new relation in Grassmann algebra is presented, corresponding naturally to the four-dimensional Pachner move 3 -> 3. This relation is obtained by deforming a known relation associated with an exotic chain complex built for a triangulated four-manifold. The possible deformation terms turn out to correspond to second (middle) homologies of one more exotic complex. Both sides of the new relation are nonlinear in the deformation: they contain terms of degrees 0, 1 and 2.

Algebraic models of local period maps and Yukawa algebras

We describe some L-infinity model for the local period map of a compact Kaehler manifold. Applications include the study of deformations with associated variation of Hodge structure constrained by certain closed strata of the Grassmannian of the de Rham cohomology. As a byproduct we obtain an interpretation in the framework of deformation theory of the Yukawa coupling.

The deformation quantization of certain super-Poisson brackets and BRST cohomology

On every split supermanifold equipped with the Rothstein even super-Poisson bracket we construct a deformation quantization by means of a Fedosov-type procedure. In other words, the supercommutative algebra of all smooth sections of the dual Grassmann algebra bundle of an arbitrarily given vector bundle E (equipped with a fibre metric) over a symplectic manifold M will be deformed by a series of bidifferential operators having first order supercommutator proportional to the Rothstein superbracket. Moreover, we discuss two constructions related to the above result, namely the quantized BRST-coh…

二维共形场论、全顶点代数与流-流形变

Two-dimensional conformal field theory, full vertex algebra and current-current deformation

本文通过引入全顶点代数概念，数学上构造了二维共形场论的非微扰流-流形变，其参数空间为正交格拉斯曼流形的商，并给出了由此产生的新顶点算子代数同构类的计数公式。

非交换代数形变中的交换泊松代数

Commutative Poisson algebras from deformations of noncommutative algebras

本文将形式形变导出泊松括号的经典结果推广至非交换代数，在中心与商空间的积上构造泊松代数结构，并以多个量子代数实例验证。