Constant solutions to Yang-Baxter equation are investigated over Grassmann algebra for the case of 6-vertex R-matrix. The general classification of all possible solutions over Grassmann algebra and particular cases wi...Constant solutions to Yang-Baxter equation are investigated over Grassmann algebra for the case of 6-vertex R-matrix. The general classification of all possible solutions over Grassmann algebra and particular cases with 2,3,4 generators are studied. As distinct from the standard case, when R-matrix over number field can have a maximum 5 nonvanishing elements, we obtain over Grassmann algebra a set of new full 6-vertex solutions. The solutions leading to regular R-matrices which appear in weak Hopf algebras are considered.展开更多
Let ∧ be the Z2-Galois covering of the Grassmann algebra A over a field k of characteristic not equal to 2. In this paper, the dimensional formulae of Hochschild homology and cohomology groups of ∧ are calculated ex...Let ∧ be the Z2-Galois covering of the Grassmann algebra A over a field k of characteristic not equal to 2. In this paper, the dimensional formulae of Hochschild homology and cohomology groups of ∧ are calculated explicitly. And the cyclic homology of∧ can also be calculated when the underlying field is of characteristic zero. As a result, we prove that there is an isomorphism from i≥1 HH^i（∧） to i≥1 HH^i（∧）.展开更多
The mid-facet of a simplex in n-dimensional Euclidean space which was introduced quite recently is an important geometric element. An analytic expression for the mid-facet area of a simplex is firstly given. In order ...The mid-facet of a simplex in n-dimensional Euclidean space which was introduced quite recently is an important geometric element. An analytic expression for the mid-facet area of a simplex is firstly given. In order to obtain the expression,the exterior differential method was presented. Furthermore, the properties of the mid-facets of a simplex analogous to median lines of a triangle (such as for all mid-facets of a simplex,there exists another simplex such that its edge-lengths equal to these mid-facets area respectively, and all of the mid-facets of a simplex have a common point) were proved. Finally, by applying the analytic expression, a number of inequalities which combine edge-lengths, circumradius, median line, bisection area and facet area with the mid-facet area for a simplex were established.展开更多
The mid-facet of a simplex in n-dimensional Euclidean space which was introduced quite recently is an important geometric element. An analytic expression for the mid-facet area of a simplex is firstly given. In order...The mid-facet of a simplex in n-dimensional Euclidean space which was introduced quite recently is an important geometric element. An analytic expression for the mid-facet area of a simplex is firstly given. In order to obtain the expression,the exterior differential method was presented. Furthermore, the properties of the mid-facets of a simplex analogous to median lines of a triangle (such as for all mid-facets of a simplex,there exists another simplex such that its edge-lengths equal to these mid-facets area respectively, and all of the mid-facets of a simplex have a common point) were proved. Finally, by applying the analytic expression, a number of inequalities which combine edge-lengths, circumradius, median line, bisection area and facet area with the mid-facet area for a simplex were established.展开更多
A path-integral representation of central spin system immersed in an antiferromagnetic environment was investigated. To carry out this study, we made use of the discrete-time propagator method associated with a basic ...A path-integral representation of central spin system immersed in an antiferromagnetic environment was investigated. To carry out this study, we made use of the discrete-time propagator method associated with a basic set involving coherent states of Grassmann variables which made it possible to obtain the analytical propagator which is the centerpiece of the study. In this study, we considered that the environment was in the low-temperature and low-excitation limit and was split into 2 subnets that do not interact with each other. The evaluation of our system was made by considering the first neighbor approximation. From the formalism of the path integrals, it is easy to evaluate the partition function and thermodynamic properties followed from an appropriate tracing over Grassmann variables in the imaginary time domain. We show that the energy of the system depends on the number of sites <em>n</em> when <em>β </em><em></em><span></span>→ 0.展开更多
In an inner-product space, an invertible vector generates a reflection with respect to a hyperplane, and the Clifford product of several invertible vectors, called a versor in Clifford algebra, generates the compositi...In an inner-product space, an invertible vector generates a reflection with respect to a hyperplane, and the Clifford product of several invertible vectors, called a versor in Clifford algebra, generates the composition of the corresponding reflections, which is an orthogonal transformation. Given a versor in a Clifford algebra, finding another sequence of invertible vectors of strictly shorter length but whose Clifford product still equals the input versor, is called versor compression. Geometrically, versor compression is equivalent to decomposing an orthogonal transformation into a shorter sequence of reflections. This paper proposes a simple algorithm of compressing versors of symbolic form in Clifford algebra. The algorithm is based on computing the intersections of lines with planes in the corresponding Grassmann-Cayley algebra, and is complete in the case of Euclidean or Minkowski inner-product space.展开更多
文摘Constant solutions to Yang-Baxter equation are investigated over Grassmann algebra for the case of 6-vertex R-matrix. The general classification of all possible solutions over Grassmann algebra and particular cases with 2,3,4 generators are studied. As distinct from the standard case, when R-matrix over number field can have a maximum 5 nonvanishing elements, we obtain over Grassmann algebra a set of new full 6-vertex solutions. The solutions leading to regular R-matrices which appear in weak Hopf algebras are considered.
基金Supported by National Natrual Science Foundation of China (Grant No.10501010)
文摘Let ∧ be the Z2-Galois covering of the Grassmann algebra A over a field k of characteristic not equal to 2. In this paper, the dimensional formulae of Hochschild homology and cohomology groups of ∧ are calculated explicitly. And the cyclic homology of∧ can also be calculated when the underlying field is of characteristic zero. As a result, we prove that there is an isomorphism from i≥1 HH^i（∧） to i≥1 HH^i（∧）.
文摘The mid-facet of a simplex in n-dimensional Euclidean space which was introduced quite recently is an important geometric element. An analytic expression for the mid-facet area of a simplex is firstly given. In order to obtain the expression,the exterior differential method was presented. Furthermore, the properties of the mid-facets of a simplex analogous to median lines of a triangle (such as for all mid-facets of a simplex,there exists another simplex such that its edge-lengths equal to these mid-facets area respectively, and all of the mid-facets of a simplex have a common point) were proved. Finally, by applying the analytic expression, a number of inequalities which combine edge-lengths, circumradius, median line, bisection area and facet area with the mid-facet area for a simplex were established.
文摘The mid-facet of a simplex in n-dimensional Euclidean space which was introduced quite recently is an important geometric element. An analytic expression for the mid-facet area of a simplex is firstly given. In order to obtain the expression,the exterior differential method was presented. Furthermore, the properties of the mid-facets of a simplex analogous to median lines of a triangle (such as for all mid-facets of a simplex,there exists another simplex such that its edge-lengths equal to these mid-facets area respectively, and all of the mid-facets of a simplex have a common point) were proved. Finally, by applying the analytic expression, a number of inequalities which combine edge-lengths, circumradius, median line, bisection area and facet area with the mid-facet area for a simplex were established.
文摘A path-integral representation of central spin system immersed in an antiferromagnetic environment was investigated. To carry out this study, we made use of the discrete-time propagator method associated with a basic set involving coherent states of Grassmann variables which made it possible to obtain the analytical propagator which is the centerpiece of the study. In this study, we considered that the environment was in the low-temperature and low-excitation limit and was split into 2 subnets that do not interact with each other. The evaluation of our system was made by considering the first neighbor approximation. From the formalism of the path integrals, it is easy to evaluate the partition function and thermodynamic properties followed from an appropriate tracing over Grassmann variables in the imaginary time domain. We show that the energy of the system depends on the number of sites <em>n</em> when <em>β </em><em></em><span></span>→ 0.
文摘In an inner-product space, an invertible vector generates a reflection with respect to a hyperplane, and the Clifford product of several invertible vectors, called a versor in Clifford algebra, generates the composition of the corresponding reflections, which is an orthogonal transformation. Given a versor in a Clifford algebra, finding another sequence of invertible vectors of strictly shorter length but whose Clifford product still equals the input versor, is called versor compression. Geometrically, versor compression is equivalent to decomposing an orthogonal transformation into a shorter sequence of reflections. This paper proposes a simple algorithm of compressing versors of symbolic form in Clifford algebra. The algorithm is based on computing the intersections of lines with planes in the corresponding Grassmann-Cayley algebra, and is complete in the case of Euclidean or Minkowski inner-product space.
