• Title, Summary, Keyword: Clifford algebra

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THE MATRIX REPRESENTATION OF CLIFFORD ALGEBRA

  • Lee, Doohann;Song, Youngkwon
    • Journal of the Chungcheong Mathematical Society
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    • v.23 no.2
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    • pp.363-368
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    • 2010
  • In this paper we construct a subalgebra $L_8$ of $M_8({\mathbb{R}})$ which is a generalization of the algebra of quaternions. Moreover we prove that the algebra $L_8$ is the real Clifford algebra $Cl_3$, and so $L_8$ is a matrix representation of Clifford algebra $Cl_3$.

VECTOR GENERATORS OF THE REAL CLIFFORD ALGEBRA Cℓ0,n

  • Song, Youngkwon;Lee, Doohann
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.4
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    • pp.571-579
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    • 2014
  • In this paper, we present new vector generators of a matrix subalgebra $L_{0,n}$, which is isomorphic to the Clifford algebra $C{\ell}_{0,n}$, and we obtain the matrix form of inverse of a vector in $L_{0,n}$. Moreover, we consider the solution of a linear equation $xg_2=g_2x$, where $g_2$ is a vector generator of $L_{0,n}$.

ON COMPLEX REPRESENTATIONS OF THE CLIFFORD ALGEBRAS

  • Song, Youngkwon
    • Communications of the Korean Mathematical Society
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    • v.35 no.2
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    • pp.487-497
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    • 2020
  • In this paper, we establish a complex matrix representation of the Clifford algebra Cℓp,q. The size of our representation is significantly smaller than the size of the elements in Lp,q(ℝ). Additionally, we give detailed information about the spectrum of the constructed matrix representation.

CLIFFORD $L^2$-COHOMOLOGY ON THE COMPLETE $K\"{A}$HLER MANIFOLDS

  • Pak, Jin-Suk;Jung, Seoung-Dal
    • Journal of the Korean Mathematical Society
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    • v.34 no.1
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    • pp.167-179
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    • 1997
  • In the study of a manifold M, the exterior algebra $\Lambda^* M$ plays an important role. In fact, the de Rham cohomology theory gives many informations of a manifold. Another important object in the study of a manifold is its Clifford algebra (Cl(M), generated by the tangent space.

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Fermi Fields, Clifford Alegebras and Path Integrals II (페르미장, 클리포드대수 그리고 경로적분 II)

  • Hwang, Cheolhoi;Lee, Haewon
    • New Physics: Sae Mulli
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    • v.67 no.6
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    • pp.733-737
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    • 2017
  • In quantum field theories with Fermi fields, one can rewrite the theories by using Hermitian Fermi fields only, which show the structure of an extended Clifford algebra. We consider quantum field theories with Fermi fields with finite degrees of freedom. In our previous work, theories with positive definite Hilbert spaces were considered, and path integral formulae were re-derived using the properties of a Clifford algebra. Here, we extend the previous result to theories with indefinite Hilbert spaces and find similar path integral formulae. The Lagrangians appearing in the path integrals are determined by the matrix elements of the Hamiltonians between two different state vectors.

CLIFFORD $L^2$-COHOMOLOGY ON THE COMPLETE KAHLER MANIFOLDS II

  • Bang, Eun-Sook;Jung, Seoung-Dal;Pak, Jin-Suk
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.669-681
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    • 1998
  • In this paper, we prove that on the complete Kahler manifold, if ${\rho}(x){\geq}-\frac{1}{2}{\lambda}_0$ and either ${\rho}(x_0)>-\frac{1}{2}{lambda}_0$ at some point $x_0$ or Vol(M)=${\infty}$, then the Clifford $L^2$ cohomology group $L^2{\mathcal H}^{\ast}(M,S)$ is trivial, where $\rho(x)$ is the least eigenvalue of ${\mathcal R}_x + \bar{{\mathcal R}}(x)\;and\;{\lambda}_0$ is the infimum of the spectrum of the Laplacian acting on $L^2$-functions on M.

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EXAMPLES OF SIMPLY REDUCIBLE GROUPS

  • Luan, Yongzhi
    • Journal of the Korean Mathematical Society
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    • v.57 no.5
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    • pp.1187-1237
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    • 2020
  • Simply reducible groups are important in physics and chemistry, which contain some of the important groups in condensed matter physics and crystal symmetry. By studying the group structures and irreducible representations, we find some new examples of simply reducible groups, namely, dihedral groups, some point groups, some dicyclic groups, generalized quaternion groups, Heisenberg groups over prime field of characteristic 2, some Clifford groups, and some Coxeter groups. We give the precise decompositions of product of irreducible characters of dihedral groups, Heisenberg groups, and some Coxeter groups, giving the Clebsch-Gordan coefficients for these groups. To verify some of our results, we use the computer algebra systems GAP and SAGE to construct and get the character tables of some examples.

CLASSIFICATION OF CLIFFORD ALGEBRAS OF FREE QUADRATIC SPACES OVER FULL RINGS

  • Kim, Jae-Gyeom
    • Bulletin of the Korean Mathematical Society
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    • v.22 no.1
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    • pp.11-15
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    • 1985
  • Manddelberg [9] has shown that a Clifford algebra of a free quadratic space over an arbitrary semi-local ring R in Brawer-Wall group BW(R) is determined by its rank, determinant, and Hasse invariant. In this paper, we prove a corresponding result when R is a full ring.Throughout this paper, unless otherwise specified, we assume that R is a commutative ring having 2 a unit. A quadratic space (V, B, .phi.) over R is a finitely generated projective R-module V with a symmetric bilinear mapping B: V*V.rarw.R which is non-degenerate (i.e., the natural mapping V.rarw.Ho $m_{R}$(V,R) induced by B is an isomorphism), and with a quadratic mapping .phi.: V.rarw.R such that B(x,y)=1/2(.phi.(x+y)-.phi.(x)-.phi.(y)) and .phi.(rx) = $r^{2}$.phi.(x) for all x, y in V and r in R. We denote the group of multiplicative units of R by U9R). If (V, B, .phi.) is a free rank n quadratic space over R with an orthogonal basis { $x_{1}$,.., $x_{n}$}, we will write < $a_{1}$,.., $a_{n}$> for (V, B, .phi.) where the $a_{i}$=.phi.( $x_{i}$) are in U(R), and denote the space by the table [ $a_{ij}$ ] where $a_{ij}$ =B( $x_{i}$, $x_{j}$). In the case n=2 and B( $x_{1}$, $x_{2}$)=1/2 we reserve the notation [a $a_{11}$, $a_{22}$] for the space. A commutative ring R having 2 a unit is called full [10] if for every triple $a_{1}$, $a_{2}$, $a_{3}$ of elements in R with ( $a_{1}$, $a_{2}$, $a_{3}$)=R, there is an element w in R such that $a_{1}$+ $a_{2}$w+ $a_{3}$ $w^{2}$=unit.TEX>=unit.t.t.t.

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