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In this chapter, we provide the foundations of quantum Clifford analysis in q-commutative variables with symmetric difference operators. We consider a q-Dirac operator on the quantum Euclidean space that factorizes the \(U_q(\mathfrak {o})\) -invariant Laplacian \(\Delta _q.\) Due to the noncommutativity of the multiplication, we need a special Clifford algebra \(C \kern -0.1em \ell _{0,n}^q.\) We define q-monogenic functions as null solutions of the q-Dirac operator and q-spherical monogenic functions. We define an inner Fischer product and decompose the space of homogeneous polynomials of degree \(k.\)

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The q-Dirac Operator on Quantum Euclidean Space

  • 会议论文

  • 发表日期：

    2026年

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  • Swanhild Bernstein,
  • Martha Lina Zimmermann,
  • Baruch Schneider

摘要

In this chapter, we provide the foundations of quantum Clifford analysis in q-commutative variables with symmetric difference operators. We consider a q-Dirac operator on the quantum Euclidean space that factorizes the \(U_q(\mathfrak {o})\) -invariant Laplacian \(\Delta _q.\) Due to the noncommutativity of the multiplication, we need a special Clifford algebra \(C \kern -0.1em \ell _{0,n}^q.\) We define q-monogenic functions as null solutions of the q-Dirac operator and q-spherical monogenic functions. We define an inner Fischer product and decompose the space of homogeneous polynomials of degree \(k.\)