The parity operator for parafermions and parabosons

Abstract In this paper we reexamine the definition of parafermions and parabosons by means of Green’s triple relations, and extend these relations by including a parity operator P which is also determined by means of triple relations. As a consequence, we are dealing with new algebraic structures. It is shown that the algebra underlying a set of n parafermions together with P is the orthogonal Lie algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">so</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . The Fock spaces correspond to particular irreducible representations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">so</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , and the action of P in these spaces leads to interesting observations. Next, we show that the algebra underlying a set of n parabosons together with P is the orthosymplectic Lie superalgebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">osp</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . In this case, the Fock spaces correspond to certain irreducible infinite-dimensional representations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">osp</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . Both for parafermions and parabosons the spectrum of P is closely related to the so-called order of statistics p , introduced by Green.