Quadratically Presented Number Operator Algebras
نویسنده
چکیده
Following an idea of D. Bennequin, we give a classiication of quadratic algebras generated by creation and destruction operators, in which the Heisenberg equations of motion for a system of harmonic oscillators hold. For simplicity of exposition, we x K = C, the eld of complex numbers, but all the results would be valid for any commutative eld of characteristic 0, endowed with an involution (possibly the identity). Let X A = fa i ji 2 Ig, X A + = fa + i ji 2 Ig, be two sets of generators, in bijection with a set of indices I. We denote by J the involution of the disjoint union X := X A ` X A + sending a i to a + i , which we extend to an anti-automorphism of the free algebra L := KhXi. We consider a two-sided ideal I of L, generated by elements of degree 2, which is stable under J, and form the quotient B = L=I. We propose to determine the algebras B in which there exist \number op-+ j)] = ij (a + i) (1) N i ; (a j)] = ? ij (a i) (2) where is the canonical quotient map. We call these \(quadratically presented) number operator algebras". We will set aside the algebras 0 and C, and consider them as trivial. The classiication of these algebras relies partly on the following lemma, which is proved by easy induction on the length of monomials :
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Number Operator Algebras Fabien Besnard 7 th February 2008
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