  This paper is essentially a short version of hep-th/9404046. We compute
multiplicative anomaly det(AB)/(detA detB) =F(A,B) for elliptic
pseudo-differential operators (PDOs) A, B on a closed manifold M in terms of
their symbols. We prove that F(A,B)=1 for elliptic differential operators close
to positive-definite ones on an odd-dimensional M. For such M we introduce a
holomorphic determinant. Its monodromy lies in a finite group of roots of
unity.
  In general case, we relate the multiplicative anomaly with a central
extension of the group of elliptic symbols and with an invariant quadratic form
on this extension. We compute the Lie algebra of the central extension in terms
of logarithmic symbols. The main tool is a new trace-type functional defined on
classical PDOs of non-integer orders. A canonical det of elliptic PDOs
generalizing the zeta-regularized det is introduced.
