  We study the low-energy approximation for calculation of the heat kernel
which is determined by the strong slowly varying background fields in strongly
curved quasi-homogeneous manifolds. A new covariant algebraic approach, based
on taking into account a finite number of low-order covariant derivatives of
the background fields and neglecting all covariant derivatives of higher
orders, is proposed. It is shown that a set of covariant differential operators
together with the background fields and their low-order derivatives generates a
finite dimensional Lie algebra. This algebraic structure can be used to present
the heat semigroup operator in the form of an average over the corresponding
Lie group. Closed covariant formulas for the heat kernel diagonal are obtained.
These formulas serve, in particular, as the generating functions for the whole
sequence of the Hadamard-\-Minakshisundaram-\-De~Witt-\-Seeley coefficients in
all symmetric spaces.
