  We develop in this paper the principles of an associative algebraic approach
to bulk logarithmic conformal field theories (LCFTs). We concentrate on the
closed $gl(1|1)$ spin-chain and its continuum limit - the $c=-2$ symplectic
fermions theory - and rely on two technical companion papers, "Continuum limit
and symmetries of the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013)
245-288] and "Bimodule structure in the periodic gl(1|1) spin chain" [Nucl.
Phys. B 871 (2013) 289-329]. Our main result is that the algebra of local
Hamiltonians, the Jones-Temperley-Lieb algebra JTL_N, goes over in the
continuum limit to a bigger algebra than the product of the left and right
Virasoro algebras. This algebra, S - which we call interchiral, mixes the left
and right moving sectors, and is generated, in the symplectic fermions case, by
the additional field $S(z,\bar{z})=S_{ab}\psi^a(z)\bar{\psi}^b(\bar{z})$, with
a symmetric form $S_{ab}$ and conformal weights (1,1). We discuss in details
how the Hilbert space of the LCFT decomposes onto representations of this
algebra, and how this decomposition is related with properties of the finite
spin-chain. We show that there is a complete correspondence between algebraic
properties of finite periodic spin chains and the continuum limit. An important
technical aspect of our analysis involves the fundamental new observation that
the action of JTL_N in the $gl(1|1)$ spin chain is in fact isomorphic to an
enveloping algebra of a certain Lie algebra, itself a non semi-simple version
of $sp(N-2)$. The semi-simple part of JTL_N is represented by $Usp(N-2)$,
providing a beautiful example of a classical Howe duality, for which we have a
non semi-simple version in the full JTL image represented in the spin-chain. On
the continuum side, simple modules over the interchiral algebra S are
identified with "fundamental" representations of $sp(\infty)$.
