By Jenca G.

We turn out that if E1 and E2 are a-complete impression algebras such that E1 is an element of E2 and E2 is an element of E1, then E1 and E2 are isomorphic.

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**Example text**

From this, the exponents of Λ∗ are seen to be −cj . Also, the exponents for π d Λ are cj + 2d. We note that a lattice is almost self-dual if and only if its exponents are all either 0 or 1. If all the exponents of Λ are the same, then we can scale Λ by an appropriate power of π, to make every exponent of the rescaled lattice either 0 or 1, according as to whether the original exponent value is even or odd. The diﬀerence between the largest and smallest exponent of Λ is a measure of how far from self-dual a scalar multiple of Λ can be.

T ) = T , 3. (ST ) = T S . The symplectic group can be deﬁned as Sp(V ) = {g ∈ GL(V ) | gu, gv = u, v = u, g gv or g g = I}, with Lie algebra sp(V ) = {T ∈ End(V ) | T = −T }. There is the usual notion of duality with respect to the symplectic form. For a subspace U of V , deﬁne U ⊥ = {v ∈ V | v, u = 0 for all u ∈ U }. Then (U ⊥ )⊥ = U and dim U + dim U ⊥ = dim V . We say U is isotropic if , |U = 0, which is equivalent to U ⊂ U ⊥ . If all subspaces in a ﬂag are isotropic then we say the ﬂag is isotropic.

Hence, we see that by looking for a maximal almost-self dual lattice in L, we will ﬁnd a lattice which is actually self-dual. Still considering the self-dual, maximal lattice ﬂag L, let Λo be a self-dual lattice in L. Since L is totally ordered by inclusion, we may label the lattices in L by integers, in a manner such that Λm+1 is the largest element of L strictly contained in Λm . This labeling is clearly unique. Then the lattices Λm for 0 ≤ m ≤ 2n are contained between Λo and πΛo = Λ2n . They therefore deﬁne a ﬂag in Λ = Λ/πΛ.