14 1. BASIC NOTIONS
m/mn+1
is maximal. Let us show that any proper ideal I a is contained in
m/mn+1.
If I
m/mn+1,
then there exists α I represented by some x R
satisfying (x) = 0, where : R R/m

= is the augmentation map. By
multiplying with a scalar if necessary, we may clearly achieve that (x) = 1 that
is, x 1 m. By Lemma 1.22, there exists y R such that xy = 1. Then, for
the equivalence classes in a = R/mn+1 we have 1 = [x][y] = α[y], therefore 1 I,
so I = a.
Since m/mn+1 contains all proper ideals, it is the unique maximal proper ideal,
which proves that a is local. The Artin property of a is also clear: any proper ideal
I a is contained in m/mn+1, so Im+1 (m/mn+1)m+1 = 0.
Let us review three most important types of deformations. As usual, for el-
ements a, b of an associative algebra A we denote by ab their product. Also the
notation := {1, 2, 3,...} of the set of natural numbers is standard.
Definition 1.24. A formal deformation is a deformation over the complete
local augmented ring [[t]].
Proposition 1.25. A formal deformation of A = (V, · ) is given by a family
(1.27) {μi : V ⊗V V | i }
satisfying, for each k 1 and a, b, c V ,
(Dk)
μk(a, b)c + μk(ab, c) +
i+j=k
μi(μj(a, b),c) =
aμk(b, c) + μk(a, bc) +
i+j=k
μi(a, μj(b, c)).
Proof. By Lemma 1.20, the deformed multiplication μ is determined by its re-
striction to V ⊗V ( [[t]] V )
[[t]]
( [[t]] V ). Now expand μ(a, b), for a, b V ,
into a power series
μ(a, b) = μ0(a, b) + tμ1(a, b) +
t2μ2(a,
b) +
t3μ3(a,
b) + · · ·
with some -bilinear functions μi : V ⊗V V , i 0. Clearly, μ0 must be the
original multiplication in A so, in fact,
(1.28) μ(a, b) = ab + tμ1(a, b) +
t2μ2(a,
b) +
t3μ3(a,
b) + · · · .
It is easy to see that μ is associative if and only if (Dk) is satisfied for each k 1.
Proposition 1.25 shows that the set
DefA( [[t]]) of formal deformations of A
consists of families (1.27) satisfying the infinite system (Dk), k 1, of quadratic
equations. The trivial deformation is the one with μi = 0 for each i 1. It is also
clear that
(1.29) GA
(
[[t]]
)

= u =
V
+ φ1t +
φ2t2
+
φ3t3
+ · · · | φi Lin(V, V ) ,
with the group multiplication
(
V
+φ1t +
φ2t2
+ φ3
t3
+ · · · )(
V
+ φ1 t + φ2
t2
+
φ3t3
+ · · · ) :=
V
+ (φ1 + φ1 )t + (φ2 + φ1φ1 + φ2
)t2
+ (φ3 + φ2φ1 + φ1φ2 + φ3
)t3
+ · · · ,
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