W -Algebras via Lax Type Operators
189
3 Linear Algebra Intermezzo
3.1 Set Up
Let g be a finite-dimensional reductive Lie algebra, let {f, 2x, e} ⊂ g be an sl 2 -
triple and let (6) be the corresponding ad x-eigenspace decomposition. In Sects. 4
and 5 we will use the projection map π ≤
1
2
: g → g ≤
1
2
= ⊕ k≤
1
2
g k with kernel
g >
1
2
= ⊕ k>
1
2
g k .
Let ϕ : g → End V be a faithful representation of g on an N -dimensional vector
space V . Throughout the paper we shall often use the following convention: we
denote by lowercase Latin letters elements of the Lie algebra g, and by the same
uppercase letters the corresponding (via ϕ) elements of End V . For example, F =
ϕ(f ) is a nilpotent endomorphism of V . Moreover, X = ϕ(x) is a semisimple
endomorphism of V with half-integer eigenvalues. The corresponding X-eigenspace
decomposition of V is
V =
k∈
1
2 Z
V [k] .
(9)
Note that
d
2 is the largest X-eigenvalue in V .
Recall that the trace form on g associated to the representation V is, by definition,
(a|b) = tr V (AB) ,
a, b ∈ g ,
(10)
and we assume that it is non-degenerate. Let {u i } i∈I be a basis of g compatible
with the ad x-eigenspace decomposition (6), i.e. I = = k I k where {u i } i∈I k is a basis
of g k . We also denote I ≤
1
2
= = k≤
1
2
I k . Moreover, we shall also need, in Sect. 5, that
{u i } i∈I contains a basis {u i } i∈I f of g f = {a ∈ g | [a, f ] = 0}, the centralizer of
f in g. Let {u i } i∈I be the basis of g dual to {u i } i∈I with respect to the form (10),
i.e. (u i |u j ) = δ i,j . According to our convention, we denote by U i = ϕ(u i ) and
U i = ϕ(u i ), i ∈ I , the corresponding endomorphisms of V .
In Sects. 4 and 5 we will consider the following important element:
U =
i∈I
u i U
i
∈ g ⊗ End V .
(11)
Here and further we are omitting the tensor product sign.
Furthermore, the following endomorphism of V , which we will call the shift
matrix, will play an important role in Sect. 4
D = −
i∈I ≥1
U
i U i ∈ End V .
(12)
189
3 Linear Algebra Intermezzo
3.1 Set Up
Let g be a finite-dimensional reductive Lie algebra, let {f, 2x, e} ⊂ g be an sl 2 -
triple and let (6) be the corresponding ad x-eigenspace decomposition. In Sects. 4
and 5 we will use the projection map π ≤
1
2
: g → g ≤
1
2
= ⊕ k≤
1
2
g k with kernel
g >
1
2
= ⊕ k>
1
2
g k .
Let ϕ : g → End V be a faithful representation of g on an N -dimensional vector
space V . Throughout the paper we shall often use the following convention: we
denote by lowercase Latin letters elements of the Lie algebra g, and by the same
uppercase letters the corresponding (via ϕ) elements of End V . For example, F =
ϕ(f ) is a nilpotent endomorphism of V . Moreover, X = ϕ(x) is a semisimple
endomorphism of V with half-integer eigenvalues. The corresponding X-eigenspace
decomposition of V is
V =
k∈
1
2 Z
V [k] .
(9)
Note that
d
2 is the largest X-eigenvalue in V .
Recall that the trace form on g associated to the representation V is, by definition,
(a|b) = tr V (AB) ,
a, b ∈ g ,
(10)
and we assume that it is non-degenerate. Let {u i } i∈I be a basis of g compatible
with the ad x-eigenspace decomposition (6), i.e. I = = k I k where {u i } i∈I k is a basis
of g k . We also denote I ≤
1
2
= = k≤
1
2
I k . Moreover, we shall also need, in Sect. 5, that
{u i } i∈I contains a basis {u i } i∈I f of g f = {a ∈ g | [a, f ] = 0}, the centralizer of
f in g. Let {u i } i∈I be the basis of g dual to {u i } i∈I with respect to the form (10),
i.e. (u i |u j ) = δ i,j . According to our convention, we denote by U i = ϕ(u i ) and
U i = ϕ(u i ), i ∈ I , the corresponding endomorphisms of V .
In Sects. 4 and 5 we will consider the following important element:
U =
i∈I
u i U
i
∈ g ⊗ End V .
(11)
Here and further we are omitting the tensor product sign.
Furthermore, the following endomorphism of V , which we will call the shift
matrix, will play an important role in Sect. 4
D = −
i∈I ≥1
U
i U i ∈ End V .
(12)