190
D. Valeri
Finally, we denote by Ω V ∈ End V ⊗ End V the permutation map:
Ω V (v 1 ⊗ v 2 ) = v 2 ⊗ v 1 for all v 1 , v 2 ∈ V .
(13)
Using Sweedler’s notation we write Ω V = Ω
V ⊗ Ω
V to denote, as usual, a sum of
monomials in End V ⊗ End V . Suppose that V has a non-degenerate bilinear form
· | ·· : V × V → F, which is symmetric or skewsymmetric:
1 |v 2 = 2 |v 1 , v 1 , v 2 ∈ V , where ∈ {±1} .
(14)
Then, we denote by
Ω
†
V = (Ω
V )
†
⊗ Ω
V ,
(15)
where A † is the adjoint of A ∈ End V with respect to (14).
3.2 The “Identity” Notation
Let U ⊂ V be a subspace of V , and assume that there is “natural” splitting V =
U ⊕ U . We shall denote, with an abuse of notation, by 1 U both the identity map
U
∼
−→ U , the inclusion map U U→ V , and the projection map V U with kernel
U . The correct meaning of 1 U should be clear from the context.
3.3 Generalized Quasideterminants
Let R be a unital associative algebra and let V be a finite-dimensional vector space
with direct sum decompositions V = U ⊕ U = W ⊕ W . Assume that A ∈
R ⊗ End(V ) and 1 W A −1 1 U ∈ R ⊗ Hom(U, W ) are invertible. The (generalized)
quasideterminant of A with respect to U and W , cf. [18, 35], is defined as
|A| U,W := (1 W A
−1 1 U )
−1
∈ R ⊗ Hom(W, U ) .
(16)
Remark 2 Provided that both A and 1 U A1 W are invertible, it is possible to
write the generalized quasideterminant (16) in the more explicit form |A| U,W =
1 U A1 W − 1 U A1 W (1 U A1 W ) −1 1 U A1 W .
D. Valeri
Finally, we denote by Ω V ∈ End V ⊗ End V the permutation map:
Ω V (v 1 ⊗ v 2 ) = v 2 ⊗ v 1 for all v 1 , v 2 ∈ V .
(13)
Using Sweedler’s notation we write Ω V = Ω
V ⊗ Ω
V to denote, as usual, a sum of
monomials in End V ⊗ End V . Suppose that V has a non-degenerate bilinear form
· | ·· : V × V → F, which is symmetric or skewsymmetric:
1 |v 2 = 2 |v 1 , v 1 , v 2 ∈ V , where ∈ {±1} .
(14)
Then, we denote by
Ω
†
V = (Ω
V )
†
⊗ Ω
V ,
(15)
where A † is the adjoint of A ∈ End V with respect to (14).
3.2 The “Identity” Notation
Let U ⊂ V be a subspace of V , and assume that there is “natural” splitting V =
U ⊕ U . We shall denote, with an abuse of notation, by 1 U both the identity map
U
∼
−→ U , the inclusion map U U→ V , and the projection map V U with kernel
U . The correct meaning of 1 U should be clear from the context.
3.3 Generalized Quasideterminants
Let R be a unital associative algebra and let V be a finite-dimensional vector space
with direct sum decompositions V = U ⊕ U = W ⊕ W . Assume that A ∈
R ⊗ End(V ) and 1 W A −1 1 U ∈ R ⊗ Hom(U, W ) are invertible. The (generalized)
quasideterminant of A with respect to U and W , cf. [18, 35], is defined as
|A| U,W := (1 W A
−1 1 U )
−1
∈ R ⊗ Hom(W, U ) .
(16)
Remark 2 Provided that both A and 1 U A1 W are invertible, it is possible to
write the generalized quasideterminant (16) in the more explicit form |A| U,W =
1 U A1 W − 1 U A1 W (1 U A1 W ) −1 1 U A1 W .