aa r X i v : . [ m a t h . R A ] A p r Invariant Algebras
Keqin LiuDepartment of MathematicsThe University of British ColumbiaVancouver, BCCanada, V6T 1Z2January, 2011
Abstract
We introduce invariant algebras and representation ( c ,...,c ) of alge-bras, and give many ways of constructing Lie algebras, Jordan algebras,Leibniz algebras, pre-Lie algebras and left-symmetric algebras in an in-variant algebras. In this paper, we introduce invariant algebras and representation ( c ,...,c ) ofalgebras. The main property of invariant algebras is that an invariant algebracarries 14 associative products, which can be used to construct many differentalgebras inside an invariant algebra. We use this paper to give 6 ways of con-structing a Lie algebra structure, 6 ways of constructing a Jordan algebra struc-ture, 4 ways of constructing a Leibniz algebra structure, 14 ways of constructinga pre-Lie algebra structure and 14 ways of constructing a left-symmetric algebrastructure inside an invariant algebra. Based on the main property of invariantalgebras, we introduce representation ( c ,...,c ) of algebras in the last section ofthis paper. Using representation ( c ,...,c ) of algebras is a natural way of de-veloping representation theory for some non-associative algebras like Leibnizalgebras, pre-Lie algebras and left-symmetric algebras.Throughout, all vector spaces are vector spaces over a field k . The notion of invariant algebras introduced in this paper is given in the following
Definition 1.1
Let A be an associative algebra with an idempotent q . The (right) invariant algebra) ( A, q ) induced by the idempotent q is defined by ( A, q ) := { x, | x ∈ A and qxq = qx } . (1)1he most important example of invariant algebras is the linear invariantalgebra over a vector space. Let V be a vector space, let End ( V ) be the asso-ciative algebra of all linear transformations from V to V , and let I (or I V ) bethe identity linear transformation of V . If W is a subspace of V , then an lineartransformation q satisfying q ( W ) = 0 and( q − I )( V ) ⊆ W (2)is an idempotent because q ( v ) − q ( v ) = q ( q − I )( v ) ∈ q ( W ) = 0 for all v ∈ V .An element q of End ( V ) satisfying (2) is called a W -idempotent .Note that ( End ( V ) , q ) = { f | f ∈ End ( V ) and f ( W ) ⊆ W } , (3)where W is a subspace of a vector space V and q is a W -idempotent. Theinvariant algebra ( End ( V ) , q ) is called the linear invariant algebra over V induced by the W -idempotent q , which consists of all linear transformations of V having W as an invariant subspace by (3).Let ( A, q ) be an invariant algebra. The set(
A, q ) ann := { qx − x | x ∈ ( A, q ) } (4)is clearly a subspace of ( A, q ). For a , x ∈ ( A, q ), we have( qx − x ) a = q ( xa ) − ( xa ) ∈ ( A, q ) ann and a ( qx − x ) = q ( − aqx + ax ) − ( − aqx + ax ) ∈ ( A, q ) ann . Hence, the subspace (
A, q ) ann is an ideal of ( A, q ), which is called the annihi-lator of (
A, q ). For x ∈ ( A, q ), we have xq − x = q ( − xq + x ) − ( − xq + x ) ∈ ( A, q ) ann , which implies that( A, q ) ann ⊇ { xq − x | x ∈ ( A, q ) } [ { qx − xq | x ∈ ( A, q ) } . (5) Definition 1.2
Let ( A, q A ) and ( B, q B ) be invariant algebras. A map φ :( A, q A ) → ( B, q B ) is called an invariant homomorphism if φ ( x + y ) = φ ( x ) + φ ( y ) , φ ( xy ) = φ ( x ) φ ( y ) for x , y ∈ ( A, q A ) and φ (1 A ) = 1 B , φ ( q A ) = q B , A and 1 B are the identities of A and B , respectively. A bijective invarianthomomorphism is called an invariant isomorphism .The next proposition shows that any invariant algebra can be embedded ina linear invariant algebra. Proposition 1.1 If ( A, q ) is an invariant algebra, then the map φ : a a L for a ∈ ( A, q ) is an injective invariant homomorphism from ( A, q ) to the linear invariant al-gebra ( End ( A, q ) , q L ) over ( A, q ) induced by the ( A, q ) ann -idempotent q L , where a L is the left multiplication defined by a L ( x ) := ax for x ∈ ( A.q ) . We now define Hu-Liu products in the following
Definition 2.1
Let ( A, q ) be an invariant algebra over a field k . For ≤ i ≤ ,the i -th Hu-Liu product ◦ i is defined by x ◦ y : = kyx + hyqx, (6) x ◦ y : = kyx − kyqx + qxy, (7) x ◦ y : = yx + kyqx − yxq, (8) x ◦ y : = yx + kqxy − yxq, (9) x ◦ y : = kyx − kyqx + qyx, (10) x ◦ y : = yx + kqyx − yxq, (11) x ◦ y : = kyx + xqy − kyxq, (12) x ◦ y : = kxy + hxqy, (13) x ◦ y : = xy − xqy + kqxy, (14) x ◦ y : = xy − xqy + kqyx, (15) x ◦ y : = xy − xyq + kqxy, (16) x ◦ y : = xy + kyqx − xyq, (17) x ◦ y : = xy − xyq + kqyx, (18) x ◦ y : = xy + kxqy − xyq, (19) where k , h are fixed scalars in the field k and x , y ∈ ( A, q ) . For i = 1 or 8, ◦ i is also denoted by ◦ i,k,h to indicate that ◦ and ◦ dependon the scalars k and h . Similarly, for 7 ≥ i ≥ ≥ i ≥ ◦ i is also denotedby ◦ i,k .The next proposition gives the main property of 14 Hu-Liu products.3 roposition 2.1 If ( A, q ) is an invariant algebra over a field k , then the i -thHu-Liu product ◦ i satisfies the associative law: ( x ◦ i y ) ◦ i z = x ◦ i ( y ◦ i z ) , (20) where ≤ i ≤ and x , y , z ∈ ( A, q ) . If (
A, q ) is an invariant algebra, then (
A, q ) can be made into a Lie algebra bythe well-known square bracket [ , ], where [ , ] is defined by[ x, y ] := xy − yx for x , y ∈ ( A, q ). (21)Except the ordinary square bracket (21), there are other six square bracketswhich also make an invariant algebra into a Lie algebra. We now introduce thesix square brackets in the following
Definition 3.1
Let ( A, q ) be an invariant algebra over a field k . For ≤ i ≤ ,the i -th square bracket [ , ] i is defined by [ x, y ] : = qxy − qyx, (22)[ x, y ] : = xqy − yqx, (23)[ x, y ] : = xy − yx + kxqy − kyqx, (24)[ x, y ] : = xy − yx − xqy + yqx − kqxy + kqyx, (25)[ x, y ] : = xy − yx − xyq + yxq − kqxy + kqyx, (26)[ x, y ] : = xy − yx − xyq + yxq + kxqy − kyqx, (27) where x , y ∈ ( A, q ) , and k is a fixed scalar in the field k . The square bracket [ , ] i with 3 ≤ i ≤ , ] i,k to indicateits dependence on the scalar k . The next proposition gives the basic propertyof the six square brackets.Each i -th square bracket with 6 ≥ i ≥ ◦ i by a few ways. One of the ways is given by[ x, y ] = x ◦ , y − y ◦ , x, [ x, y ] = x ◦ , , y − y ◦ , , x, [ x, y ] ,k = x ◦ , ,k y − y ◦ , ,k x, [ x, y ] ,k = x ◦ ,k y − y ◦ ,k x, [ x, y ] ,k = x ◦ ,k y − y ◦ ,k x, [ x, y ] ,k = x ◦ ,k y − y ◦ ,k x. Proposition 3.1
Let ( A, q ) be an invariant algebra. i) The -st and -nd square brackets satisfy the Jacobi identity; that is, [[ x, y ] i , z ] i + [[ y, z ] i , x ] i + [[ z, x ] i , y ] i = 0 for x, y, z ∈ ( A, q ) , (28) where i = 1 and . (ii) The -rd square bracket satisfies the following long Jacobi-like identity − st : [[ x, y ] h , z ] k + [[ y, z ] h , x ] k + [[ z, x ] h , y ] k ++ [[ x, y ] k , z ] h + [[ y, z ] k , x ] h + [[ z, x ] k , y ] h = 0 (29) for x, y, z ∈ ( A, q ) and h , k ∈ k . (iii) Let x, y, z ∈ ( A, q ) and k , h ∈ k . If i = 4 , and , then the i -th anglebracket satisfies the Jacobi-like identity − st : [[ x, y ] i,h , z ] i,k + [[ y, z ] i,h , x ] i,k + [[ z, x ] i,h , y ] i,k = 0 . (30) Moreover, we have [[ x, y ] i,h , z ] i,k = [[ x, y ] i,k , z ] i,h for i = 4 , and . (31)Since the six square brackets are anti-commutative, each of the six squarebrackets makes an invariant algebra into a Lie algebra. We begin this section by recalling the definition of a Jordan algebra from [2].
Definition 4.1 A Jordan algebra J is an algebra over a field k of character-istic = 2 with a product composition ⊙ satisfying x ⊙ y = y ⊙ x (Commutative Law) (32) and (cid:16) ( x ⊙ x ) ⊙ y (cid:17) ⊙ x = ( x ⊙ x ) ⊙ ( y ⊙ x ) (Jordan Identity) (33) for all x , y ∈ J . Based on the i -th square brackets in Definition 3.1, we have the following Proposition 4.1
An invariant algebra ( A, q ) over a field k of characteristic = 2 is a Jordan algebra under each of the following products: x ⊙ y = 12 ( qxy + qyx ) , (34) x ⊙ y = 12 ( xqy + yqx ) , (35)5 ⊙ y = 12 ( xy + kxqy + yx + kyqx ) , (36) x ⊙ y = 12 ( xy − xqy − kqxy + yx − yqx − kqyx ) , (37) x ⊙ y = 12 ( xy − xyq − kqxy + yx − yxq − kqyx ) , (38) x ⊙ y = 12 ( xy − xyq + kxqy + yx − yxq + kyqx ) , (39) where x , y ∈ ( A, q ) , and k is a fixed scalar in the field k . We begin this section with the definition of i -th angle brackets. Definition 5.1
Let ( A, q ) be an invariant algebra over a field k . For i = 1 , , and , the i -th angle bracket h , i i is defined by h x, y i : = xqy − qyx, (40) h x, y i : = xy − yx + yqx − xyq + kqyx − kqxy, (41) h x, y i : = xy − yx − xyq + yxq + kxqy − kqyx, (42) h x, y i : = xy − yx + yqx − xyq + kxqy − kqyx, (43) where x , y ∈ ( A, q ) , and k is a fixed scalar in the field k . The angle bracket h , i i with 2 ≤ i ≤ h , i i,k to indicateits dependence on the scalar k . The next proposition gives the basic propertyof the four angle brackets. Proposition 5.1
Let ( A, q ) be an invariant algebra. (i) The -st angle bracket h x, y i satisfies the (right) Leibniz identity; that is, h x, h y, z i i = hh x, y i , z i − hh x, z i , y i (44) for x, y, z ∈ ( A, q ) . (ii) Let x, y, z ∈ ( A, q ) and k , h ∈ k . If i = 2 , and , then the i -th anglebracket satisfies the Jacobi-like identity − nd : h x, h y, z i i,k i i,h = hh x, y i i,h , z i i,k − hh x, z i i,k , y i i,h . (45) Moreover, we have h x, h y, z i i,k i i,h = h x, h y, z i i,h i i,k for i = 2 , and and hh x, y i i,k , z i i,h = hh x, y i i,h , z i i,k for i = 2 and . (47)6ollowing [3], the notion of (right) Leibniz algebras is given in the following Definition 5.2
A vector space L is called a (right) Leibniz algebra if thereexists a binary operation h , i : L × L → L , called the angle bracket , such thatthe (right) Leibniz identity holds: hh x, y i , z i = h x, h y, z ii + hh x, z i , y i for x, y, z ∈ L . (48)By Proposition 5.1, each of the four angle brackets makes an invariant alge-bra into a (right) Leibniz algebra. We begin this section by recalling the definition of a pre-Lie algebra from [1].
Definition 6.1 A pre-Lie algebra A is a vector space equipped with a dotoperation · : A ⊗ A → A which satisfy the following identity: ( x · y ) · z − x · ( y · z ) = ( x · z ) · y − x · ( z · y ) for x , y , z ∈ A . (49)We use ( A, · ) to denote a pre-Lie algebra A equipped with a dot operation · . Clearly, the square bracket( x, y ) [ x, y ] · := x · y − y · x for x , y ∈ A (50)satisfies the Jacobi identity; that is, a pre-Lie algebra is a Lie-admissible algebra.The square bracket [ x, y ] · defined by (50) is called the accompanying squareproduct . The following proposition gives 14 ways of introducing a pre-Liealgebra structure in an invariant algebra and shows that each correspondingaccompanying square product only differs from a square product in Definition3.1 by a scalar. Proposition 6.1
Let ( A, q ) be an invariant algebra over a field k . Let k be afixed scalar in the field k . (( A, q ) , · i ) is a pre-Lie algebra if the dot operation · i is chosen in one of the following ways: (1) x · y = kyx + xqy − kyqx , [ x, y ] · = [ x, y ] for k = 0 and − k [ x, y ] · = [ x, y ] for k = 0 ; (2) x · y = kyx + xqy − kyqx − qyx − qxy , [ x, y ] · = [ x, y ] for k = 0 and − k [ x, y ] · = [ x, y ] for k = 0 ; (3) x · y = kyx + xqy + (1 − k ) yqx − qyx , [ x, y ] · = [ x, y ] for k = 0 and − k [ x, y ] · = [ x, y ] for k = 0 ; x · y = kyx + xqy − qyx − qxy − kyxq , [ x, y ] · = [ x, y ] for k = 0 and − k [ x, y ] · = [ x, y ] for k = 0 ; (5) x · y = kyx + xqy + yqx − qyx − kyxq , − [ x, y ] · = [ x, y ] for k = 0 and − k [ x, y ] · = [ x, y ] for k = 0 ; (6) x · y = yx − xqy − yqx + xyq and − [ x, y ] · = [ x, y ] ; (7) x · y = yx + kxqy − yqx + xyq and − [ x, y ] · = [ x, y ] ; (8) x · y = yx + kxqy − yqx + xyq − ( k +1) qyx − ( k +1) qxy and − [ x, y ] · = [ x, y ] ; (9) x · y = xy + kyqx − xyq − yxq and [ x, y ] · = [ x, y ] ; (10) x · y = xy + yqx − xyq + kqyx − yxq and [ x, y ] · = [ x, y ] ; (11) x · y = xy + yqx − xyq + kqxy − yxq and [ x, y ] · = [ x, y ] ; (12) x · y = xy + kxqy − xyq − kqyx − kqxy and [ x, y ] · = [ x, y ] ; (13) x · y = xy + kxqy + kyqx − xyq − kqyx and [ x, y ] · = [ x, y ] ; (14) x · y = xy + kxqy − ( k + 1) qyx − ( k + 1) qxy and [ x, y ] · = [ x, y ] ,where x , y ∈ ( A, q ) , and [ x, y ] i is the i -th square product. We begin this section by recalling the definition of a left symmetric algebra from[1].
Definition 7.1 A left symmetric algebra A is a vector space equipped witha dot operation · : A ⊗ A → A which satisfy the following identity: x · ( y · z ) − ( x · y ) · z = y · ( x · z ) − ( y · x ) · z for x , y , z ∈ A . (51)We use ( A, · ) to denote a left symmetric algebra A equipped with a dotoperation · . It is easy to check that the square bracket( x, y ) [ x, y ] · := x · y − y · x for x , y ∈ A (52)satisfies the Jacobi identity; that is, a left symmetric algebra is a Lie-admissiblealgebra. The square bracket [ x, y ] · defined by (52) is called the accompanyingsquare product . The following proposition gives 14 ways of introducing aleft symmetric algebra structure in an invariant algebra and shows that eachcorresponding accompanying square product only differs from a square productin Definition 3.1 by a scalar. 8 roposition 7.1 Let ( A, q ) be an invariant algebra over a field k . Let k bea fixed scalar in the field k . (( A, q ) , · i ) is a left symmetric algebra if the dotoperation · i is chosen in one of the following ways: (1) x · y = kxy − kxqy + yqx , − [ x, y ] · = [ x, y ] for k = 0 and k [ x, y ] · = [ x, y ] for k = 0 ; (2) x · y = kxy + (1 − k ) xqy + yqx − qxy , − [ x, y ] · = [ x, y ] for k = 0 and k [ x, y ] · = [ x, y ] for k = 0 ; (3) x · y = xy + yqx − qxy and [ x, y ] · = [ x, y ] ; (4) x · y = kxy − kxqy + yqx − qyx − qxy , − [ x, y ] · = [ x, y ] for k = 0 and k [ x, y ] · = [ x, y ] for k = 0 ; (5) x · y = kyx + (1 − k ) yqx − qyx − qxy , − [ x, y ] · = [ x, y ] for k = 0 and k [ x, y ] · = [ x, y ] for k = 0 ; (6) x · y = kyx + yqx − qyx − qxy − kyxq , − [ x, y ] · = [ x, y ] for k = 0 and − k [ x, y ] · = [ x, y ] for k = 0 ; (7) x · y = kyx + xqy + yqx − qxy − kyxq , − [ x, y ] · = [ x, y ] for k = 0 and − k [ x, y ] · = [ x, y ] for k = 0 ; (8) x · y = kxy + yqx − kxyq − qyx − qxy , − [ x, y ] · = [ x, y ] for k = 0 and k [ x, y ] · = [ x, y ] for k = 0 ; (9) x · y = kxy + xqy + yqx − kxyq − qxy , − [ x, y ] · = [ x, y ] for k = 0 and k [ x, y ] · = [ x, y ] for k = 0 ; (10) x · y = yx + xqy − xyq + kqxy − yxq and [ x, y ] · = [ x, y ] ; (11) x · y = yx + kxqy − xyq − yxq and − [ x, y ] · = [ x, y ] ; (12) x · y = yx + kxqy + ( k − yqx − xyq − ( k − qxy − yxq and − [ x, y ] · =[ x, y ] ; (13) x · y = yx + xqy − xyq + kqyx − yxq and − [ x, y ] · = [ x, y ] ; (14) x · y = yx + xqy + kyqx − xyq − kqyx − kqxy − yxq and − [ x, y ] · = [ x, y ] ,where x , y ∈ ( A, q ) , and [ x, y ] i is the i -th square product. Representation ( c ,...,c ) of Algebras Let ( A , ⋆ ) be an algebra (not necessarily associative algebra) over k with aproduct ⋆ . Based on invariant algebras, we introduce a representation ( c ,...,c ) of an algebra ( A , ⋆ ) in the following Definition 8.1
Let W be a subspace of a vector space V over a field k and let q be a W -idempotent. A linear map ϕ from an algebra ( A , ⋆ ) to the invariantalgebra ( End ( V ) , q ) is called a representation ( c ,...,c ) of ( A , ⋆ ) induced by ( q, W ) if there exist the scalars c , . . . , c ∈ k such that ϕ ( x ⋆ y ) = c ϕ ( x ) ϕ ( y ) + c ϕ ( y ) ϕ ( x ) + c qϕ ( x ) ϕ ( y ) + c qϕ ( y ) ϕ ( x ) ++ c ϕ ( x ) qϕ ( y ) + c ϕ ( y ) qϕ ( x ) + c ϕ ( x ) ϕ ( y ) q + c ϕ ( y ) ϕ ( x ) q (53) for all x , y ∈ A . The language of modules ( c ,...,c ) is more convenient to state some facts aboutrepresentations ( c ,...,c ) . We now introduce a module ( c ,...,c ) over an algebra inthe following Definition 8.2
Let W be a subspace of a vector space V over a field k and let q be a W -idempotent. V is called a module ( c ,...,c ) over an algebra ( A , ⋆ ) ora A -module ( c ,...,c ) induced by ( q, W ) if there is a map: ( x, v ) x · v from A × V to V such that ( ax + by ) · v = a ( x · v ) + b ( y · v ) , (54) x · ( av + bu ) = a ( x · v ) + b ( x · u ) , (55)( x ⋆ y ) · v = c x · y · v + c y · x · v + c q · x · y · v + c q · y · x · v ++ c x · q · y · v + c y · q · x · v + c x · y · q · v + c y · x · q · v, (56) x · W := { x · w | w ∈ W } ⊆ W, (57) where x , y ∈ A , v , u ∈ V , a , b ∈ k and q · v := q ( v ) . A module ( c ,...,c ) over an algebra ( A , ⋆ ) induced by ( q, W ) is also denotedby V ( q,W ) . A subspace U of an A -module ( c ,...,c ) V = V ( q,W ) is called a submodule ( c ,...,c ) of V if x · u ∈ U for x ∈ A and u ∈ U (58)and q · u − u ∈ U for u ∈ U . (59)Every A -module ( c ,...,c ) V = V ( q,W ) has at least three submodules ( c ,...,c ) :0, W and V . An A -module ( c ,...,c ) with c = . . . = c = 0 is called an A -module ( c c ). 10 roposition 8.1 If V ( q,W ) is module ( c ,...,c ) over an algebra ( A , ⋆ ) , then thesubspace W is an A -module ( c c ) by restriction and the quotient space VW is a A -module ( c c c c c c c c ) under the following action: x · ( v + W ) := x · v + W for x ∈ A and v ∈ V . (60)We now define the building blocks for modules ( c ,...,c ) over an algebra. Definition 8.3
Let V ( q,W ) be a module ( c ,...,c ) over an algebra ( A , ⋆ ) . (i) V ( q,W ) is said to be -irreducible if V ( q,W ) = 0 and V ( q,W ) has no submodules ( c ,...,c ) which are not equal to and V ( q,W ) . (ii) V ( q,W ) is said to be -irreducible if W = 0 , V ( q,W ) = W and V ( q,W ) hasno submodules ( c ,...,c ) which are not equal to , W and V ( q,W ) . The next proposition gives the basic properties of the building blocks.
Proposition 8.2
Let V = V ( q,W ) be a module ( c ,...,c ) over an algebra ( A , ⋆ ) . (i) If V ( q,W ) is -irreducible, then either V = W in which case q = 0 on V and V is a A -module ( c c ) , or W = 0 in which case q = 1 on V and V is a A -module ( c c c c c c c c ) . (ii) If V ( q,W ) is -irreducible, then both the A -module ( c c ) W and the A -module ( c c c c c c c c ) VW are -irreducible. Definition 8.4
Let ( V ) ( q ,W ) and ( V ) ( q ,W ) be two modules ( c ,...,c ) over analgebra ( A , ⋆ ) . (i) A linear map ϕ : V → V is called a homomorphism ( c ,...,c ) if ϕ ( x · v ) = x · ϕ ( v ) for x ∈ L and v ∈ V (61) and ϕ ( q · v ) = q · ϕ ( v ) for v ∈ V . (62) (ii) If a homomorphism ( c ,...,c ) ϕ : V → V is bijective, then we say that ϕ isan isomorphism ( c ,...,c ) and V is isomorphic to V . Let V = V ( q,W ) be a module ( c ,...,c ) over an algebra ( A , ⋆ ) induced by ( q, W ).If U is a submodule ( c ,...,c ) of V , then q | U is a ( W ∩ U )-idempotent and ¯ q : VU → VU is a U + WU -idempotent, where ¯ q is defined by¯ q · ( v + U ) = ¯ q · v + U for v ∈ V . (63)11ence, U q | U,W ∩ U is an A -module ( c ,...,c ) induced by ( q | U, W ∩ U ), and thequotient space VU is also a A -module ( c ,...,c ) under the module ( c ,...,c ) actionconsisting of (63) and the following x · ( v + U ) := x · v + U for x ∈ A and v ∈ V . (64)The A -module ( c ,...,c ) (cid:18) VU (cid:19) ( ¯ q, U + WU ) is called the quotient A -module ( c ,...,c ) with respect to submodule ( c ,...,c ) U . Proposition 8.3
Let ( V ) ( q ,W ) and ( V ) ( q ,W ) be two modules ( c ,...,c ) overan algebra ( A , ⋆ ) , and suppose that ϕ : V → V is a homomorphism ( c ,...,c ) . (i) ϕ ( W ) ⊆ W . (ii) The kernel
Kerϕ := { v ∈ V | ϕ ( v ) = 0 } is a submodule ( c ,...,c ) of A -module ( c ,...,c ) V . (iii) The image
Imϕ := { ϕ ( v ) | v ∈ V } is a submodule ( c ,...,c ) of A -module ( c ,...,c ) V . (iv) The map ¯ ϕ : V Kerϕ → Imϕ defined by ϕ ( v + Kerϕ ) := ϕ ( v ) for v ∈ V (65) is an isomorphism ( c ,...,c ) from the quotient A -module ( c ,...,c ) (cid:18) V Kerϕ (cid:19) induced by (cid:16) ¯ q, Kerϕ + W Kerϕ (cid:17) to the submodule ( c ,...,c ) U ( q | U,W ∩ U ) . References [1] Dietrich Burde,
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