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GradedLieAlgebras :: Homomorphisms and derivations

Homomorphisms and derivations

A Lie algebra homomorphism $M \ \to\ L$ is defined using map(LieAlgebra,LieAlgebra,List) by giving the values in $L$ of the generators of $M$. A homomorphism preserves weight and sign, and M#Field must be the same as L#Field.

i1 : M = lieAlgebra({x,y},Weights => {2,2})

o1 = M

o1 : LieAlgebra
i2 : L = lieAlgebra({a,b},Signs => 1)

o2 = L

o2 : LieAlgebra
i3 : f1 = map(L,M,{a a,b b})

o3 = f1

o3 : LieAlgebraMap
i4 : describe f1

o4 = x => (a a)
     y => (b b)
     source => M
     target => L

Like the situation for ring maps, the meaning of map(L,M) is that a generator in $M$ is sent to the generator in $L$ with the same name, weight and sign if there is such a generator, otherwise it is sent to zero.

i5 : M = lieAlgebra{a,b,c}

o5 = M

o5 : LieAlgebra
i6 : L = lieAlgebra({a,b,d},Weights => {2,1,1})

o6 = L

o6 : LieAlgebra
i7 : f2 = map(L,M)

o7 = f2

o7 : LieAlgebraMap
i8 : describe f2

o8 = a => 0
     b => b
     c => 0
     source => M
     target => L

Another similarity with ring maps is that a map $M \ \to\ L$ need not be well defined, in the sense that that the relations in $M$ need not be sent to zero in $L$. It may also happen that the map does not commute with the differentials in $M$ and $L$. All this can be checked up to a certain degree using isWellDefined(ZZ,LieAlgebraMap). If $M$ is finitely presented, see Quotient Lie algebras and subspaces, then it is possible to get the information that the map is well defined and commutes with the differentials for all degrees, if the first input $n$ in isWellDefined(n,f) is big enough.

i9 : F=lieAlgebra({a,b},Weights => {{1,0},{2,1}},Signs => 1,
         LastWeightHomological => true)

o9 = F

o9 : LieAlgebra
i10 : D=differentialLieAlgebra{0_F,a a}

o10 = D

o10 : LieAlgebra
i11 : f=map(D,F)
warning: the map might not be well defined, 
           use isWellDefined

o11 = f

o11 : LieAlgebraMap
i12 : isWellDefined(2,f)
the map is well defined for all degrees
the map does not commute with the differential

o12 = false
i13 : use F
i14 : Q=F/{a a}

o14 = Q

o14 : LieAlgebra
i15 : g=map(Q,D)
warning: the map might not be well defined, 
           use isWellDefined

o15 = g

o15 : LieAlgebraMap
i16 : isWellDefined(2,g)
the map is well defined for all degrees
the map commutes with the differential for all degrees

o16 = true

Surjectivity for a Lie algebra map may be tested using isSurjective(LieAlgebraMap). The input map might not be well defined. The method function isIsomorphism(LieAlgebraMap) may be used to test if a Lie algebra map $f: M \ \to\ L$ is an isomorphism. Here $M$ and $L$ must be equal, but not necessarily identical. Also, $M$ must be finitely presented. It is tested that the map is well defined, commutes with the differentials and is surjective. Injectivity follows from this by dimension reasons. See Holonomy Lie algebras and symmetries for applications where the map is a permutation of the variables.

i17 : isSurjective f

o17 = true
i18 : use F
i19 : Q1=F/{a a}

o19 = Q1

o19 : LieAlgebra
i20 : Q1===Q

o20 = false
i21 : Q1==Q

o21 = true
i22 : h=map(Q1,Q)

o22 = h

o22 : LieAlgebraMap
i23 : isIsomorphism h

o23 = true

A derivation $d: M \ \to\ L$ is defined using lieDerivation by giving a Lie algebra map $f: M \ \to\ L$ and a list of elements in $L$ that are the values of $d$ on the generators of $M$. One may use isWellDefined(ZZ,LieDerivation) to test if a derivation is well defined, which means that the relations in $M$ are sent to zero (the derivation need not commute with the differentials).

i24 : use Q
i25 : d=lieDerivation(g,{a b,b b})

o25 = d

o25 : LieDerivation
i26 : isWellDefined(2,d)
the derivation is well defined for all degrees

o26 = true
i27 : use D
i28 : f=map(D,F)
warning: the map might not be well defined, 
           use isWellDefined

o28 = f

o28 : LieAlgebraMap
i29 : d=lieDerivation(f,{a b,b b})
warning: the derivation might not be well defined, use isWellDefined

o29 = d

o29 : LieDerivation
i30 : isWellDefined(2,d)
the map defining the 
            derivation is not well defined

o30 = false

Omitting the first input in lieDerivation gives derivations $d: L \ \to\ L$ with the identity map on $L$ as the defining map.

The following example shows a way to determine the derivations of a Lie algebra studied by David Anick, which may be seen as the positive part of the twisted loop algebra on sl_2. This also explains the periodic behaviour of the Lie algebra.

i31 : L = lieAlgebra{a,b}/{a a a b,b b b a}

o31 = L

o31 : LieAlgebra
i32 : dims(1,20,L)

o32 = {2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1}

o32 : List

The space of derivations of degree 0 is 2-dimensional, and contains the Euler derivation, see euler(LieAlgebra), which is the identity in degree 1.

i33 : deuler = euler L

o33 = deuler

o33 : LieDerivation
i34 : deuler b a b a b a b a

o34 = 8 (b a b a b a b a)

o34 : L

We will now prove that the space of derivations of degree 6 is 2-dimensional. The space of linear maps from degree 1 to degree 7 is 4-dimensional. Not all of them define derivations.

i35 : basis(7,L)

o35 = {(a b a b a b a), (b b a b a b a)}

o35 : List
i36 : da61 = lieDerivation{a b a b a b a,0_L}
warning: the derivation might not be well defined, use isWellDefined

o36 = da61

o36 : LieDerivation
i37 : isWellDefined(4,da61)
the derivation is well defined for all degrees

o37 = true
i38 : db61 = lieDerivation{0_L,a b a b a b a}
warning: the derivation might not be well defined, use isWellDefined

o38 = db61

o38 : LieDerivation
i39 : isWellDefined(4,db61)

o39 = false
i40 : da62 = lieDerivation{b b a b a b a,0_L}
warning: the derivation might not be well defined, use isWellDefined

o40 = da62

o40 : LieDerivation
i41 : isWellDefined(4,da62)

o41 = false
i42 : db62 = lieDerivation{0_L,b b a b a b a}
warning: the derivation might not be well defined, use isWellDefined

o42 = db62

o42 : LieDerivation
i43 : isWellDefined(4,db62)
the derivation is well defined for all degrees

o43 = true

The output displayed above shows that da61 and db62 are derivations. To determine whether a linear combination of db61 and da62 is well defined (i.e., maps the relations in $L$ to zero), we consider derivations from the free Lie algebra $M$ on $a,b$ to $L$.

i44 : M = lieAlgebra{a,b}

o44 = M

o44 : LieAlgebra
i45 : f = map(L,M)

o45 = f

o45 : LieAlgebraMap
i46 : use L
i47 : dMb61 = lieDerivation(f,{0_L,a b a b a b a})

o47 = dMb61

o47 : LieDerivation
i48 : dMa62 = lieDerivation(f,{b b a b a b a,0_L})

o48 = dMa62

o48 : LieDerivation
i49 : use M
i50 : dMb61 a a a b

o50 = 0

o50 : L
i51 : dMa62 a a a b

o51 = 2 (b a b a b a b a b a)

o51 : L

It follows from the output displayed above that the only linear combination of dMb61 and dMa62 that is zero on (a a a b) is a multiple of dMb61, but we have seen that dMb61 is not a derivation on $L$. Hence, the space of derivations of degree 6 is 2-dimensional. Also, da61 + db62 is the inner derivation corresponding to right multiplication with the basis element of degree 6, (b a b a b a). This is seen by using innerDerivation.

i52 : use L
i53 : da61+db62===innerDerivation(b a b a b a)

o53 = true

Since the dimension of the Lie algebra in degree 8 is 1, the dimension of the space of derivations of degree 7 is at most 2.

i54 : da7=lieDerivation({b a b a b a b a,0_L})
warning: the derivation might not be well defined, use isWellDefined

o54 = da7

o54 : LieDerivation
i55 : isWellDefined(4,da7)
the derivation is well defined for all degrees

o55 = true
i56 : db7=lieDerivation({0_L,b a b a b a b a})
warning: the derivation might not be well defined, use isWellDefined

o56 = db7

o56 : LieDerivation
i57 : isWellDefined(4,db7)
the derivation is well defined for all degrees

o57 = true
i58 : da7===innerDerivation(b b a b a b a)

o58 = true
i59 : db7===innerDerivation(a b a b a b a)

o59 = true

It follows from the output displayed above that the space of derivations of degree 7 is also 2-dimensional, but consists only of inner derivations. The conclusion is that the space of derivations of $L$ of positive degree modulo the inner derivations is 1-dimensional in all even degrees, and 0 in all odd degrees. We may also use LieDerivation LieDerivation to examine the structure of this quotient Lie algebra.

i60 : d2 = lieDerivation({a b a,0_L})
warning: the derivation might not be well defined, use isWellDefined

o60 = d2

o60 : LieDerivation
i61 : d4 = lieDerivation({a b a b a,0_L})
warning: the derivation might not be well defined, use isWellDefined

o61 = d4

o61 : LieDerivation
i62 : describe d2 d4

o62 = a => (a b a b a b a)
      b => 0
      map => id_L
      sign => 0
      weight => {6, 0}
      source => L
      target => L

Define $dn$ ($n\ \ge\ 2$, $n$ even) as the derivation which maps $a$ to (a b a b ... a) of length $n+1$ and $b$ to 0. It follows from the output displayed above that [ $d2$, $d4$ ] = $d6$.

i63 : d6 = lieDerivation({a b a b a b a,0_L})
warning: the derivation might not be well defined, use isWellDefined

o63 = da61

o63 : LieDerivation
i64 : describe d2 d6

o64 = a => 2 (a b a b a b a b a)
      b => 0
      map => id_L
      sign => 0
      weight => {8, 0}
      source => L
      target => L
i65 : d16 = lieDerivation({a b a b a b a b a b a b a b a b a,0_L})
warning: the derivation might not be well defined, use isWellDefined

o65 = d16

o65 : LieDerivation
i66 : describe d2 d16

o66 = a => 7 (a b a b a b a b a b a b a b a b a b a)
      b => 0
      map => id_L
      sign => 0
      weight => {18, 0}
      source => L
      target => L

It follows from the output displayed above that [ $d2$, $d6$ ] = $2d8$ and [ $d2$, $d16$ ] = $7d18$. In fact, this Lie algebra is the infinite dimensional filiform Lie algebra, which is the Witt algebra in positive degrees (with a degree doubling).

See also