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# 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 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

- map(LieAlgebra,LieAlgebra,List) -- make a Lie algebra homomorphism
- lieDerivation -- make a graded derivation
- isWellDefined(ZZ,LieAlgebraMap) -- whether a Lie map is well defined
- isWellDefined(ZZ,LieDerivation) -- whether a Lie derivation is well defined
- Differential Lie algebra tutorial

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GradedLieAlgebras/hom.m2:236:0.