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Abstract
For any decomposition of a Lie superalgebra into a direct sum
of a subalgebra
and a subspace
without any further resctrictions on
and
we construct a nonlinear realization of
on
The result generalizes a theorem by Kantor from Lie algebras to Lie superalgebras. When
is a differential graded Lie algebra, we show that it gives a construction of an associated
-algebra.
1. Introduction
Representations of Lie algebras can be generalized to nonlinear realizations. This means that the elements in the Lie algebra are mapped to operators that are not necessarily linear, but constant, quadratic or of higher order. In many important applications, the operators act on a vector space which can be identified with a subspace of the Lie algebra itself, complementary to a subalgebra. One example is the conformal realization of the Lie algebra on a D-dimensional vector space, based on the decomposition of
as a 3-graded Lie algebra
where
and
are D-dimensional subspaces. In this realization, the subalgebra
acts linearly, whereas the two D-dimensional subspaces can be considered as consisting of constant and quadratic operators, respectively. In other examples, the linearly realized subalgebra is not the degree-zero subalgebra in a
-grading, but defined by being pointwise invariant under an involution.
For any decomposition of a Lie algebra G into a direct sum of a subalgebra H and a complementary subspace E, there is formula for a nonlinear realization of G on E given by Kantor [Citation10]. The conformal realization of a semisimple Lie algebra with a 3-grading
is obtained from this formula in the special case where
and
The corresponding application to a semisimple Lie algebra with a 5-grading
leads to a quasiconformal realization if the subspaces
are one-dimensional [Citation7, Citation17]. In these cases the subspace E is actually also a subalgebra, but this need not be the case in the general formula. There are no restrictions on
nor on
the only requirement is
In this paper, we generalize Kantor’s formula from Lie algebras to Lie superalgebras. Also in the restriction to Lie algebras, our proof is very different from Kantor’s, being purely algebraic, without references to homogeneous spaces for Lie groups.
We expect our generalization to be useful in applications to physics, in particular to models where a Lie superalgebra can be used to organize the field content or to encode the gauge structure. In such cases it might be interesting to investigate whether the Lie superalgebra also can be realized as a symmetry. We also expect the result to be relevant for applications of other related structures, such as Leibniz algebras, differential graded Lie algebras and -algebras, for which a renewed interest has appeared recently in the context of gauge theories, see for example Refs. [Citation2–4, Citation11, Citation14–16]. In fact, our framework illuminates the relations between these structures. In particular, our main result leads to the construction of an
-algebra associated to any differential graded Lie algebra.
The paper is organized as follows.
We start in Section 2 with an arbitrary vector space U1. We associate a
-graded Lie algebra U to it, from which we in turn construct the Lie algebra S of symmetric operators on U1. The
-graded Lie algebra U associated to a vector space U1 was introduced in Ref. [Citation9], but here we use a different recursive approach, following Refs. [Citation17, Citation18].
In Section 3, we modify the construction: we then start with a vector space
that is equipped with a
-grading, to which we associate a
-graded Lie superalgebra
[Citation18]. From
we construct the Lie superalgebra
of symmetric operators on
(where the symmetry is now actually a
-graded symmetry).
In Section 4, we furthermore assume that the vector space
itself is a Lie superalgebra
This means that it is equipped with a Lie superbracket, consistent with the
-grading already present in Section 3. We show that it extends to a Lie superbracket on
(different from the one defined in Section 3).
In Section 5, we still assume
but also that this Lie superalgebra decomposes into a direct sum
where
is a subalgebra. We show that it extends to a corresponding direct sum
As our main result, Theorem 5.4, we show that there is a Lie superalgebra homomorphism from
to
This result generalizes the main theorem in Ref. [Citation10] from Lie algebras to Lie superalgebras.
In Section 6, we assume that
itself has a
-grading consistent with the
-grading, and is equipped with a differential, turning it into a differential graded Lie algebra. As an example of an application of our main result, we use it in order to construct an
-algebra from
and show that the brackets agree with those given explictly in Ref. [Citation6].
2. The ![](//:0)
-graded Lie algebra associated to a vector space
We start with an arbitrary vector space U1 over some field of characteristic zero, from which we define vector spaces recursively by
(2.1)
(2.1)
for
Thus
consists of all linear maps from U1 to
and in particular
Let for some
and let
Then
and if
this means that
is an element in
Continuing in this way, we finally find that
is an element in U1, which we may also write as
Thus we have a vector space isomorphism
(2.2)
(2.2)
and we may consider elements in
not only as linear maps from U1 to
but also as linear maps from
to U1, or as p-linear operators on U1. We will refer to elements in
simply as operators of order p, even for p = 0, so that the elements in U1 are considered as operators of order zero.
2.1. The Lie algebra ![](//:0)
![](//:0)
Next we let be the direct sum of the vector spaces defined in the previous section,
For any
(where
) and any
we write
(2.3)
(2.3)
We then define a map
(2.4)
(2.4)
for any
recursively by
(2.5)
(2.5)
and extend it to a bilinear operation on
by linearity. For
this is the usual composition of (linear) maps,
(2.6)
(2.6)
where the last equality follows from Equation(2.3)
(2.3)
(2.3) since
and
are elements in U1. We give two more examples,
(2.7)
(2.7)
(2.8)
(2.8)
which are easily generalized to
(2.9)
(2.9)
(2.10)
(2.10)
In these examples, the subscripts of the operators indicate their orders (whereas the subscripts on elements x in U1 are just labels used to distinguish them from each other).
The property means that
is a
-graded algebra with respect to
(with vanishing subspaces corresponding to positive integers). The following proposition says that this algebra furthermore is associative.
Proposition 2.1.
The vector space together with the bilinear operation
is an associative algebra.
Proof.
We will show that
(2.11)
(2.11)
for any triple of operators
of order
respectively, and any
We do this by induction over
When
we have
and the assertion follows by Equation(2.6)
(2.6)
(2.6) . Suppose now that it holds when
for some
and set
We then have (omitting the subscripts)
(2.12)
(2.12)
using the induction hypothesis in the third step, and the proposition follows by the principle of induction. □
Note that the identity Equation(2.11)(2.11)
(2.11) is not satisfied when r = 0 and
Then (omitting the subscripts and setting
) we instead have the (right) Leibniz identity
(2.13)
(2.13)
For any
we now set
(2.14)
(2.14)
and we have the following obvious consequence of Proposition 2.1.
Corollary 2.2.
The vector space together with the bracket
is a Lie algebra.
2.2. Extending ![](//:0)
to U
Let be the free Lie algebra generated by the vector space U1 (with the natural
-grading) and set
(2.15)
(2.15)
We will use the notation
(2.16)
(2.16)
for any
We use the same notation for the two Lie brackets on
and
respectively, and we will now unify them into one Lie bracket on the whole of U, the direct sum of these two vector spaces. We thus have to define brackets
for any
and
For
we set
(2.17)
(2.17)
If
then we may assume that
for some
We then define recursively
(2.18)
(2.18)
and extend the bracket by linearity to the case when u is a sum of such terms
In order to ensure that the definition is meaningful, we have to show that it respects the Jacobi identity in the sense that
(2.19)
(2.19)
for any
and
Indeed, we get
(2.20)
(2.20)
using Jacobi identities like
(2.21)
(2.21)
Such Jacobi identities follow either (if
) by the fact that
is a Lie algebra or (if
) by the definition Equation(2.18)
(2.18)
(2.18) .
Proposition 2.3.
The vector space together with the bracket
is a Lie algebra.
Proof.
The Jacobi identities with either all three elements in or all three elements in
are satisfied, by Corollary 2.2 and by the construction of
as a free Lie algebra. Also the Jacobi identities with one element in
and two elements in
are satisfied, by the definition Equation(2.18)
(2.18)
(2.18) . It only remains to check the Jacobi identities with two elements
and one element
Assuming that u is homogeneous with respect to the
-grading,
this can be done by induction over
For k = 1, we have
(2.22)
(2.22)
For
we may (as above), assume that
where
Assuming furthermore (as the induction hypothesis) that
(2.23)
(2.23)
we get
(2.24)
(2.24)
and it follows by induction that also these Jacobi identities are satisfied. □
2.3. The Lie algebra S of symmetric operators
For let
be the subspace of
consisting of elements
such that
for all
and set
(2.25)
(2.25)
Because of the
-grading, if
then
consists of all operators Ap such that
for all
whereas
and S1 = U1. Furthermore,
is the idealizer (or normaliser) of
in U, and S can be identified with the quotient space obtained by factoring out
from its idealizer in U.
We will refer to elements in as symmetric operators of order p, and a linear combination of symmetric operators will also be called a symmetric operator, even if it is not homogenous with respect to the
-grading. Note that we consider all elements in U0 as symmetric operators of order one, and even all elements in U1 as symmetric operators of order zero.
It follows easily by the Jacobi identity that if A is a symmetric operator of order one or higher, then is a symmetric operator as well, for any
The operators in U of order two or higher included in S are indeed precisely those that are symmetric in the following sense. If and
(so that
), then
(2.26)
(2.26)
so that
It is straightforward to show that generally, the condition
for all
is equivalent to the condition that
(2.27)
(2.27)
where
is any permutation of
We write this (as usual) as
(2.28)
(2.28)
where the right hand side denotes
times the sum of
over all permutations
of
For any symmetric operator Ap there is a unique corresponding map (non-linear if
) given by
In order to characterize a symmetric operator Ap it is thus sufficient to set
in
In particular for symmetric operators, it is convenient to replace the bilinear operation on
by another one, which differs from
by normalization. We define a bilinear operation
on
by
(2.29)
(2.29)
for
and
If Ap and Bq are symmetric operators, we then get
(2.30)
(2.30)
Since
is an operator of order
the linear map
given by
(2.31)
(2.31)
satisfies
(2.32)
(2.32)
for any two operators A and B and thus the two algebras obtained by equipping the vector space
with
and
respectively, are isomorphic to each other.
We now extend the bilinear operation from
to
First we set
(2.33)
(2.33)
for
(where
and
Thus the definition Equation(2.29)
(2.29)
(2.29) is still valid if we allow one of Ap and Bq to be an operator of order zero, that is, an element in U1. For example, we have
(2.34)
(2.34)
whereas
(2.35)
(2.35)
Second, we set
(2.36)
(2.36)
for
in order to close
under
This makes the operation
really different from
(not only up to normalization), since we kept
undefined.
For any we set
(2.37)
(2.37)
It follows that the vector space S equipped with this bracket is a Lie algebra isomorphic to the quotient algebra obtained by factoring out
from the idealizer of
in U. Moreover, if U1 is n-dimensional, it is straightforward to show that S is isomorphic to the Lie algebra Wn of formal vector fields
where fi are formal power series in n variables
3. Generalization from Lie algebras to Lie superalgebras
We will now repeat the steps in the preceding section in a more general case. Instead of starting with an arbitrary vector space U1 we now start with an arbitrary -graded vector space
Thus
can be decomposed into a direct sum
of two subspaces
and
Like for any
-graded vector space, these subspaces (and their elements) are said to be even and odd, respectively. This leads to a corresponding decomposition
(3.1)
(3.1)
of each vector space
by refining Equation(2.1)
(2.1)
(2.1) to
(3.2)
(3.2)
Now, let
be the free Lie superalgebra generated by
(with the natural
-grading) and set
(3.3)
(3.3)
We thus have a
-graded vector space
If
for i = 0, 1, we use the notation
for the
-degree of u.
We can now repeat the steps in the preceding section, carrying over notation and terminology in a straightforward way. The formulas will however differ from those in the preceding section by factors of powers of where we (without loss of generality) have to assume that the elements in
that appear are homogeneous with respect to the
-grading.
Thus, we equip with an associative bilinear operation
from which we define a Lie superbracket
on
In these definitions, we modify Equation(2.5)
(2.5)
(2.5) to
(3.4)
(3.4)
and Equation(2.14)
(2.14)
(2.14) to
(3.5)
(3.5)
When we then unify the brackets on
and
to one on the whole of
we keep the definition
in Equation(2.17)
(2.17)
(2.17) , but modify Equation(2.18)
(2.18)
(2.18) to
(3.6)
(3.6)
We do not give the proofs here, since they differ from those given in the preceding section only by factors of powers of
In fact, the modifications made here are actually generalizations, since the Lie superalgebra reduces to the original Lie algebra U in the special case where
has a trivial odd subspace,
Thus, starting with a vector space
we can decompose it in different ways into a direct sum of an even and an odd subspace, which lead to different associated
-graded Lie superalgebras
The decomposition where
is considered as an even vector space (coinciding with its even subspace) leads to the associated
-graded Lie algebra described in the preceding section. But we can also consider it as an odd vector space. Only in this case the
-grading of the Lie superalgebra is consistent, which means that
if i is even and
if i is odd.
The Lie superalgebra constructed from
in the same way as S is constructed from U, now consists of operators with a
-graded symmetry, rather than purely symmetric ones. However, for simplicity we will still refer to them as symmetric operators. Generalizing the notation Equation(2.28)
(2.28)
(2.28) , we denote
-graded symmetry with angle brackets rather than ordinary parentheses, so that
(3.7)
(3.7)
if
where the right hand side denotes
times the sum of
over all permutations
of
where ε is the number of transpositions of two odd elements.
3.1. Leibniz algebras
In the next section we will assume that the -graded vector space
is a Lie superalgebra. Before that, we will briefly give another example of a case where
is an algebra. In many such cases, identities for elements in this algebra can be reformulated as identities for elements in the associated
-graded Lie superalgebra
including the bilinear operation of the algebra as an element in
A (left) Leibniz algebra is an algebra where the bilinear operation
satisfies the (left) Leibniz identity
(3.8)
(3.8)
If we now consider
as a
-graded vector space with trivial even subspace and let Θ be the element in
associated to
by
(3.9)
(3.9)
then the Leibniz identity Equation(3.9)
(3.9)
(3.9) is equivalent to the condition
(3.10)
(3.10)
(Since
is odd,
is odd as well, and the condition
is not trivially satisfied, but equivalent to
). Indeed, by the Jacobi identity (keeping in mind that
are all odd),
(3.11)
(3.11)
Now let
be the one-dimensional subspace of
spanned by Θ. Since
the subspace
of
is a subalgebra. This Lie superalgebra can also be considered as a differential graded Lie algebra
with a differential
Thus any Leibniz algebra gives rise to a differential graded Lie algebra [Citation2, Citation11, Citation14]. In Section 6 we will see how in turn any differential graded Lie algebra gives rise to an
-algebra.
4. The case when ![](//:0)
is a Lie superalgebra ![](//:0)
![](//:0)
We now assume not only that is a
-graded vector space, but furthermore that
is a Lie superalgebra
with a bracket
We extend the bracket to the whole of
recursively by
(4.1)
(4.1)
We recall that any operator Ap of order p is defined by its action on
and that
If B = y is an element in
that is, an operator of order zero, then
so that
(4.2)
(4.2)
Proposition 4.1.
The -graded vector space
together with the bracket
is a Lie superalgebra.
Proof.
We will show that the Jacobi identity
(4.3)
(4.3)
is satisfied for any triple of operators A, B, C of order p, q, r, respectively, by induction over
When
we have
and the Jacobi identity is satisfied since
is a Lie superalgebra. If we assume that it is satisfied when
for some
and set
we then get
(4.4)
(4.4)
using the induction hypothesis in the second and third steps, and the proposition follows by the principle of induction. □
It is furthermore easy to see that this Lie algebra is -graded, but the
-grading is different from the one that is respected by
(on the subspace
). We have
(4.5)
(4.5)
so the relevant
-degree of an operator is just (the negative of) its order.
We will now show that the subspace of the Lie superalgebra
closes under the bracket Equation(4.1)
(4.1)
(4.1) and thus form a subalgebra.
Proposition 4.2.
If , then
as well.
Proof.
Since all operators of order zero or one are included in and because of the
-grading Equation(4.5)
(4.5)
(4.5) , we can assume that both A and B are of order one or higher, so that
is of order two or higher.
We have to show that for any
We first show this for
and in particular when
for
Thus we have to show that
(4.6)
(4.6)
under the assumption that
(4.7)
(4.7)
and
(4.8)
(4.8)
The first term in Equation(4.6)
(4.6)
(4.6) is equal to
(4.9)
(4.9)
Now the second and third term on the right hand side cancel the corresponding contributions from the second term in Equation(4.6)
(4.6)
(4.6) . Furthermore, the first and fourth term cancel the corresponding contributions from the second term in Equation(4.6)
(4.6)
(4.6) by Equation(4.7)
(4.7)
(4.7) and Equation(4.8)
(4.8)
(4.8) . When
for
we can assume
where
If we then assume that
(as induction hypothesis), we get
(4.10)
(4.10)
which is proportional to
(4.11)
(4.11)
Now, since A and B are symmetric,
and
are symmetric as well, and the proposition can be proven by induction. □
For and
considering the operator
as a linear map
we have
(4.12)
(4.12)
The next proposition says that the identity Equation(4.1)
(4.1)
(4.1) can be generalized in the sense that
can be replaced by any
We omit the proof since the steps are the same as in Proposition 4.1.
Proposition 4.3.
For any , we have
(4.13)
(4.13)
4.1. Multiple brackets involving the identity map
The identity map on is an even symmetric operator of order one. We denote it simply by 1, so that
and
(4.14)
(4.14)
We now generalize this notation and, for any integer
write
(4.15)
(4.15)
where the identity map 1 appears k times on the right hand side. We will furthermore from now on use multibrackets to denote nested brackets (for any elements in any Lie superalgebra) and write Equation(4.15)
(4.15)
(4.15) as
(4.16)
(4.16)
Note that
Proposition 4.4.
If , then
as well.
Proof.
By induction, using that it suffices to show this in the case when k = 1, which is a special case of Proposition 4.2. □
For considering
as linear map
we have
(4.17)
(4.17)
In calculations with multiple brackets involving the identity map, we will need the rules in the next proposition. They are more or less obvious when reformulated in the notation Equation(4.17)
(4.17)
(4.17) and also straightforward to prove rigorously in the more compact notation that we have demonstrated here. However, since the calculations are rather lengthy, and we have already given similar proofs, we omit this one.
Proposition 4.5.
Let A and B be operators and an integer. Then we have
(4.18)
(4.18)
and
(4.19)
(4.19)
In the summations in Equation(4.19)
(4.19)
(4.19) , the summation variables i and j take all non-negative integer values (such that
), and we set
for any operator A. Also in all summations below, the summation variables are allowed to be zero, unless otherwise stated.
5. Main theorem
Suppose that the Lie superalgebra decomposes into a direct sum
of a subalgebra
and a subspace
For any
we write
where
and
Since
is a subalgebra, we thus have
(5.1)
(5.1)
and
for any
Then this decomposition of
extends to a decomposition of the Lie superalgebra
into a corresponding direct sum
where
is a subalgebra. For any
we define
and
recursively by
(5.2)
(5.2)
for any
It follows immediately that
and also that if
then
and
as well. We let
and
be the subspaces spanned by all
and
respectively, such that
We then have the following proposition, which can be proven in the same way as Proposition 4.1, by induction over the sum of the orders of A and B.
Proposition 5.1.
For any , we have
(5.3)
(5.3)
Thus the subspace
of the Lie superalgebra
is a subalgebra.
The next proposition says that the identity Equation(5.2)(5.2)
(5.2) can be generalized in the sense that
can be replaced by any
Again, it can be proven in the same way as Proposition 4.1, by induction over the order of B.
Proposition 5.2.
For any we have
(5.4)
(5.4)
We thus obtain the following generalization of Proposition 4.5 by projecting all outermost brackets on
Proposition 5.3.
Let A and B be operators and an integer. Then we have
(5.5)
(5.5)
and
(5.6)
(5.6)
Proof.
This follows directly from Propositions 4.5 and 5.2. □
In particular, when n = 1 we have the identity
(5.7)
(5.7)
which we will use below (in the case where A and B are operators of order zero, so that the second term vanishes).
For any and any integer
we define
recursively by
(5.8)
(5.8)
and
(5.9)
(5.9)
for
In particular, we have
(5.10)
(5.10)
and thus
(5.11)
(5.11)
For example,
(5.12)
(5.12)
For any
we also define
(5.13)
(5.13)
This is however not a recursive definition, since it is a(q), not
that appears in the second term, and
is replaced by
Note also that the bracket is not projected on
In fact,
is projected on
since
(5.14)
(5.14)
and
(5.15)
(5.15)
for
It follows that
(5.16)
(5.16)
for any a, b and
We are now ready to formulate and prove our main theorem.
Theorem 5.4.
The map
(5.17)
(5.17)
is a Lie superalgebra homomorphism.
Proof.
We will show that
(5.18)
(5.18)
for any
The left hand side is equal to
(5.19)
(5.19)
Thus it suffices to show that
(5.20)
(5.20)
for
We will do this by induction. When r = 0, the left hand side in Equation(5.20)
(5.20)
(5.20) equals
(5.21)
(5.21)
where we have used Equation(5.7)
(5.7)
(5.7) and Equation(5.10)
(5.10)
(5.10) , whereas the right hand side equals
Thus the right hand side minus the left hand side equals
(5.22)
(5.22)
In the induction step, we need to study
(5.23)
(5.23)
for some
and show that this the expression equals
(5.24)
(5.24)
under the assumption that
(5.25)
(5.25)
for
We will first study the first term in the summand on the right hand side of Equation(5.23)(5.23)
(5.23) , and a particular part of it. Its counterpart, the corresponding part of the second term in the summand, is then obtained by interchanging a and b, and multiplying with
In the summations in Equation(5.26)
(5.26)
(5.26) and Equation(5.27)
(5.27)
(5.27) below where the summation variables add up to m − 1, the sum should be read as zero if m = 0.
We have
(5.26)
(5.26)
The contribution to the sum Equation(5.23)
(5.23)
(5.23) from the last term in Equation(5.26)
(5.26)
(5.26) , and its counterpart, is
(5.27)
(5.27)
Taking all terms into account, we get
(5.28)
(5.28)
Here we have used Equation(5.26)
(5.26)
(5.26) in (a). In (b) we have used the definition of
and Equation(5.27)
(5.27)
(5.27) . The second and the fifth term on the right hand side of (a) go into the third term of the right hand side of (b), whereas the third and sixth term of (a) go into the fourth term of the right hand side of (b), by Equation(5.27)
(5.27)
(5.27) . In (c) we have used the definition Equation(5.13)
(5.13)
(5.13) of
and
and the induction hypothesis. In (d) we have used Equation(5.16)
(5.16)
(5.16) and in (e) we have used Proposition 5.3. The theorem now follows by the principle of induction. □
Considering as a linear map
we have
(5.29)
(5.29)
where the inner sum goes over all k-tuples of integers
such that
(5.30)
(5.30)
If
(and k > 0), the factor in the second and third line should be read as
(5.31)
(5.31)
If k = 0, the inner sum in Equation(5.29)
(5.29)
(5.29) should be read as
(5.32)
(5.32)
Here
but since
is a subspace of
we can as well assume
and consider
as a linear map
6. Getzler’s theorem
As an example of an application, we end this paper by proving a theorem which says that any differential graded Lie algebra (a Lie superalgebra with a consistent -grading and a differential) gives rise to an
-algebra (a generalization of a differential graded Lie algebra including also higher brackets [Citation12, Citation13]). Combined with the result described in Section 3.1 that any Leibniz algebra gives rise to a differential graded Lie algebra, it leads to the conclusion that any Leibniz algebra gives rise to an
-algebra [Citation11, Citation15, Citation16]. The theorem has already been proven in at least two different ways in the literature. It follows from the results in Ref. [Citation8] by Fiorenza and Manetti, and has been proven more directly in Ref. [Citation6] by Getzler. Here we follow Getzler’s formulation of the it, and prove it using our main result, Theorem 5.4.
Suppose that the Lie superalgebra has a consistent
-grading,
Then this
-grading induces a
-grading on each subspace
of
and thus a
-grading of
different from the one that
comes with by construction,
(6.1)
(6.1)
The two
-gradings form together a
-grading,
(6.2)
(6.2)
If there is an element
such that
then
together with Q constitutes an
-algebra. The element Q can then be decomposed as a sum of elements
for
each of which can be considered as a linear map
called a p- bracket. Following Ref. [Citation6], we use curly brackets for the p-brackets,
The condition
decomposes into infinitely many identities for these p-brackets, similar to the usual Jacobi identity.
We note that there are different conventions for -algebras. The fact that we consider the p-brackets as elements
means that we use the convention where they are graded symmetric and have degree −1.
Theorem 6.1.
[Citation6] Let be a differential graded Lie algebra with differential δ of degree –1 and bracket
. Let D be the linear operator on L which equals δ on L1 but vanishes on Li for
. Then the subspace
is an
-algebra with p-brackets given by
(6.3)
(6.3)
for p = 1 and
(6.4)
(6.4)
for
where
are the Bernoulli numbers (
In Ref. [Citation6] there appears to be a sign error that we have here corrected by inserting a minus sign on the right hand side of Equation(6.4)(6.4)
(6.4) [Citation15]. The occurrence of Bernoulli numbers in this context was observed in Ref. [Citation1], and it was also shown in Ref. [Citation5] that they similarly show up in extended geometry, encoding the gauge structure of generalized diffeomorphisms.
Proof.
Let be a one-dimensional vector space spanned by an element Θ and set
for
Then
(6.5)
(6.5)
is a consistently
-graded Lie superalgebra, where the bracket in the subalgebra
of L is extended by
and
for
Furthermore, set
and
Then
is a subalgebra of
and
We can thus use Theorem 5.4, which in particular says that the element
in
satisfies
since
in
Also, it follows by the construction of
and the
-grading that
since
and the identity map
If we write
it thus follows that
together with the element
in
(note that
) is an
-algebra, with the p-brackets
(6.6)
(6.6)
The right hand side here is given by Equation(5.29)
(5.29)
(5.29) with
that is
(6.7)
(6.7)
where the inner sum goes over all k-tuples of integers
such that
(6.8)
(6.8)
It remains to show that Equation(6.7)
(6.7)
(6.7) equals the expressions on the right hand side of Equation(6.3)
(6.3)
(6.3) and Equation(6.4)
(6.4)
(6.4) when p = 1 and
respectively. When p = 1, we indeed get
(6.9)
(6.9)
When
all terms in Equation(6.7)
(6.7)
(6.7) with
are zero, since
Furthermore, all the subscripts
but the first one can be removed, since
is a subalgebra in this case. Also, when
even the first subscript
can be removed. Thus, for
we have
(6.10)
(6.10)
where the inner sums go over all k-tuples of integers
such that
(6.11)
(6.11)
Since we have removed the subscripts
the factor in the second and third line of each term above is actually independent of the choice of
and also of the integer k. Setting
(6.12)
(6.12)
where the term with k = 0 should be read as
we thus get
(6.13)
(6.13)
Now Cp can also be written
(6.14)
(6.14)
where the inner sum goes over all k-tuples of positive integers
such that
Written this way, it is easily shown (by induction, using recursion formulas for the Bernoulli numbers) that
(6.15)
(6.15)
and we arrive at Equation(6.4)
(6.4)
(6.4) . □
Acknowledgments
I would like to thank Martin Cederwall, Sylvain Lavau and Arne Meurman for discussions. I am particularly grateful to Sylvain Lavau, who have also given many useful comments on the manuscript.
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