1. Introduction

In[4], the theory of Lie superalgebras was given and in[5], theory of Kac- Moody Lie superalgebras was described. Borcherds[2] initiated the study of generalized Kac-Moody algebras(GKM algebras). Wakimoto[19] introduced BKM superalgebras(BKM Lie superalgebras). The existence of special imaginary roots for Kac-Moody algebras(KM algebras) were shown in[1] and the concept of special imaginary roots was extended from KM algebras to GKM algebras in[7]. In[11], some properties of roots of GKM algebras were studied and in[12],[14], special imaginary roots of these classes were found out and finally in[15], a complete classification of GKM algebras possessing special imaginary roots was found out.

The notion of special imaginary roots of BKM algebras was generalized to BKM superalgebras in[16] and certain classes of BKM Lie superalgebras possessing special imaginary roots were found out in[16]. In[18], a complete classification of BKM Lie superalgebras possessing special imaginary roots was given. The concept of strictly imaginary roots for KM algebras was introduced by Kac([5],[6]). Casperson[3] gave a complete classification of KM algebras possessing strictly imaginary property. The concept of purely imaginary roots for KM algebras was introduced in[10] and therein the KM algebras possessing purely imaginary property were completely classified.

Again in[13], the concept of purely imaginary roots from KM algebras to GKM was extended, and the GKM algebras possessing purely imaginary property were completely classified. In[14], the properties of strictly imaginary roots and purely imaginary roots of GKM algebras were compared and using the classification of GKM algebras possessing purely imaginary property, the algebras whose purely imaginary roots are strictly imaginary roots were found. Complete classification of GKM algebras possessing special imaginary roots and strictly imaginary property were given in[15].

The concepts of strictly imaginary roots and purely imaginary roots of Borcherds Kac-Moody algebras(BKM algebras) were extended to BKM superalgebras in[17]. A complete classification of those BKM superalgebras with purely alien imaginary property and purely imaginary property were given in[17]. Moreover, the properties of strictly imaginary roots and purely imaginary roots of BKM superalgebras were compared and the BKM superalgebras whose purely imaginary roots are also strictly imaginary were found out in[17].

Aim of this paper is to give a complete classification of BKM Lie superalgebras possessing strictly imaginary property.

2. Preliminaries

2.1. Basic Definitions

In this section, we briefly recall the fundamental definitions regarding BKM Lie superalgebras, their Weyl groups and root systems as given in[19].

For the definition of Generalized Generalized Cartan matrix(GGCM) one can see[9].

Definition 2.1.1:[19].

Let be a finite index set and let be an real matrix. Let be a subset of I. If A satisfies the following conditions, then is called a BKM super matrix.

if then

if and then

Define, subsets and of I by .

Let be a collection of positive integers such that for all We call a charge of A.

Also set and .

Remarks:

(1) If is an empty set then the BKM super matrix coincides with the corresponding BKM matrix( or GKM matrix).

(2) For description of the quasi- Dynkin diagram, one can refer to[19]. A Generalized Generalized Cartan Matrix is called indecomposable if it cannot be reduced to a block diagonal form by shuffling rows and columns[8].

For the sake of completeness we repeat the following fundamentals already explained in[17].

Definition 2.1.2:[6]

Let I be an index set. be an indecomposable BKM super matrix where and . Then one and only one of the following three possibilities holds forA.

(Fin) det; there exists such that and or .

(Aff) Corank =1; there exists such that and

(Ind) det; there exists such that and

Referring to the above three cases, we say that A is of finite, affine or indefinite type respectively and write or respectively.

Definition 2.1.3:[18]

We say that a BKM super matrix is of hyperbolic type, if it is indefinite type and every principal submatrix of A is either finite or affine type BKM super matrix.

Definition 2.1.4[19]:

If a BKM super matrix decomposes as where, a diagonal matrix and is a symmetric matrix, then A is said to be symmetrizable.

If A is a symmetrizable BKM supermatrix, then taking the diagonal matrix D satisfying by we have and for all i and j.

We assume that is a symmetrizable and indecomposable BKM supermatrix.

Definition 2.1.5:[19]

For any BKM supermatrix, where , we have a triple , where and satisfying the following relations:

(i) is a finite dimensional (complex) vector space such that rank A.

(ii) is linearly independent and is linearly independent, where

iii) , where denotes a duality pairing between and This triple is called a realization of A.

Call an element of (respectively ) a fundamental root or a simple root (respectively fundamental coroot or a simple coroot).

Moreover, set and . We call an element of (resp. ) a real simple root (resp. an imaginary simple root).

Also divide as , the set of all even simple roots and , the set of all odd simple roots.

Let is the residue class ring mod 2 with elements and

Definition 2.1.6:[19]

A -graded vector space possessing the operation called the bracket product, is called a Lie superalgebra if it satisfies the following conditions:

for all .

Definition 2.1.7:[19]

The Borcherds Kac-Moody Lie superalgebra (abbreviated as BKM Lie superalgebra or BKM superalgebra) associated to a symmetrizable BKM super matrix is the Lie superalgebra generated by the vector space and the elements satisfying the following relations:

1.

2.

3.

4.

5. if then

6. if , then

7. if then .

Remarks:

As we are assuming that the matrix is symmetrizable, the associated BKM superalgebra is simple (for a proof one can see[6], and also[19] ),which we will denote by . So for a BKM supermatrix , is called BKM Lie superalgebra or BKM superalgebra associated to

In[5], Dynkin diagrams were defined for Lie superalgebras. Dynkin diagrams were already extended from KM algebras to GKM algebras in[11] and then extended to BKM Lie superalgebras in[17], which are again given below.

Definition 2.1.8:[17]

To every BKM super matrix , where , the index set I, is associated with a Dynkin diagram defined as follows:

has n vertices and vertices i and j are connected by number of lines if and there is an arrow pointing towards i if . If , i and j are connected by a bold faced edge equipped with the ordered pair Moreover

1. if , the i-th vertex will be denoted by a white circle.

2. if , the i-th vertex will be denoted by a white circle with (od) written within parentheses and below the circle to denote the vertex corresponding to an odd simple root in this case.

3. if , the i -th vertex will be denoted by a crossed circle.

4. if , the i -th vertex will be denoted by a crossed circle with (od) written within parentheses and below the circle to denote the vertex corresponding to an odd simple root in this case.

5. if , the i -th vertex will be denoted by a white circle with written within parentheses and above the circle.

6. if , the i -th vertex will be denoted by a white circle with written within parentheses and above the circle with (od) written within parentheses and below the circle to denote the vertex corresponding to an odd simple root in this case.

With these definitions, the Dynkin diagrams of all BKM superalgebras can be drawn.

Some examples of Dynkin diagrams of BKM superalgebras were drawn in[17].

A BKM Lie superalgebra , like a KM or BKM algebra, has the following natural root space decomposition:

where

is called the root space associated to An element is called a root, if and The number mult is called the multiplicity of the root α. A root α of can be expressed as where 's are all or all . Corresponding to whether 's are all or all , α is called a positive root or a negative root respectively. Also is called the height of α and is denoted by We denote by and the set of all roots, positive roots and negative roots respectively. Also note that and .

Definition 2.1.9:[19]

Let be a BKM Lie superalgebra. Set and take a subspace of satisfying .

Define the symmetric bilinear form on as follows:

. Then is non-degenerate on and this induces the linear isomorphism,

We completely identify and via this map v and omit the symbol v in the following results. The proofs of these results are in[19].

Lemma 2.1.10:[19]

For , one has the following:

1.

2.

3.

4. If then , in particular if , then

Remark:

For . In terms of inner product , we have and .

Definition 2.1.11:[17]

For each we define the simple reflection by

The Weyl group W of is the subgroup of generated by the 's . Note that is a coxeter system. So for a real root we define the reflection of with respect to α by

where is the dual real root of Note that .

Lemma 2.1.12:[19].

The bilinear form on and is invariant under the action of the Weyl group.

In particular, we have for

Definition 2.1.13:[17]

The set of all real roots of a BKM Lie superalgebra is defined as

Then the set of all imaginary roots is We have, and .

Definition 2.1.14:[19]

Let , then we have . is called the root lattice and is called the positive root lattice.The root lattice becomes a (partially) ordered set by putting for .

Now, for , support of α is defined as . If is a connected subset of the Dynkin diagram of A, we say that is connected.

Definition 2.1.15:[19]

Imaginary roots of BKM superalgebras are basically of two types, domestic-type and alien-type.

Domestic-type imaginary root:

An imaginary root which is conjugate to a fundamental root under the action of the Weyl group is called domestic-type imaginary root. We denote by , the set of all domestic-type imaginary roots.

Alien-type imaginary root:

An imaginary root which is not a conjugate to a fundamental root under the action of the Weyl group is called alien-type imaginary root. We denote by , the set of all alien-type imaginary roots.

Since an imaginary root is either conjugate or not conjugate to a fundamental root under the action of Weyl group, each imaginary root is either domestic imaginary or alien imaginary.

We have

Lemma 2.1.16:[19]

1. is invariant under the action of the Weyl group.

2. If then there exists satisfying

3. For

Theorem 2.1.17:[19]

For a symmetrizable BKM supermatrix if we set

then concerning and we have the following results: 1. .

2. .

where and

Also by notation

and

.

Lemma 2.1.18:[19]

For and one has the following:

1. The set is a finite set. Let p be the minimum contained in this set, and let q be the maximum in this set. Then,

(c) the sequence is bilaterally symmetric, and the left half of this sequence is monotone nondecreasing. Namely,

1.

Strictly domestic type imaginary roots, strictly alien type imaginary roots, strictly imaginary roots, purely imaginary roots, purely domestic type imaginary roots and purely alien imaginary root were already explained in[17]. We repeat the following definitions which we need here.

Definition 2.1.19:[`17]

A domestic-type imaginary root in a BKM super algebra is said to be strictly domestic-type imaginary, if for every either or is a root. Let denote the set of all strictly domestic-type imaginary roots, positive and negative strictly domestic-type imaginary roots respectively.

Definition 2.1.20:[17]

An alien-type imaginary root in a BKM super algebra is said to be strictly alien-type imaginary, if for every either or is a root. Let denote the set of all strictly alien-type imaginary roots, positive and negative strictly alien-type imaginary roots respectively.

Definition 2.1.21:[17]

An imaginary root in a BKM super algebra is said to be strictly imaginary, if for every either or is a root. The set of all strictly imaginary roots is denoted by Let denote the set of all strictly imaginary roots, positive and negative strictly imaginary roots respectively.

Remark:

As it was noticed in[5],

(1)If then

(2) is a semigroup.

Definition 2.1.22:[17]

A BKM super matrix is said to have strictly imaginary property, if

If a BKM supermatrix satisfies strictly imaginary property, we say that corresponding BKM Lie superalgebra satisfies strictly imaginary property.

Purely alien imaginary roots, purely domestic imaginary roots were already explained in Sthanumoorthy et al.(2009).

Definition 2.1.23:[17]

Let we say that α is purely imaginary, if for any We say that the BKM super algebra has the purely imaginary property, if satisfies this property. We have,

Similarly we say that a negative root is purely imaginary if is a purely imaginary root. Denote byand

Then, the set of all purely imaginary roots is

We omit the proof of the following theorem for BKM Lie superalgebras which can be directly verified using the proof for KM algebras already proved in[3].

Theorem 2.1.24:[17]

For BKM Lie superalgebras, the following results are true:

(a) If and for all then

(b) If for all then

(c) If and for all then for all

(d) If then

(e) is a semigroup.

In addition to the above results, we prove the following results for BKM Lie superalgebras.

Theorem 2.1.25:

a) If and is connected, then

b) If for all then

Proof:

a) Let and be connected. Then

So by lemma 2.1.18.

b) Let . If for all then or

So, by lemma 2.1.18., we have or Hence α is a strictly imaginary root.

Remark:

From the property of the Theorem (2.1.24), we have

3. Complete Classification of BKM Lie Superalgebras Possessing Strictly Imaginary Property

Remark:

In[17], a complete classification of BKM Lie superalgebras possessing purely imaginary property was given.

3.1. First we Give the following Results from[3].

Definition 3.1.1:[3]

We say that the generalized Cartan matrix A has the property SIM(more briefly: ) if

Definition 3.1.2:[3]

A is said to satisfy NC1, if there exists no subsets such that is affine or indefinite type, and is decomposable.

The following theorem proved by Casperson (1994) is for the indefinite Kac-Moody algebras possessing strictly imaginary property. Casperson (1994) gave a complete classification of Kac-Moody algebras possessing strictly imaginary property.

Theorem 3.1.3:[3]

A GCM lies in SIM if and only if it satisfies the condition NC1 and has no principal submatrix contained in the following list:

1. The matrices of the form with and or

2. The matrices of the following Dynkin diagrams of twisted affine type:

3. The strictly hyperbolic matrices associated with the Dynkin diagrams of the form:

4. The hyperbolic matrices associated with the Dynkin diagrams of the form:

From[3], we can conclude the following for affine Kac-Moody algebras:

5. If an algebra is affine, we have that and hence where is the unique minimal positive imaginary root and k is the order of the diagram automorphism used to construct the algebra.

Again for the case of matrices, the following theorem gives a complete classification of the non-strictly imaginary roots:

Theorem 3.1.4:[3]

Suppose, for the GCM, , that

Then, either

1. and (or)

2. where and

Corollary 3.1.5:[3]

The GCM is not in SIM if and only if and either or

Proposition 3.1.6:[3]

A GCM of affine type is a member of SIM if and only if it is of non-twisted affine type.

Remarks:

From[17], the set of all strictly imaginary roots of any BKM superalgebra is a subset of set of all purely imaginary roots, that is , may be an empty set or non-empty set depending upon the algebras. So, for the BKM superalgebras possessing purely imaginary property, we verify whether the set is empty or not. In the case where all purely imaginary roots are strictly imaginary as is always true. BKM superalgebras which satisfy the condition will be in the class of BKM algebras possessing strictly imaginary property.Hence the condition is equivalent to which is equivalent to SIM property.

3.2. As in the Cases of Special And Purely Imaginary Roots We Divide The Classes of BKM Superalgebras Into Two Categories. We Divide these BKM Lie Superalgebras into Two Categories.

Category 1: BKM Lie superalgebras without odd roots(GKM algebras only)

Category 2: BKM Lie superalgebras with a non-empty set of odd roots:

We discuss category 1 below.

Category 1: BKM Lie superalgebras without odd root:(GKM algebras only) Complete classification of GKM algebras possessing Strictly imaginary property was already given in[17].

Category 2: BKM Lie superalgebras with a non-empty set of odd roots:

We divide this category 2 into two classes, which are

Category2: Class(I): BKM Lie superalgebras of finite order Cartan matrices and with a non-empty set of odd roots

Category2: Class(II): BKM Lie superalgebras of infinite order Cartan matrices and with a non-empty set of odd roots

We discuss below these two classes separately .

Category 2:Class(I): BKM Lie superalgebras with a non-empty set of odd roots

We classify these BKM superalgebras into three subclasses (i),(ii) and (iii).

(i).BKM superalgebras with all simple roots being real with a non-empty set of odd roots:

These are BKM superalgebras which do not have any imaginary root. So this set of BKM superalgebras do not possess strictly imaginary property.

(ii).BKM superalgebras all whose simple roots are imaginary with a non-empty set of odd roots:

These are BKM superalgebras whose supermatrices do not appear as the extensions of KM matrices. So all the diagonal elements are negative. Hence there is no real simple root and all the roots are imaginary and also strictly imaginary.

(iii).BKM superalgebras with finite (non-zero) number of real simple roots and finite (non-zero) number of imaginary simple roots with a non-empty set of odd roots:

Remark:

Hereafter we denote by GGX, a Generalized Generalized Cartan matrix ( BKM super matrix or BKM matrix). We prove the following theorem for this case.

Theorem 3.2.1:

Let (the symmetrizable GGX)

Here are positive integers . Moreover, GX is KM matrix of finite, affine or indefinite type of order and GGX is a supermatrix of finite, affine or indefinite type with r simple imaginary roots added to that of GX. Then the following results are true for BKM superalgebras with odd roots.

1. GX is of finite type:

(a) If with (for all) is true for all then the corresponding BKM superalgebra satisfies SIM property.

(b)If with in the above GGX for is true for all then the corresponding BKM superalgebra satisfies SIM property.

Here in (a) and (b), is the set of all odd roots and is the set of all positive imaginary roots.

2. GX is of untwisted affine type:

(a) If with (for all ) is true for all then the corresponding BKM Lie superalgebra satisfies SIM property.

(b)If with in the above GGX for is true for all then the corresponding BKM Lie superalgebra satisfies SIM property.

3. If GX is of twisted affine type, then Strictly imaginary property does not hold.

4. If GX is of indefinite type, then Strictly imaginary property does not always hold.

Proof:

In the usual notation, let with and is the set of all simple roots with is the set of all simple imaginary roots and is the set of all simple real roots.

Ingeneraland

(1) Let GX be of finite type GCM and be an positive imaginary root. Then

Here

We discuss below Case(a), Case(b) and Case(c) separately.

Case(a): If then we have

By Theorem 2.1.24.(a), it is clear that if and then Strictly imaginary property holds.

Case(b): If then can be written as

Finally,

We divide this case(b) into Case(b)(1), Case(b)(2) (Case(b)(2)(i),Case(b)(2)(ii)) and Case(b)(3) separately.

Case(b)(1): Let

We have for

Since s are always negative integers and are always positive, by theorem 2.1.25., it is clear that

Case(b)(2):Let

Case(b)(2)(i): Let with

If by the theorem 2.1.25,

On the contrary, if Then

As is a real simple root, This is also not true, because are negative integers. So by theorem 2.1.25.,

Case(b)(2)(ii): Let with Then

But

Here because is a simple real root and for with and So

with Hence by theorem 2.1.25., if

Case(b)(3):Let

We have

As all the s are negative integers , by theorem 2.1.25.,

Case(c): If then can be written as

Here

Finally,

We discuss below Case(c)(1), Case(c)(2) and Case(3) separately in Case(c).

Case(c)(1): Let

We have for

Since s are always negative integers and are always positive, by theorem 2.1.25., it is clear that

Case(c)(2): Let

Case(c)(2)(i):Let with

If then

We have,

But

As is a real simple root with This is not true, because are negative integers. So by theorem 2.1.25.,

Case(c)(2)(ii): Let with Then

But

Here because is a simple real root and for with and So

with By theorem 2.1.25., if

Case(c)(3):Let We have

As all the s are negative integers, by theorem 2.1.25.,

(2) Let GX be of untwisted affine type. For KM algebras of untwisted affine type, SIM property holds as per Casperson(1994). For BKM algebras with odd roots which we get as extensions of KM algebras untwisted affine type, the proof is exactly same to case(1) and hence SIM property holds.

(3) Let GX be of twisted affine type. As per Casperson(1994) mentioned above, SIM property does not hold for KM algebras and the same is true for BKM Lie superalgebras which appear as extension of KM algebras of twisted affine type. Hence SIM property does not hold.

(4)Let GX be of indefinite type. As far as indefinite BKM Lie superalgebras are concerned, extension of finite and untwisted affine type of KM algebras will hold SIM property, where as other algebras do not hold.

The following example will illustrate the above theorem.

Example: Extension of finite type

Let ( the symmetrizable GGX)

This is a BKM supermatrix of indefinite type denoted by , which is an extension of finite type .

If and then the Dynkin diagram can be drawn as follows:

Figure 1. Dynkin diagram of SBGA2

The Weyl group for corresponding BKM Lie superalgebra is .

Here

Case(a): . The following relations (i), (ii),...,(vii) can be directly verified.

(i)Ifwith then

(ii) If with then

(iii) If with then

(iv)If with then

(v)If with then

(vi)If

with then

By Theorem 2.1.24., and from the above results (i), (ii), (iii), (iv), (v) and (vi), it is clear that if and that is if (in general), then for all

Case (b): Let The following relations (i), (ii),...,(vi) can be easily verified.

(i)If we get

By Theorem 2.1.25., if then and which implies

(ii) If we get

(iii) If we get

(iv) If we get

(v) If we get

(vi) If we get

From the above results (ii), (iii), (iv), (v) and (vi) with it is clear that for in (ii), in (iii), in (iv), in (v), in (vi). Hence, by Theorem 2.1.25., SIM property holds.

In general, if then for

Case(c): Let The following relations (i),(ii),...,(vi) can be directly verified.

(i)If we get

By Theorem 2.1.25., if then and which implies

(ii) If we get

(iii) If we get

(iv) If we get

(v)If we get

(vi) If we get

From the above results (ii), (iii), (iv), (v) and (vi) with it is clear that for in (ii), in (iii), in (iv), in (v), in (vi). Hence, by theorem 2.1.25., SIM property holds.

In general, for SIM property holds if

Example: Extension of untwisted affine type

Let (the symmetrizable GGX) This is a BKM supermatrix of indefinite type denoted by , which is an extension of untwisted affine type .

If k=0, and then the Dynkin diagram can be drawn as follows:

Figure 2. Dynkin diagram of SBGA1(1).

The Weyl group of the corresponding BKM Lie superalgebra is

Then

Here

Case(a):Let The following relations (i),(ii), ....,(vi) can be easily verified.

(i) If with and then

(ii) If with and then

(iii) If with and then

(iv) If

with and then

(v) If

with and then

(vi)If

with and then

Similarly, we can find for different and

Hence by Theorem 2.1.24., and by the above results (i),(ii), ....,(vi) and others, it is clear that for all if and That is if SIM property holds.

Case(b): Let

The following relations (i), (ii),...,(iv) can be easily verified.

(i)If we get

By Theorem 2.1.25., if then which implies

(ii) If we get

(iii) If we get

we get

Similarly, we can find for and

Hence by Theorem 2.1.25., and by the above results (ii), (iii) and others, with it is clear that for , ,..... In general, if we get for

Case(C): If The following relations (i), (ii),...,(iv) can be easily verified.

(i)If we get and

By Theorem 2.1.25., if then which implies

(ii) If we get

(iii) If we get

(iv)If we get

Similarly, we can find for and

Hence by Theorem 2.1.25 and by the above results (ii), (iii),...,(iv) and others, with it is clear that for .....,..... In general, if we get for

Class(II): BKM Lie superalgebras of infinite order and with a finite non-empty set of odd roots

We divide this class into three subclasses.

(i) All simple roots are imaginary(odd or even)

(ii) One simple real root(odd or even) and infinite number of imaginary roots(odd or even)

(iii) Finite number of simple real roots and infinite number of imaginary roots.

We discuss these cases below

(i) All simple roots are imaginary(odd or even):

For this class, all the roots are imaginary. So these algebras satisfy strictly imaginary property.

(ii) One simple real root(odd or even) and infinite number of imaginary roots(odd or even):

We prove the following theorem for this case.

Theorem 3.2.2: Let (the symmetrizable GGX)

Here are positive integers and GGX is the BKM supermatrix with one real simple root and infinite number of imaginary roots. If with is true for all then the corresponding BKM Lie superalgebra satisfies Strictly imaginary property.

Proof:

In the usual notation with and with and We define and

Let and . Then by Theorem 2.1.24.,

If which is same as then Strictly imaginary property holds.

Remark:

For BKM superalgebras which appear as extension of twisted affine type(Case 3) and extension of indefinite type(Case 4), examples were given in[17]( set)(Section 4, Case 3, subcase 2).

Remarks:

We have seen above that in the case of BKM Lie superalgebras of infinite order with one simple real root (odd or even ) and infinite number of imaginary roots (odd or even) for SIM property holds only when and are all greater than one. As a counter example, we consider Monster Lie superalgebra with one simple real root and infinite number of simple imaginary roots(odd or even). Consider with and for (c(1): multiplicity of the root corresponding to -2 ). As , the corresponding BKM Lie superalgebra does not satisfy SIM property. We prove this below. Consider Monster Lie superalgebra which has the following supermatrix as defined below:

Let be an index set and consider the Borcherds-Cartan super matrix with charge where are the coefficients of the elliptic modular function

Here

is the BKM supermatrix and We define is the real root corresponding to the diagonal element 2 and are the imaginary roots corresponding to the diagonal element -2

We consider any then

This implies does not satisfy the strictly imaginary property for . Hence Strictly imaginary property does not hold for a Monster Lie superalgebra.

(iii) Finite number(atleast two) of simple real roots and infinite number of imaginary roots:

We prove the following theorem for this case.

Theorem 3.2.3:

Let ( the symmetrizable GGX)

Here are positive integers and GGX is the BKM supermatrix with real simple roots and infinite number of imaginary roots. If with is true for all then the corresponding BKM Lie superalgebra satisfies Strictly imaginary property.

Proof:

In the usual notations with () and Here, the set of all simple roots with the set of simple real root and the set of all simple imaginary roots.

We define and

Let and

As all are negative integers and we have Hence and SIM property holds.

Remarks:

As in the case of Monster Lie superalgebra with one simple real root and infinite number of imaginary simple roots with the condition we can consider BKM Lie superalgebras with two simple real roots and infinite number of imaginary simple roots with the condition,

In this case, as and for some (similar to Monster Lie superalgebra as in theorem 3.2.3. ) SIM property does not hold . Hence we understand that for the infinite order case, the SIM property depends on the non-diagonal non-zero entries of the corresponding BKM supermatrix.

4. Conclusions

In this paper, a complete classification of Borcherds Kac-Moody Lie superalgebras possessing strictly imaginary property is given. From this classification, one can understand that strictly imaginary property depends mainly on the coefficients of the corresponding BKM supermatrix. With these findings, different complete classifications of Borcherds Kac-Moody Lie superalgebras possessing special imaginary roots, purely imaginary roots and strictly imaginary roots were separately found out in different research papers. In fact, these classifications will be very much helpful to the researchers to extend these classes of root systems to other types of finite and infinite dimensional Lie (super)algebras. Moreover, other characteristics of these classes of Borcherds Kac-Moody Lie superalgebras possessing these root systems can also be studied. These findings may also lead to many other applications.

ACKNOWLEDGEMENTS

The research has been financially supported by the University Grants Commission (UGC), Govt. of India, through the Major research Project, F.No.36-272/2008(SR). The authors, N.Sthanumoorthy (Principal Investigator of the Project) and K.Priyadharsini(Project Fellow for the Project) are thankful to the UGC for the same.