Determinants, Differential Forms and Bracket Algebras

If V is a free k-module with basis e1, . . . , en, and if f : V →V is a khomomorphism, then the determinant of f , denoted by det( f ), is the element det( f ) ∈k for which <n( f ): e1 ∧· · · ∧en ↦f (e1) ∧· · · ∧f (en) = det( f )(e1 ∧· · · ∧en). If A = [ai j] is an n × n matrix with entries in k, then A defines a k-map f : kn →kn by f (x) = Ax, where x ∈kn is a column vector. If e1, . . . , en is the standard basis of kn, then f (ei) = j a jie j, and the matrix A = [ai j] associated to f has ith column the coordinates of f (ei) = Aei. We define det(A) = det( f ): Ae1 ∧· · · ∧Aen = det(A)(e1 ∧· · · ∧en). Thus, the wedge of the columns of A in ;n(kn) is a constant multiple of e1 ∧· · · ∧en, and det(A) is that constant.

Let A be an n × n matrix with entries in a commutative ring k. If H = i1, . . . , i p and L = j1, . . . , jp are increasing p ≤n-lists, then AH L is the p× p submatrix [ast], where (s, t) ∈H × L. A minor of order p is the determinant of a p × p submatrix. For example, every entry ai j is a minor of A = [ai j] of order 1. If A =   a11 a12 a13 a21 a22 a23 a31 a32 a33  , then some minors of order 2 are det a11 a12 a21 a22 and det a12 a13 a32 a33 . In particular, if 1 ≤i ≤n, let i′ denote the increasing n −1 ≤n-list in which i is omitted; thus, an (n −1) × (n −1) submatrix has the form Ai′ j′, and its determinant is a minor of order n −1. Note that Ai′ j′ is the submatrix obtained from A by deleting its ith row and jth column.

An n × n matrix A with entries in a commutative ring k is invertible over k if there is a matrix B with entries in k such that AB = I = B A. If k is a field, then invertible matrices are usually called nonsingular, and they are characterized by having a nonzero determinant. Consider the matrix with entries in Z: A = . Now det(A) = 2 ̸= 0, but it is not invertible over Z. Suppose a c b d = 3a + b 3c + d a + b c + d . If this product is I, then 3a + b = 1 = c + d 3c + d = 0 = a + b. Hence, b = −a and 1 = 3a + b = 2a; as there is no solution to 1 = 2a in Z, the matrix A is not invertible over Z. Of course, A is invertible over Q.

Let k be a commutative ring and let A be a k-algebra. A derivation of A is a homomorphism d : A →A of k-modules for which d(ab) = (da)b + a(db). In words, a derivation acts like ordinary differentiation in calculus, for we are saying that the product rule, ( f g)′ = f ′g + f g′, holds.

Let k be a commutative ring. A not necessarily associative k-algebra A is a k-module equipped with some multiplication A × A →A, denoted by (a, b) ↦ab, such that (i) a(b + c) = ab + ac and (b + c)a = ba + ca for all a, b, c ∈A; (ii) ua = au for all u ∈k and a ∈A; (iii) a(ub) = (au)b = u(ab) for all u ∈k and a, b ∈A. A derivation of A is a k-map d : A →A for which d(ab) = (da)b + a(db). Aside from ordinary differentiation in calculus, which is a derivation because the product rule holds, ( f g)′ = f ′g + f g′, another example is provided by the R-algebra A of all real valued functions f (x1, . . . , xn) of several variables. The partial derivatives ∂/∂xi are derivations, for i = 1, . . . , n. The composite of two derivations need not be a derivation. For example, if d : A →A is a derivation, then d2 = d ◦d : A →A satisfies the equation d2( f g) = d2( f )g + 2d( f )d(g) + f d2(g); the mixed term 2d( f )d(g) is the obstruction to d2 being a derivation. If f, g ∈A, then dn( f g) = n i=0 n i di f · dn−ig. It is still worthwhile to compute the composite of two derivations d1 and d2. If A is a not necessarily associative algebra and f, g ∈A, then d1d2( f g) = d1 [(d2 f )g + f (d2g)] = (d1d2 f )g + (d2 f )(d1g) + (d1 f )(d2g) + f (d1d2g). Of course, d2d1( f g) = (d2d1 f )g + (d1 f )(d2g) + (d2 f )(d1g) + f (d2d1g). If we denote d1d2 −d2d1 by [d1, d2], then subtraction gives [d1, d2]( f g) = ([d1, d2] f )g + f ([d1, d2]g); that is, [d1, d2] = d1d2 −d2d1 is a derivation.

If k is a field, then a bracket algebras over k is a vector space L over k equipped with a bilinear operation L × L →L, denoted by (a, b) ↦[a, b] (and called bracket), such that (i) [a, a] = 0 for all a ∈L; (ii) For each a ∈L, the function ada : b ↦[a, b] is a derivation. bracket algebras For all u, v ∈L, bilinearity gives [u + v, u + v] = [u, u] + [u, v] + [v, u] + [v, v], which, when coupled with the first axiom [a, a] = 0, gives [u, v] = −[v, u]; that is, bracket is anticommutative. The second axiom is often written out in more detail. If b, c ∈L, then their product in L is denoted by [b, c]; that ada is a derivation is to say [a, [b, c]] = [[a, b], c] + [b, [a, c]]; rewriting, [a, [b, c]] −[b, [a, c]] −[[a, b], c] = 0. The anticommutativity from the first axiom now gives the bracket identity: [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0 for all a, b, c ∈L. Thus, a vector space L is a bracket algebras if and only if [a, a] = 0 for all a ∈L and the bracket identity holds. Here are some examples of bracket algebras.

An ideal of a bracket algebras L is a subspace I such that [x, a] ∈I for every x ∈L and a ∈I. Even though a bracket algebras need not be commutative, its anticommutativity shows that every left ideal (as just defined) is necessarily a right ideal; that is, every ideal is two-sided. A bracket algebras L is called simple if L ̸= {0} and L has no nonzero proper ideals.

If I is an ideal in L, then the quotient L/I is the quotient space (considering L as a vector space and I as a subspace) with bracket defined by [a + I, b + I] = [a, b] + I. It is easy to check that this bracket on L/I is well-defined. If a′ + I = a + I and b′ + I = b + I, then a −a′ ∈I and b −b′ ∈I, and so [a′, b′] −[a, b] = [a′, b′] −[a′, b] + [a′, b] −[a, b] = [a′, b′ −b] + [a′ −a, b′] ∈I. bracket algebras

Let k be a commutative ring. (i) If I is the identity matrix, then det(I) = 1. (ii) If A and B are n × n matrices with entries in k, then det(AB) = det(A) det(B).

If k is a commutative ring, then det: Matn(k) →k is the unique alternating multilinear function with det(I) = 1.

Let e1, . . . , en be a basis of a free k-module, where k is a commutative ring. If σ is a permutation of 1, 2, . . . , n, then eσ(1) ∧· · · ∧eσ(n) = sgn(σ)(e1 ∧· · · ∧en).

Let e1, . . . , en be a basis of a free k-module, where k is a commutative ring. If A = [ai j] is an n × n matrix with entries in k, then det(A) = σ∈Sn sgn(σ)aσ(1),1aσ(2),2 · · · aσ(n),n.

If A is an n × n matrix with entries in a commutative ring k, then det(At) = det(A), where At is the transpose of A.

If A = [ai j] is an n × n matrix with entries in a field k, then tr(A) = α1 + α2 + · · · + αn.

If A and B are similar n × n matrices with entries in a field k, then A and B have the same characteristic polynomial.

Let k be a commutative ring, and let xi1,. . ., xi p ∈kn be regarded as columns of an n × p matrix A, where H = i1, . . . , i p is an increasing p ≤n-list. Then xi1 ∧· · · ∧xi p = L det(AL,H)eL, where L varies over all increasing p ≤n-lists.

Let A = [ai j] be an n×n matrix with entries in a commutative ring k. (i) If I = i1, . . . , i p is an increasing p-list and xi1, . . . , xi p are the corresponding columns of A, then denote xi1 ∧· · · ∧xi p by xI. If J = j1, . . . , jq is an increasing q-list, then xI ∧xJ = H,K ρH,K det(AH,I) det(AK,J)eH∗K , where H ∗K is the increasing (p + q)-list formed from H ∪K when H ∩K = ∅. (ii) Laplace23 expansion down the jth column: For each fixed j, det(A) = (−1)1+ ja1 j det(A1′ j′) + · · · + (−1)n+ janj det(An′ j′), where Ai′ j′ is the (n −1) × (n −1) submatrix obtained from A by deleting its ith row and jth column. (iii) cofactor expansion across the ith row: For each fixed i, det(A) = (−1)i+1ai1 det(Ai′1′) + · · · + (−1)i+nain det(Ai′n′).

If A is an n × n matrix with entries in a commutative ring k, then A adj(A) = det(A)I = adj(A)A. 24There is no connection between the adjoint of a matrix as just defined and the adjoint of a matrix defined on an inner product space. Determinants

Let k be a commutative ring, and let X = A C B be an (m + n) × (m + n) matrix with entries in k, where A is an m × m submatrix, and B is an n × n submatrix. Then det(X) = det(A) det(B).

If A = [ai j] is a triangular n × n matrix, that is, ai j = 0 for all i < j (lower triangular) or ai j = 0 for all i > j (upper triangular), then det(A) = n → i=1 aii; that is, det(A) is the product of the diagonal entries.

Let k be a commutative ring, and let M be a k-module. (i) If ϕ : M →M is a k-map, then there exists a unique derivation Dϕ : T (M) → T (M), where T (M) is the tensor algebra on M, which is a graded map (of degree 0) with Dϕ|M = ϕ; that is, for all p ≥0, Dϕ ( T p(M) ) ⊆T p(M). (ii) If ϕ : M →M is a k-map, then there exists a unique derivation dϕ : ;(M) → ;(M), which is a graded map (of degree 0) with dϕ|M = ϕ; that is, for all p ≥0, dϕ ( p < (M) ) ⊆ p < (M).

Let k be a commutative ring, and let M be a finitely generated free k-module with basis e1, . . . , en. If ϕ : M →M is a k-map and dϕ : ;(M) →;(M) is the derivation it determines, then dϕ n< (M) = tr(ϕ)eL, where eL = e1 ∧· · · ∧en.

If A = a c b d , then the wedge product of the columns of A is (ae1 + be2) ∧(ce1 + de2) = ace1 ∧e1 + ade1 ∧e2 + bce2 ∧e1 + bde2 ∧e2 = ade1 ∧e2 + bce2 ∧e1 = ade1 ∧e2 −bce1 ∧e2 = (ad −bc)(e1 ∧e2). Therefore, det(A) = ad −bc. The next example considers triple integrals.

(i) If f : L →L′ is a bracket-algebra homomorphisms, then its kernel is defined as usual: ker f = {a ∈L : f (a) = 0}. Conversely, the natural map ν : L →L/I, defined by a ↦a + I, is a bracket-algebra homomorphisms whose kernel is I. Thus, a subspace of L is an ideal if and only if it is the kernel of some bracket-algebra homomorphisms. (ii) If I and J are ideals in a bracket algebras L, then I J = r [ir, jr] : ir ∈I and jr ∈J . In particular, L2 = LL is the analog for bracket algebras of the commutator subgroup of a group: L2 = {0} if and only if L is abelian. (iii) There is an analog for bracket algebras of the derived series of a group. The derived series of a bracket algebras L is defined inductively: L(0) = L; L(n+1) = (L(n))2. A bracket algebras L is called solvable if there is some n ≥0 with L(n) = {0}. (iv) There is an analog for bracket algebras of the descending central series of a group. The descending central series is defined inductively: L1 = L; Ln+1 = LLn. A bracket algebras L is called nilpotent if there is some n ≥0 with Ln = {0}. ◀ We merely mention the first two theorems in the subject. If L is a bracket algebras and a ∈L, then ada : L →L, given by ada : x ↦[a, x], is a linear transformation on L (viewed merely as a vector space). We say that a is ad-nilpotent if ada is a nilpotent operator; that is, (ada)m = 0 for some m ≥1. (i) If L is a finite-dimensional bracket algebras over any field k, then L is nilpotent if and only if every a ∈L is ad-nilpotent. (ii) If L is a bracket subalgebra of gl(n, k) all of whose elements A are nilpotent matrices, then L can be put into strict upper triangular form (all diagonal entries are 0); that is, there is a nonsingular matrix P so that P AP−1 is strictly upper triangular for every A ∈L.