Package 'jordan'

A Suite of Routines for Working with Jordan Algebras

Description

A Jordan algebra is an algebraic object originally designed to study observables in quantum mechanics. Jordan algebras are commutative but non-associative; they satisfy the Jordan identity. The package follows the ideas and notation of K. McCrimmon (2004, ISBN:0-387-95447-3) "A Taste of Jordan Algebras". To cite the package in publications, please use Hankin (2023) <doi:10.48550/arXiv.2303.06062>.

Details

A Jordan algebra is a non-associative algebra over the reals with a multiplication that satisfies the following identities:

$xy=yx$

$(xy)(xx) = x(y(xx))$

(the second identity is known as the Jordan identity). In literature one usually indicates multiplication by juxtaposition but one sometimes sees $x\circ y$. Package idiom is to use an asterisk, as in x*y. There are five types of Jordan algebras:

1. Real symmetric matrices, class real_symmetric_matrix, abbreviated in the package to rsm

2. Complex Hermitian matrices, class complex_herm_matrix, abbreviated to chm

3. Quaternionic Hermitian matrices, class quaternion_herm_matrix, abbreviated to qhm

4. Albert algebras, the space of $3\times 3$ octonionic matrices, class albert

5. Spin factors, class spin

(of course, the first two are special cases of the next). The jordan package provides functionality to manipulate jordan objects using natural R idiom.

Objects of all these classes are stored in dataframe (technically, a matrix) form with columns being elements of the jordan algebra.

The first four classes are matrix-based in the sense that the algebraic objects are symmetric or Hermitian matrices (the S4 class is “jordan_matrix”). The fifth class, spin factors, is not matrix based.

One can extract the symmetric or Hermitian matrix from objects of class jordan_matrix using as.list(), which will return a list of symmetric or Hermitian matrices. A function name preceded by a “1” (for example as.1matrix() or vec_to_qhm1()) means that it deals with a single (symmetric or Hermitian) matrix.

Algebraically, the matrix form of jordan_matrix objects is redundant (for example, a real_symmetric_matrix of size $n\times n$ has only $n(n+1)/2$ independent entries, corresponding to the upper triangular elements).

Author(s)

Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>)

Maintainer: Robin K. S. Hankin <[email protected]>

References

K. McCrimmon 1978. “Jordan algebras and their applications”. Bulletin of the American Mathematical Society, Volume 84, Number 4.

Examples

rrsm()     # Random Real Symmetric matrices
rchm()     # Random Complex Hermitian matrices
rqhm()     # Random Quaternionic Hermitian matrices
ralbert()  # Random Albert algebra
rspin()    # Random spin factor

x <- rqhm(n=1)  
y <- rqhm(n=1)  
z <- rqhm(n=1)  

x/1.2 + 0.3*x*y     # Arithmetic works as expected ...
x*(y*z) -(x*y)*z    # ... but '*' is not associative


## Verify the Jordan identity for  type 3 algebras:

LHS <- (x*y)*(x*x)
RHS <- x*(y*(x*x))

diff <- LHS-RHS  # zero to numerical precision

diff[1,drop=TRUE]  # result in matrix form

---

Methods for Function Arith in package Jordan

Description

Methods for Arithmetic functions for jordans: +, -, *, /, ^

Usage

jordan_negative(z)
jordan_plus_jordan(e1,e2)
jordan_plus_numeric(e1,e2)
jordan_prod_numeric(e1,e2)
jordan_power_jordan(e1,e2)
albert_arith_albert(e1,e2)
albert_arith_numeric(e1,e2)
albert_inverse(e1)
albert_power_albert(...)
albert_power_numeric(e1,e2)
albert_power_single_n(e1,n)
albert_prod_albert(e1,e2)
chm_arith_chm(e1,e2)
chm_arith_numeric(e1,e2)
chm_inverse(e1)
chm_power_numeric(e1,e2)
chm_prod_chm(e1,e2)
numeric_arith_albert(e1,e2)
numeric_arith_chm(e1,e2)
numeric_arith_qhm(e1,e2)
numeric_arith_rsm(e1,e2)
qhm_arith_numeric(e1,e2)
qhm_arith_qhm(e1,e2)
qhm_inverse(x)
qhm_power_numeric(e1,e2)
qhm_prod_qhm(e1,e2)
rsm_arith_numeric(e1,e2)
rsm_arith_rsm(e1,e2)
rsm_inverse(e1)
rsm_power_numeric(e1,e2)
rsm_prod_rsm(e1,e2)
spin_plus_numeric(e1,e2)
spin_plus_spin(e1,e2)
spin_power_numeric(e1,e2)
spin_power_single_n(e1,n)
spin_power_spin(...)
spin_prod_numeric(e1,e2)
spin_prod_spin(e1,e2)
spin_inverse(...)
spin_negative(e1)
vec_albertprod_vec(x,y)
vec_chmprod_vec(x,y)
vec_qhmprod_vec(x,y)
vec_rsmprod_vec(x,y)

Arguments

z, e1, e2	Jordan objects or numeric vectors
n	Integer for powers
...	Further arguments (ignored)
x, y	Numeric vectors, Jordan objects in independent form

Details

The package implements the Arith group of S4 generics so that idiom like A + B*C works as expected with jordans.

Functions like jordan_inverse() and jordan_plus_jordan() are low-level helper functions. The only really interesting operation is multiplication; functions like jordan_prod_jordan().

Names are implemented and the rules are inherited (via onion::harmonize_oo() and onion::harmonize_on()) from rbind().

Value

generally return jordans

Author(s)

Robin K. S. Hankin

Examples

x <- rspin()
y <- rspin()
z <- rspin()

x*(y*(x*x)) - (x*y)*(x*x) # should be zero

x + y*z

---

Concatenation

Description

Combines its arguments to form a single jordan object.

Usage

## S4 method for signature 'jordan'
c(x,...)

Arguments

x, ...

Jordan objects

Details

Returns a concatenated jordan of the same type as its arguments. Argument checking is not performed.

Value

Returns a Jordan object of the appropriate type (coercion is not performed)

Note

Names are inherited from the behaviour of cbind(), not c().

Author(s)

Robin K. S. Hankin

Examples

c(rqhm(),rqhm()*10)

---

Coercion

Description

Various coercions needed in the package

Usage

as.jordan(x,class)
vec_to_rsm1(x)
vec_to_chm1(x)
vec_to_qhm1(x)
vec_to_albert1(x)
rsm1_to_vec(M)
chm1_to_vec(M)
qhm1_to_vec(M)
albert1_to_vec(H)
as.real_symmetric_matrix(x,d,single=FALSE)
as.complex_herm_matrix(x,d,single=FALSE)
as.quaternion_herm_matrix(x,d,single=FALSE)
as.albert(x,single=FALSE)
numeric_to_real_symmetric_matrix(x,d)
numeric_to_complex_herm_matrix(x,d)
numeric_to_quaternion_herm_matrix(x,d)
numeric_to_albert(e1)
as.list(x,...)
matrix1_to_jordan(x)

Arguments

x, e1	Numeric vector of independent entries
M, H	A matrix
d	Dimensionality of algebra
single	Boolean, indicating whether a single value is to be returned
class	Class of object
...	Further arguments, currently ignored

Details

The numeral “1” in a function name means that it operates on, or returns, a single element, usually a matrix. Thus function as.1matrix() is used to convert a jordan object to a list of matrices. Length one jordan objects are converted to a matrix.

Functions vec_to_rsm1() et seq convert a numeric vector to a (symmetric, complex, quaternion, octonion) matrix, that is, elements of a matrix-based Jordan algebra.

Functions rsm1_to_vec() convert a (symmetric, complex, quaternion, octonion) matrix to a numeric vector of independent components. The upper triangular components are used; no checking for symmetry is performed (the lower triangular components, and non-real components of the diagonal, are discarded).

Functions as.real_symmetric_matrix(), as.complex_herm_matrix(), as.quaternion_herm_matrix() and as.albert() take a numeric matrix and return a (matrix-based) Jordan object.

Functions numeric_to_real_symmetric_matrix() have not been coded up yet.

Function matrix1_to_jordan() takes a matrix and returns a length-1 (matrix based) Jordan vector. It uses the class of the entries (real, complex, quaternion, octonion) to decide which type of Jordan to return.

Value

Return a coerced value.

Author(s)

Robin K. S. Hankin

Examples

vec_to_chm1(1:16)  # Hermitian matrix

as.1matrix(rchm())

as.complex_herm_matrix(matrix(runif(75),ncol=3))

matrix1_to_jordan(cprod(matrix(rnorm(35),7,5)))
matrix1_to_jordan(matrix(c(1,1+1i,1-1i,3),2,2))
matrix1_to_jordan(Oil + matrix(1,3,3))

---

Methods for compare S4 group

Description

Methods for comparison (equal to, greater than, etc) of jordans. Only equality makes sense.

Usage

jordan_compare_jordan(e1,e2)

Arguments

e1, e2

Jordan objects

Value

Return a boolean

Examples

# rspin() > 0 # meaningless and returns an error

---

Extract and replace methods for jordan objects

Description

Extraction and replace methods for jordan objects should work as expected.

Replace methods can take a jordan or a numeric, but the numeric must be zero.

Value

Generally return a jordan object of the same class as the first argument

Methods

[

signature(x = "albert", i = "index", j = "missing", drop = "logical"): ...

[

signature(x = "complex_herm_matrix", i = "index", j = "missing", drop = "logical"): ...

[

signature(x = "jordan", i = "index", j = "ANY", drop = "ANY"): ...

[

signature(x = "jordan", i = "index", j = "missing", drop = "ANY"): ...

[

signature(x = "quaternion_herm_matrix", i = "index", j = "missing", drop = "logical"): ...

[

signature(x = "real_symmetric_matrix", i = "index", j = "missing", drop = "logical"): ...

[

signature(x = "spin", i = "index", j = "missing", drop = "ANY"): ...

[

signature(x = "spin", i = "missing", j = "index", drop = "ANY"): ...

[<-

signature(x = "albert", i = "index", j = "missing", value = "albert"): ...

[<-

signature(x = "complex_herm_matrix", i = "index", j = "ANY", value = "ANY"): ...

[<-

signature(x = "complex_herm_matrix", i = "index", j = "missing", value = "complex_herm_matrix"): ...

[<-

signature(x = "jordan_matrix", i = "index", j = "missing", value = "numeric"): ...

[<-

signature(x = "quaternion_herm_matrix", i = "index", j = "missing", value = "quaternion_herm_matrix"): ...

[<-

signature(x = "real_symmetric_matrix", i = "index", j = "missing", value = "real_symmetric_matrix"): ...

[<-

signature(x = "spin", i = "index", j = "index", value = "ANY"): ...

[<-

signature(x = "spin", i = "index", j = "missing", value = "numeric"): ...

[<-

signature(x = "spin", i = "index", j = "missing", value = "spin"): ...

Author(s)

Robin K. S. Hankin

Examples

showClass("index")  # taken from the Matrix package

a <- rspin(7)
a[2:4] <- 0
a[5:7] <- a[1]*10
a

---

Multiplicative identities

Description

Multiplying a jordan object by the identity leaves it unchanged.

Usage

as.identity(x)
rsm_id(n,d)
chm_id(n,d)
qhm_id(n,d)
albert_id(n)
spin_id(n=3,d=5)

Arguments

n	Length of vector to be created
d	Dimensionality
x	In function as.identity(), a jordan object. Return value will be a jordan object of the same dimensionality but entries equal to the identity

Details

The identity object in the matrix-based classes (jordan_matrix) is simply the identity matrix. Function as.identity() takes an object of any of the five types (rsm, chm, qhm, spin, or albert) and returns a vector of the same length and type, but comprising identity elements.

Class spin has identity $\left(1,\mathbf{0}\right)$.

Value

A jordan object is returned.

Author(s)

Robin K. S. Hankin

Examples

x <- as.albert(matrix(sample(1:99,81,replace=TRUE),nrow=27))
I <- as.identity(x)
x == x*I  # should be TRUE


rsm_id(6,3)

---

Create jordan objects

Description

Creation methods for jordan objects

Arguments

M	A matrix with columns representing independent entries in a matrix-based Jordan algebra
a, V	Scalar and vector components of a spin factor

Details

The functions documented here are the creation methods for the five types of jordan algebra.

• quaternion_herm_matrix()

• complex_herm_matrix()

• real_symmetric_matrix()

• albert()

• spin()

(to generate quick “get you going” Jordan algebra objects, use the rrsm() family of functions, documented at random.Rd).

Value

Return jordans or Boolean as appropriate

Author(s)

Robin K. S. Hankin

See Also

random

Examples

A <- real_symmetric_matrix(1:10)  # vector of length 1
as.1matrix(A)                     # in matrix form

complex_herm_matrix(cbind(1:25,2:26))
quaternion_herm_matrix(1:15)

albert(1:27)
spin(-6,cbind(1:12,12:1))


x <- rrsm() ; y <- rrsm() ; z <- rrsm()  # also works with the other Jordans

x*(y*z) - (x*y)*z         # Jordan algebra is not associative...
(x*y)*(x*x) - x*(y*(x*x)) # but satisfies the Jordan identity

---

Classes in the "jordan" package

Description

Various classes in the jordan package.

Author(s)

Robin K. S. Hankin

References

K. McCrimmon 1978. “Jordan algebras and their applications”. Bulletin of the American Mathematical Society, Volume 84, Number 4.

Examples

showClass("jordan")

---

Miscellaneous Jordan functionality

Description

Miscellaneous Jordan functionality that should be documented somewhere

Usage

harmonize_spin_numeric(e1,e2)
harmonize_spin_spin(e1,e2)

Arguments

e1, e2

Objects to harmonize

Details

Miscellaneous low-level helper functions.

The harmonize functions harmonize_spin_numeric() and harmonize_spin_spin() work for spin objects for the matrix-based classes onion::harmonize_oo() and onion::harmonize_on() are used.

Value

These are mostly low-level helper functions; they not particularly user-friendly. They generally return either numeric or Jordan objects.

Author(s)

Robin K. S. Hankin

---

Sizes of Matrix-based Jordan algebras

Description

Given the number of rows in a (matrix-based) Jordan object, return the size of the underlying associative matrix algebra

Usage

r_to_n_rsm(r)
r_to_n_chm(r)
r_to_n_qhm(r)
r_to_n_albert(r=27)
n_to_r_rsm(n)
n_to_r_chm(n)
n_to_r_qhm(n)
n_to_r_albert(n=3)

Arguments

n	Integer, underlying associative algebra being matrices of size $n\times n$
r	Integer, number of rows of independent representation of a matrix-based jordan object

Details

These functions are here for consistency, and the albert ones for completeness.

For the record, they are:

• Real symmetric matrices, rsm, $r=n(n+1)/2$, $n=(\sqrt{1+4r}-1)/2$

• Complex Hermitian matrices, chm, $r=n^2$, $n=\sqrt{r}$

• Quaternion Hermitian matrices, qhm, $r=n(2n-1)$, $n=(1+\sqrt{1+8r})/4$

• Albert algebras, $r=27$, $n=3$

Value

Return non-negative integers

Note

I have not been entirely consistent in my use of these functions.

Author(s)

Robin K. S. Hankin

Examples

r_to_n_qhm(nrow(rqhm()))

---

Random Jordan objects

Description

Random jordan objects with specified properties

Usage

ralbert(n=3)
rrsm(n=3, d=5)
rchm(n=3, d=5)
rqhm(n=3, d=5)
rspin(n=3, d=5)

Arguments

n	Length of random object returned
d	Dimensionality of random object returned

Details

These functions give a quick “get you going” random Jordan object to play with.

Value

Return a jordan object

Author(s)

Robin K. S. Hankin

Examples

rrsm()
ralbert()
rspin()

---

Print methods

Description

Show methods, to display objects at the prompt

Usage

albert_show(x)
spin_show(x)
jordan_matrix_show(x)
description(x,plural=FALSE)

Arguments

x	Jordan object
plural	Boolean, indicating whether plural form is to be given

Details

The special algebras use a bespoke show method, jordan_matrix_show() or spin_show(). If the number of elements is small, they display a concise representation and modify the row and column names of the underlying matrix slightly; spin factors are displayed with the scalar component offset from the vector component.

Print methods for special algebras are sensitive to the value of option head_and_tail, a two-element integer vector indicating the number of start lines and end lines to print.

Function description() gives a natural-language description of its argument, used in the print method.

Value

Returns the argument

Author(s)

Robin K. S. Hankin

Examples

rspin()

rqhm()

rchm()

---

Validity methods

Description

Validity methods, to check that objects are well-formed

Usage

valid_rsm(object)
valid_chm(object)
valid_qhm(object)
valid_albert(object)
is_ok_rsm(r)
is_ok_chm(r)
is_ok_qhm(r)
is_ok_albert(r)
is_ok_rsm(r)

Arguments

object	Putative jordan object
r	Integer, number of rows in putative jordan object

Details

Validity methods. The validity_foo() functions test for an object to be the right type, and the is_ok_foo() functions test the number of rows being appropriate for a jordan object of some type; these functions return an error if not appropriate, or, for jordan_matrix objects, the size of the matrix worked with.

Value

Return a Boolean

Author(s)

Robin K. S. Hankin

Examples

is_ok_qhm(45)   # 5x5 Hermitian quaternionic matrices
#is_ok_qhm(46)   # FALSE

---

The zero Jordan object

Description

Package idiom for the zero Jordan object, and testing

Usage

is.zero(x)
is_zero_jordan(e1,e2=0)

Arguments

x, e1	Jordan object to test for zeroness
e2	Dummy numeric object to make the Arith method work

Details

One often wants to test a jordan object for being zero, and natural idiom would be rchm()==0. The helper function is is_zero_jordan(), and the generic is is.zero().

Value

Returns a Boolean

Author(s)

Robin K. S. Hankin

Examples

rrsm()*0 == 0

---