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Errors in mathematical formulas or arguments

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15 Eq. (1.33) (P⇒Q) ∧ (Q⇒R) = (P⇒R) (P⇒Q) ∧ (Q⇒R) ⇒ (P⇒R) Viktor Skorniakov, Vilnius University
22 Table in Eq. (2.17), 2nd row T T T T T T F F F F Viktor Skorniakov, Vilnius University
25 Eq. (2.26) f(G) = {y ∈ Y | ∃ x: f(x)=y} f(G) = {y ∈ Y | ∃ x ∈ G: f(x)=y} Viktor Skorniakov, Vilnius University
28 Example 3 Matrix multiplication is a binary operation on the said set. Matrix addition is a binary operation on the set of all n×n matrices. Viktor Skorniakov, Vilnius University
39 Eq. (3.4) The expression on the right side should be placed under a square root sign. Viktor Skorniakov, Vilnius University
51 Next to last sentence in the proof of Theorem 3.10 Now, one can write an arbitrary combination of a and b as ax + by = (ak+bl)g Now, one can write an arbitrary combination of a and b as ax + by = (kx+ly)g Viktor Skorniakov, Vilnius University
69 2nd line after Eq. (4.7) something that is more commonly denoted by f(v) something that is more commonly denoted by f̃(v) Viktor Skorniakov, Vilnius University
70 Example 2 The example given is a valid example of a linear map, which is an element in the dual space. It does not describe the full dual space. There are other maps, of course. Charanjit Singh Aulakh, IISER Mohali
73 Eq. (4.22), right side xi2 |xi|p
98 Eq. (5.60) e'(j) = ∑k (S-1)ik e(i) e'(k) = ∑i (S-1)ik e(i) Charanjit Singh Aulakh, IISER Mohali
104 After Eq. (5.98) The proof of the Cayley-Hamilton theorem is not trivial. The result is an operator relation, as given in Eq. (5.98), and cannot be inferred from Eq. (5.95) which is a determinant equation. The proof mentioned in the book is often called the bogus proof of the theorem. Just ignore the "proof". The theorem has been rightly applied in what follows immediately, and elsewhere. Charanjit Singh Aulakh, IISER Mohali
124 Eq. (5.222) U = ∑i |R(i)⟩ ⟨e(i)|,   V = ∑i |L(i)⟩ ⟨e(i)|. U = ∑i |e(i)⟩ ⟨R(i)|,   V = ∑i exp(-iαi) |e(i)⟩ ⟨L(i)|,
where αi is the phase of λi. 		Charanjit Singh Aulakh, IISER Mohali
128 2nd line after Eq. (6.5) If the Wronskian vanishes, the functions are linearly independent. If the Wronskian vanishes, the functions are linearly dependent. Joydeep Sarkar, Chennai Mathematical Institute
174 after Eq. (7.63) hi−1 also commutes with g hi−1 also commutes with h Sayan Chakrabarti, IIT Guwahati
176 Th. 7.16 The theorem, as stated, is wrong. It is best to disregard the statement as well as the proof. Charanjit Singh Aulakh, IISER Mohali
185 Eq. 7.100 img(fk) = ker(fk+1) img(fk-1) = ker(fk) Charanjit Singh Aulakh, IISER Mohali
207 2nd line after Eq. (8.77) imply b=a−k imply b=a−k+1
222 last line one 2-cycle and three 1-cycles one 2-cycle and two 1-cycles Inayat R. Bhat
229 Eq. (9.5), 1st line U†Ai†AiU† U†Ai†AiU Trinesh Sana, Calcutta University
239 Eq. (9.64) The right side of the equation should have an additional factor of 1/|G|. Inayat R. Bhat
263 row for A3 in Eq. (9.149)
1 1 1 −1 −1
1 −1 1 1 −1
Inayat R. Bhat
3rd line of text after Eq. (9.149) the elements of the first three classes form a subgroup of rotations the elements of the first, third and fourth classes form a subgroup of rotations Inayat R. Bhat
rows for A3 and A4 in Eq. (9.150)
1 1 1 −1 −1
1 −1 1 1 −1
1 −1 1 1 −1
1 1 1 −1 −1
Inayat R. Bhat
265 just before Eq. (9.155) classes of S3 classes of S4 Luis Odin Estrada Ramos, UNAM, Mexico
270 Eq. (9.178) The places of A3 and A4 should be interchanged. Inayat R. Bhat
296 End of 6th line after Eq. (10.50) and α3 = 1 + 1 + 2 = 4. and α6 = 1 + 1 + 2 = 4.
301, 302 Eqs. (10.73), (10.74) The arguments of all sin and cos functions should be 2πk/M; there should not be any factor of i. Charanjit Singh Aulakh, IISER Mohali
316 Heading of Section 11.3 The group denoted by SL(2,Z) in the heading has been called SL(2Z) within the text. The notations with and without the comma mean the same thing. Augniva Ray, Saha Institute of Nuclear Physics, Calcutta
334 1st paragraph There are two different uses of the word "semisimple", as applied to algebras. In this paragraph, the word "semisimple" has been used in the exclusive sense, i.e., these are algebras which are neither simple, nor do they have any abelian ideal. Beginning from the next paragraph, the same word has been used in an inclusive sense, i.e., by simply demanding that they do not have any abelian ideal. In the latter sense, simple algebras also fall in the class of semisimple algebras. One should interpret Theorem 12.2 in this way, as well as Theorems 12.7 and 12.8 later in the chapter.

Alternatively, one can use the inclusive definition throughout by making the following changes in this paragraph:

Augniva Ray, Saha Institute of Nuclear Physics, Calcutta
If an algebra is non-simple but not in this trivial way, i.e., not by the presence of any U(1) generator that commutes with every generator, then it is called a semisimple algebra. We will see in Chapter 15 that SO(4), the algebra of the group of 4×4 orthogonal matrices, is semisimple. If an algebra is not non-simple in this trivial way, i.e., if an algebra does not have any U(1) ideal, then it is called a semisimple algebra. The definition therefore includes simple algebras which do not have any non-trivial ideal at all, as well as other algebras which have non-trival ideals which are all non-abelian. We will see in Chapter 15 that SO(4), the algebra of the group of 4×4 orthogonal matrices, is of this last kind, i.e., is semisimple but not simple.
337 midway in the 2nd paragraph But the total exponent must be Hermitian so that R(ξ) is unitary, which means that the exponent must also contain the Hermitian conjugate of −iξX, which is +iξ*X†. But the total exponent must be of the form of the imaginary unit i times a Hermitian operator so that R(ξ) is unitary, which means that the terms involving ξ in the exponent must be of the form −i(ξX + ξ*X†). Charanjit Singh Aulakh, IISER Mohali
338 Eq. (12.35), right side KI aIKI Charanjit Singh Aulakh, IISER Mohali
339 Eq. (12.45), second line [A,[B,C]+] + [B,[C,A]+]+ − [C,[A,B]+]+ = 0 [A,[B,C]+] + [B,[C,A]]+ − [C,[A,B]]+ = 0 Trinesh Sana, Calcutta University
350 Eq. (12.95), last term on the right side Ta†(2) ⊗ Tb(1) Tb(1) ⊗ Ta†(2) Charanjit Singh Aulakh, IISER Mohali
351 Eq. (12.98), 3rd term on the left side (2⁄d(ad)) tr (Ta†(1)) tr (Ta(2)) (1⁄d(ad)) [ tr (Ta†(1)) tr (Ta(2)) + tr (Ta(1)) tr (Ta†(2)) ] Charanjit Singh Aulakh, IISER Mohali
356 Section 12.10.4 The entire argument, leading to Eq. (12.124), can be taken as an argument for the normalization constant of a representation of a direct product group, replacing C2 by K. For the Casimir invariant, the factors of the dimensions of the representations should be omitted in all three equations of this section. It means that the final formula should be

C2(R,R') = C2(R) + C2(R'),

with appropriate changes in the two earlier equations. The dimensions are important only to the extent that the unit matrix is d(R)d(R')-dimensional in these earlier equations.

Charanjit Singh Aulakh, IISER Mohali
383 Eq. (13.162), under the square root sign s(s + 1) ± m(m − 1) s(s + 1) − m(m ± 1) Augniva Ray, Saha Institute of Nuclear Physics, Calcutta
386 Table 13.2, first table In the column marked m1, the entry for the second row should be −½, not ½. Augniva Ray, Asia-Pacific Center for Theoretical Physics
422 2nd line after Eq. (15.16) if V is if M is Augniva Ray, Saha Institute of Nuclear Physics, Calcutta
426 Eq. (15.31) There should not be a minus sign on the right side of this equation. Trinesh Sana, Calcutta University
439 line after Eq. (15.80) Recalling that C−1 = C−1 Recalling that (C*C)−1 = C−1 CT Augniva Ray, Saha Institute of Nuclear Physics, Calcutta
474 first line of text denote by Λ a matrix denote by Λ a matrix Augniva Ray, Saha Institute of Nuclear Physics, Calcutta
482 2nd line of Eq. (17.57), left side t' − z' ct' − z' Augniva Ray, Saha Institute of Nuclear Physics, Calcutta
498 Eq. (17.143) The right sides of Eqs. (17.143b) and (17.143d) should have overall negative signs. Amitabha Lahiri, S N Bose Centre for Basic Sciences
577 Eq. (19.148) xi → x'i + εi(x) xi → x'i = xi + εi(x) Augniva Ray, Saha Institute of Nuclear Physics, Calcutta
587 Eq. (19.211) In the expression for h(t), the sum should be over n. Augniva Ray, Saha Institute of Nuclear Physics, Calcutta
589 Eq. (19.221), the first set of nested brackets [Tam, [Tbn, Tpc]] [Tam, [Tbn, Tcp]] Augniva Ray, Saha Institute of Nuclear Physics, Calcutta
589 Eq. (19.222) fabd δdc 𝜙(m + n) + fbcd δda 𝜙(n + p) fcad δdb 𝜙(p + m) fabd δdc 𝜙(m + n) + fbcd δda 𝜙(n + p) + fcad δdb 𝜙(p + m)
604 3rd line before Eq. (20.19) For the open ball G in Y, we find For the open set G in Y, we find Viktor Skorniakov, Vilnius University
616 1st line of text the rectangle marked 'B' is (150,185)× the rectangle marked 'B' is (120,185)× Viktor Skorniakov, Vilnius University
617 Eq. (21.23) {G ∩ S   ∀G ∈ T} {G ∩ S ∣ ∀G ∈ T} Viktor Skorniakov, Vilnius University
619 Eq. (21.31) U∈Y U⊂Y Viktor Skorniakov, Vilnius University
620 Eq. (21.38) Subset U∈Y Subset U⊂Y Viktor Skorniakov, Vilnius University
626 Definition (21.18) there exist non-empty subsets there exist non-empty open subsets Viktor Skorniakov, Vilnius University
637 Eq. (22.23) In the last line of the defintion of H(s,t), the function should be f̅ instead of f. Viktor Skorniakov, Vilnius University
655 Definition (23.2) is a map is a continuous map Viktor Skorniakov, Vilnius University
661 Eqs. (23.25) to (23.27) Summation over the index i should range from 0 to d. Joydeep Sarkar, Channai Mathematical Institute
662 Text before Eq. (23.30) The most general element of C1(X) is given already in Eq. (23.26). The most general element of C1(X) is given already in Eq. (23.25). Viktor Skorniakov, Vilnius University
664 Eqs. (23.43) & (23.45) The right side of Eq. (23.43) is really the image of ∂2 and not its kernel. The kernel can be obtained by equating the right side of Eq. (23.35) to zero, which gives p=0. Ths means that the kernel is trivial: ker (∂2) = {0}. Using Eq. (23.44) now, one obtains H2(D2) = {0}/{0} = {0}. Joydeep Sarkar, Channai Mathematical Institute
684 Answer of Ex. 14.8 8, 10, 10, 6 8, 15, 15, 6 Augniva Ray, Saha Institute of Nuclear Physics, Calcutta

Errors in other places

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16 line before Eq. (2.7) assume the statement is true of all values assume the statement is true for all values Viktor Skorniakov, Vilnius University
19 line before Eq. (2.7) means the elements means the element Viktor Skorniakov, Vilnius University
25 3rd line of the penultimate paragraph consists only elements consists only of elements Viktor Skorniakov, Vilnius University
28 1st sentence of Sec. 2.2.3 algbraic structures. algebraic structures. Viktor Skorniakov, Vilnius University
48 1st sentence of 2nd paragraph all integers are also algbraic inegers. all integers are also algebraic inegers. Viktor Skorniakov, Vilnius University
50 Ex. 3.12, 2nd line form a ring. forms a ring. Viktor Skorniakov, Vilnius University
54 4th line above Eq. (3.54) This scalars These scalars Viktor Skorniakov, Vilnius University
69 2nd paragraph, 3rd line and is also linearly independent and are also linearly independent Viktor Skorniakov, Vilnius University
115 Line after Eq. (5.166) are obviously Hermitian. Both matrices are obviously Hermitian. Viktor Skorniakov, Vilnius University
164 1st line of text rpresentation representation Augniva Ray, Saha Institute of Nuclear Physics, Calcutta
211 3rd line in Section 8.8 ABCD form a permutation ABCD forms a permutation
237 Section 9.5.1, end of 1st paragraph the matrices URiU† does the same the matrices URiU† do the same
249 Section 9.5.1, end of 1st paragraph the commutators in generals the commutators in general Inayat R. Bhat
319 The sentence leading to Eq. (11.40) A closely related group are groups of transformations A closely related group contains groups of transformations Viktor Skorniakov, Vilnius University
320 second line before Eq. (11.45) given in Eq. (11.29) given in Eq. (11.28)
341 last paragraph of the inset symmetry properties of Eq. (12.52) holds. symmetry properties of Eq. (12.52) hold. Viktor Skorniakov, Vilnius University
349 4th line after Eq. (12.89) this information is uselss this information is useless Viktor Skorniakov, Vilnius University
388 last line the reduced d-matirx. the reduced d-matrix. Viktor Skorniakov, Vilnius University
402 line after Eq. (14.51) assignemnt assignment
667 3rd line in the paragraph after the definition Similarly, a 2-simples Similarly, a 2-simplex Viktor Skorniakov, Vilnius University